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#741 From: "Michael Donovan" <michael1@...>
Date: Fri Nov 5, 2004 12:20 am
Subject: Back to the weird.
jayennseth
Send Email Send Email
 
Below is a thread going back to the argument on this list that Gerald Hawkins, who showed the diatonic in crop circles, did not discover new Euclidian Theorums.  Dee's attachement I will copy and paste to the next, subject 'Back to the weird 2'.
**********thread read from bottom**************************
Dee,
    You stated below, in regards to the new theorems, (quote), "...It was a lot of fun doing them." (unquote)  What an understatement, Dee, considering that you absolutly must, as do I, feel like the real 'Indiana Jones'.
    Hoping that my foot is not jamming up my mouth I am going to say that there could be, might be, possibly be a problem in that the 'new theorums' supposed are covered by, incorporated in, or 'all a part of' known, (not new), properties of regualar polygons.  Perhaps.  Maybe.  And to be frank I will go beyond 'not sure'.  Dee, I don't know.  That is why I separated the issues in the book.  The diatonic observations are good enough for me.
    I am on about 15 Internet math discussions groups.  The attack on Gerald Hawkins claims came from Polytopia, a Yahoo group.  And the guy who shot a hole in the concept of new Euclidian theorums is a great guy, sharp mind, but thinks crops circles a hoax or joke.  Talked about mowing them in the lawn for his kids.  In most all areas of math I consider this group 'my betters'.  So I am going to post both your file and this thread to it and see what happens.
----- Original Message -----
From: "Dee Gragg" <deegragg@...>
Sent: Thursday, November 04, 2004 5:53 PM
Subject: Re: Analysis of The Miamisburg Formation; 3 Nov 2004

> Hi Michael:
>
>    Thanks for your reply and very kind words.  Let me
> first address the Euclidean theorems.  Dr. Hawkins
> discovered and proved four theorems. 
>
>    After his death I wanted to read his proofs.  But I
> could find no one who had the proofs or knew how to do
> them.  So I undertook to prove them myself.  I was
> successful in proving all four theorems (not without
> considerable struggle). 
>
>    In the process I discovered five new theorems and
> proved them as well making a total of nine; all
> unknown to the mathematics world in general. 
>
>    Dr Hawkins checked Euclid's 12 volumes of theorems
> and found that they were not in his work.  I too
> checked and did not find them nor the five new ones
> which I discovered.  Since I feel that this is of
> interest to you I am sending the paper I wrote proving
> all the theorems.
>
>   Now as to the Musical notes encoded in the circles.
> They are not unrelated to the theorems.  In fact, all
> four of Dr. Hawkins theorems are related to musical
> notes.  Alas, I found that only two of mine were; F
> below middle C and F two octaves below Middle C.
>
>   I hope you enjoy the theorems and the crop circles
> as well.  It was a lot of fun doing them.
>
>    Please let me know if you have any further
> questions or comments.
>
> Kindest Regards,
> Dee
> --- Michael Donovan <michael1@...> wrote:
>
> > This is an eyeopener,
> >     I have much of a book, The Crop Circle Message,
> > complete.  Over the past
> > month I have recieved, (and seems that I am going
> > with), an offer from a
> > computer programer to put the material into a
> > interactive computer program.
> >     I am going to make one comment, and hope for
> > some feedback and
> > clairification.
> >     But, first, this is just wonderful work.  The
> > timing for me personally
> > is mindblowing.
> >     I have in front of me page 441 of The
> > Mathematics Teacher, Vol 91, No.
> > 5, May 1998 where Gerald Hawkins posted some of his
> > concepts with a paid
> > advertisment giving Boston University Research as
> > the source.  I feel that
> > it might, (I stress 'might'), be unfortunate that
> > two issues have become
> > intertwined.  One being the observation of diatonic
> > (and now other...,)
> > scales and notes being observed in crop circles, and
> > the other the
> > possibility of additional Euclidian theorms.  There
> > has been a serious
> > challange regarding the additional Euclidian theorm.
> >  This came through the
> > Polytopia yahoo math group, and these guys are no
> > slouches.  Nor am I versed
> > enough to form an opinion, particularly without more
> > information.  I simply
> > forwarded that information to Freddy Silva and never
> > got a reply.  However,
> > I was very unconcerned because I separate the
> > issues.  The observation that
> > the diationic 'crops' up again and again is for me
> > far more exciting that
> > the possibility of a new Euclidian theorm.  And I am
> > suggesting that it
> > 'might' be wise for all to make that separation
> > until the new theorm
> > question is resolved.  At least that is what I am
> > doing.
> >     That said, let me explain more why the timing of
> > this was so
> > mindblowing.  I am assuming for a moment that it is
> > true that a 'sub' note
> > (black key) now arrives.  I 'feel' there will be
> > more.  Here are my
> > reasons....:
> >     1- The crop circles are a program of learning.
> > As we 'get' the
> > diationic 'they/it/whatever' can begin to add to it.
> >     2- The new geometry is balls, lines and points
> > don't exist.  It is akin
> > to spherepacking, but you must also assume the balls
> > are both vibrating and
> > moving.  The key 'shape' it only in the 'center' in
> > movement.  It must move
> > to keep center.  The key shape is balls around
> > balls, making a ball, a ball
> > of 12 balls around one.  Say you start with the size
> > of a pea.  13 of these
> > can make another ball the size of a golf ball.  13
> > of these can make another
> > ball the size of a softball, and on and on in a
> > geometric (logrithmic)
> > progression.  If you examine any one of these groups
> > you see seven 'rings'
> > of vibration.  The seven rings are identical in
> > angle, but not size (of
> > course) to the next jump of balls, and THIS is the
> > 'octave'.
> >     3- There is a subset of 12 planes.  !!!!!!
> > Harder to see , but now come
> > the black keys.
> >     Perhaps some of you have the September-October
> > issue of Nexus.  Look on
> > page 49, An Introduction to Global Scaling Theory by
> > Dr. Harmut Muller.
> > First he starts out by saying that there was a time
> > that math was leading
> > physics.  And it seems that for the past decades
> > math has just been a
> > stepchild of physics.  Except for some hints.... and
> > here (as he calls it a
> > 'goldmine')  is a biggie....;
> >     (quote from pg 49, Dr. Muller) "...The first
> > indication of the existence
> > of this scientific goldmine came from biology.  As a
> > rresult of 12 years of
> > research, Cislenko published his 'Structurer of
> > Fauna and Flora with Regard
> > to Body Size of Organisms' (Moscow 1980).  He work
> > documents what is
> > probably the most important biological discovery in
> > the 20th century.
> > Cislenko was able to prove that segments of incresed
> > species representation
> > are repeated on the logrithmic line of body sized in
> > equal intervals (aprox.
> > 0.5 units of the decadic logarithm).  The phenomenon
> > is not explicable from
> > a biological point of view.  Why should mature
> > individuals of amphibians,
> > reptiles, fish, birds and mammals of different
> > species find it similarly
> > advantageous to have a body size in the range of
> > 8-12 centimeters, 33-55
> > centimeters or 1.5-2.4 meters?" (unquote, Dr.
> > Muller)
> >     Why indeed?
> >     Unless...
> >     Unless the organisms are jumping in maturity to
> > octave fullnesses.
> > Think again of the pea, golf ball, softball?
> >     Dee, and all that worked on this new discovery,
> > and there seem to be
> > many involved.  Wonderful, wonderful, wonderful,
> > wonderful.  Thank you,
> > thank you, thank you.
> >     Michael Donovan, Camden, ME
> >     The New Geometry, www.midcoast.com/~michael1
> >
> > ----- Original Message -----
> > From: "Dee Gragg" <deegragg@...>
> > To: <cbajis@...>; <exopolitics@...>;
> > <JDAIntiRa@...>;
> > <jhbos5@...>; <marchu@...>;
> > <Mark@...>;
> > <markthurston@...>; <michael1@...>;
> > <mntnhiker@...>;
> > <mstewart@...>;
> > <order@...>;
> > <qala@...>; <rgarner@...>;
> > <volconsumer@...>;
> > <zy@...>
> > Sent: Wednesday, November 03, 2004 7:11 PM
> > Subject: Analysis of The Miamisburg Formation; 3 Nov
> > 2004
> >
> >
> > > Hi All:
> > >
> > >   First, Thank you all for your comments and
> > > encouragement on the previous two papers.  You
> > were
> > > most kind and I appreciate it.
> > >
> > >    This paper presents the analysis of the United
> > > States   Miamisburg Formation of 2004.  It is a
> > sister
> > > formation to the Locust Grove Formation of 2003. I
> > > believe you will find it important for at least
> > two
> > > reasons.
> > >
> > >    (1) It strengthens the existing relationship
> > > between crop circles and musical notes.
> > >
> > >    (2) It gives the first nondiatonic ratio found
> > in
> > > the formations.  For the first time we have found
> > one
> > > of the black piano keys!
> > >
> > >    I would be happy to have any of your comments
> > and I
> > > will try to answer any questions.
> > >
> > >    I hope to see many of you at the SIGNS of
> > DESTINY
> > > 2004 in Tempe November 19-22.  Dr. Snow has put
> > > together another great program for us.
> > >
> > > Kindest Regards,
> > > Dee
> >
> >
 

#742 From: "Michael Donovan" <michael1@...>
Date: Fri Nov 5, 2004 12:23 am
Subject: back to the weird 2
jayennseth
Send Email Send Email
 

Crop Circle Theorems
 
Their Proofs and Relationship to Musical Notes
 

   This research began with a simple and rather limited objective: to prove the crop circle theorems of Dr. Gerald Hawkins.  In fact if I could have found the proofs in the literature of the field, this research would never have taken place at all.  Fortunately, I couldn’t find them because once I started, I could see that further work that needed to be done.
 
   As I proved Dr. Hawkins theorems, I discovered five new ones and proved them as well.  I then took the diatonic ratios of all the theorems and related them to the frequencies of the musical scale.  With some rather startling results I might add.
 
   Beginning with Theorem IA I need to make some observations that apply to all of the theorems.  In Euclidean Geometry one almost always has to see the end before making a beginning.   Also, since we are looking for diatonic ratios, we need to find an equation or equations which will let us divide one diameter or radius by the other.  Remember too, that because we are working with ratios, the constants divide out leaving diameter ratios equal to radius ratios.  And if we square them they are equal to each other and to the ratio of the areas.
 
   Applying this to Theorem IA the equation we need to write is for the diameter of the circumscribing circle.  It contains both the radii of the initial and the circumscribed circle.  So from the equation we are able to divide it and find the diatonic ratio of 4 to 3. 
 
   Although, I have proved three more Theorem I’s, I believe this is the one Dr Hawkins meant when he said Theorem I.  See Circular Relationships for The Theorems in Appendix A.  I base this belief on the 4 to 3 diatonic relationship which is related directly to Note F above Middle C.  See Frequencies In The Fields in Appendix B.
 
   Theorem IB is like Theorem IA except that the equilateral triangle is inscribed rather than being circumscribed.  It can be proved by Theorem IA and Theorem II.  The equations already exist so just divide them for the proof.  This gives a new diatonic ratio which is also the Note F, one octave lower than the previous.
This theorem is such a simple and logical extension of the first two that I am puzzled as to why Dr Hawkins did not discover and publish it.
 
   Theorem IC is also often referred to as Theorem I although it is quite different.  Sometimes both Theorem IA and Theorem IC appear in the same article as if they were identical.  They aren’t.  The proof of Theorem IC shows that it contains no diatonic ratio that can be related to a musical frequency.  I believe that this was not the theorem Dr. Hawkins was referring to when he said Theorem I.  In my mind the origin of Theorem IC is rather murky.
 
      Theorem ID would have never been discovered if I had read the instructions for constructing Theorem IA a little closer.  Instead of circumscribing the equilateral triangle, I circumscribed the three circles and then proved the theorem before realizing my mistake.  It has a nice 7 to 3 relationship but it would need to be 8 to 3 to be Note F in the next higher octave.
 
   Theorem II is easy to prove by constructing the appropriate similar triangles and remembering their relationships.  It may be proved a number of different ways I have shown just one of them.  It has the nice diatonic ratio of 4 to 1 which relates directly to the Note C which is two octaves above Middle C.
 
   Theorem III is the simplest of all proofs.  Just remember the Pythagorean Theorem. It also has a nice diatonic ratio of 2 to 1 which relates directly to the Note C which is an octave above Middle C.
 
   Again using the Pythagorean Theorem, Theorem IVA is shown to have a nice 4 to 3 relationship.  We have previously related this to Note F using Theorem IA.
 
   While proving Theorem II, it occurred to me that there should be a similar theorem related to the hexagon. There was and that led me to discover Theorem IVB by connecting the diameters at the hexagon corners.  Again by using similar triangles and writing and dividing the proper equations it is shown to have a diatonic ratio of 1 to 3 which relates directly to the Note F.  This Note F is yet another octave lower.
 
   I have included Theorem IVC mostly for completeness as it does not have a diatonic ratio which can be related to a specific note.  If I hadn’t included it you might have wondered why since it can be proved by simply dividing Theorem IVB by Theorem IVA. 
 
   There is a Theorem V which can be used for deriving (not proving) the other theorems.  However it does not of its self have diatonic ratios and therefore was not a part of this research.
 
      Appendix A Circular Relationships for The Theorems shows a summary of all the results.  Note that to go from one column to the other, you simply square or take the square root.  But how do you know which column to use?  I have followed the lead of Dr. Hawkins in that if the circles are not concentric, you use the ratio of diameters, if they are concentric you use the ratio of areas.  This means diameters for Theorems IA, IB, IC, ID, IVB, and IVC and areas for Theorems II, III, and IVA.  Why did he pick this convention?  Certainly I don’t know, perhaps he was a practical man and he did it because it works.
 
   Frequencies In The Fields in Appendix B gives four octaves: two above and two below Middle C.   This does not encompass the full 27.5 to 4,186 Hz of a piano but does include all the frequencies found so far.  Notice that all the notes are either F or C.  Coincidence or a message?  Perhaps as we discover more notes, this will become clear.
 
   Theorem T in Appendix C is not really a part of this research, but is included as help for anyone wanting to compute circle and regular polygon ratios.  It includes all cases and relies on trigonometry rather than Euclidian Geometry.
 
   Finally, if you’re wondering about me, I have a Short Bio in Appendix D.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Copyright 2004 by C. D. Gragg. All rights reserved
 
Theorem IA
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is circumscribed about the triangle, the ratio of the diameters is 4 to 3.
              
 
 
 
 
 
Theorem IB
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is inscribed within the triangle, the ratio of the diameters is  2:3.
 
              
 

Theorem IC
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is constructed using the single circle as a center and drawing the circle through the other two centers, the ratio of the diameters is 4 to Sqrt 3.
              
 

Theorem ID
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is constructed circumscribing the three circles, the ratio of the diameters is 7 to 3.
              
 
 
 
 
 
Theorem II
   If an equilateral triangle is inscribed and circumscribed the ratio of the circles’ areas is 4:1.
              
 
 
 
 
 
 
 

Theorem III
   If a square is inscribed and circumscribed the ratio of the circles’ areas is 2:1.
               
 
 
 
 
 
 
 
 
 
Theorem IVA
   If a hexagon is inscribed and circumscribed the ratio of the circles’ areas is 4:3.
              
 
 
 
 
 
 
 
 
 
Theorem IVB
      If a hexagon is inscribed and circumscribed and the corners connected by diameters, the inscribed circles of the created equilateral triangles have a diameter ratio to the inscribed circle of 1:3.
              
 
 
 
 
 
 
 
Theorem IVC
      If a hexagon is inscribed and circumscribed and the corners connected by diameters, the inscribed circles of the created equilateral triangles have a diameter ratio to the circumscribed circle of 1:2Sqrt3.
              
 
 
 
 
 
 
 
Appendix A
 
                          Circular Relationships for
                                     The Theorems
 Theorem Ratio of Diameters and
Radii Ratio of Areas, Diameters Squared, and Radii Squared
 Theorem IA 4:3 16:9
 Theorem IB  2:3  4:9
 Theorem IC 4:Sqrt3 16:3
 Theorem ID 7:3 49:9
 Theorem II 2:1 4:1
 Theorem III Sqrt2:1 2:1
 Theorem IVA 2:Sqrt3 4:3
 Theorem IVB 1:3 1:9
 Theorem IVC 1:2 Sqrt3 1:12
  
  
  
  
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Copyright 2004 by C. D. Gragg, All rights reserved
 
 
 
 
 
Appendix B
 
Frequencies In The Fields
 
Note Name C D E F G A B C
Diatonic Ratio 1/4 9/32 5/16 1/3 3/8 5/12 15/32 1/2
Frequency (Hz) 66 74.25 82.5 88 99 110 123.75 132
 
Note Name C D E F G A B C
Diatonic Ratio 1/2 9/16 5/8 2/3 3/4 5/6 15/16 1
Frequency (Hz) 132 148.5 165 176 198 220 247.5 264
 
Note Name C* D E F G A B C
Diatonic Ratio 1 9/8 5/4 4/3 3/2 5/3 15/8 2
Frequency (Hz) 264 297 330 352 396 440 495 528
 
Note Name C D E F G A B C
Diatonic Ratio 2 9/4 5/2 8/3 3 10/3 15/4 4
Frequency (Hz) 528 594 660 704 792 880 990 1056
* Middle C
 Denotes  found in the fields
 
Theorem Summary
 
Frequency (Hz) Theorem Used For Proof
88 Theorem IVB, Gragg
176 Theorem IB, Gragg
352 Theorem IA, Theorem IVA, Hawkins
528 Theorem III, Hawkins
1056 Theorem II, Hawkins
 

 
 
 
 
 
 
 
 
 
 
                    Copyright 2004 by C. D. Gragg, All rights reserved
Appendix C
Theorem T
 
   Trigonometry can be used to solve circular relationships for inscribed and circumscribed regular  polygons for polygons of any number of sides from 3 to infinity.
 
 Proof:                                      
                                                  Where:  α = 3600          and n = number of sides    
                                                  2n
                                                          
                                                                 cos α  = R1 
                                                              R2
 
                                                                    R2 =   _1     Proving the Theorem
Figure 1. Regular polygon with                  R1       cos α
               any number of sides                              
                                                           Further:   ( R2)2 =  (1)2    
                                                                                                  ( R1)2      ( cos α)2
Table of Some Common Polygons
Figure (All are equiangular) Number of Equal Sides  Ratio of Diameters and
Radii Ratio of Areas, Diameters Squared, and Radii Squared
Triangle (1)(4)         3       2.000 4      4.000
Square (2)         4       1.414 2      2.000
Pentagon         5       1.236          1.527
Hexagon (3)          6       1.155 4/3   1.333
Heptagon         7       1.110         1.232
Octagon (4)         8       1.082         1.172
Nonagon         9       1.064         1.132
Decagon       10       1.051         1.106
       15       1.022         1.045
       20       1.012         1.025
       50       1.002         1.004
     100       1.000         1.001
     200       1.000         1.000
       ∞       1.000         1.000
(1) Theorem II, by Dr. Hawkins using Euclidian Geometry
(2) Theorem III, by Dr. Hawkins using Euclidian Geometry 
(3) Theorem IV, by Dr. Hawkins using Euclidian Geometry
(4) Found in the Kekoskee/Mayville, Wisconsin Crop Circle
             Formation July 9, 2003    
                                                                           Copyright 2004 by C. D. Gragg, All rights reserved
 
 
 
 
 
 
 

Appendix D
 
Short Bio
 
   My name is Dee Gragg.  I am a retired, mechanical engineer.  My career was spent in research, testing and evaluation. My main areas of research were automotive air bags, jet aircraft ejection seats and high speed rocket sleds.  I have published 33 technical papers as either the principal author or a co-author.  They form a part of the body of literature in their respective fields.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

#743 From: "Michael Donovan" <michael1@...>
Date: Fri Nov 5, 2004 3:21 pm
Subject: Robt. Webb. how difficult to program.
jayennseth
Send Email Send Email
 
I have been approached by a sucessful programer to put a system I have into a computer program.  He wants to make a deal.  The deal is starting to be envisioned as a cooperative owenership of the program but I would retain rights to a copywrite symbol set which navigates a new geometry.  When talk comes up with things like Delphi vs. C+ I am completely in the dark.
    On reflection I think perhaps the programing might not be anywhere as difficult as I thought.  There are just a few 3-D graphics.  Most is file management, if/then flowcharts.  I would be very good at scripting, I know exactly what I want.  What I am just realizing is that most likely, (and this is what I would want to know - what I am asking), all the 'surrounds' of the program are already done in Delphi or C+.  That is to say part of the 'blank screen' of a new program would already have the spaces to be filled across the top, as with this email, File, Edit, View, etc.  That there is a package already to adjust, rename etc.  So that the program is mostly file management.  I am taking a wild stab and saying that it would be far easier than Stella.
About how long did that take.
On the open market, how much would a package of that progarming go far.  For example, if you just had the ideas of Stella, did not program.  Ballpark, how much are we looking at?

#744 From: John Berglund <anisohedral@...>
Date: Fri Nov 5, 2004 4:33 pm
Subject: Re: back to the weird 2
anisohedral
Send Email Send Email
 
There are some interesting relationships in shapes. Theorems 1A to 1D can be seen in the following picture. (Theorem 1C requires that you know that a 30-60-90 degree triangle has the edge ratios of 1:2:sqrt(3)... the others can be seen just by counting distances.) Were we to place another circle inside the three given circles, just tangent to them, its radius would be in a ratio of 1 to 3 with the original circles. There are numerous other relationships that we could point out. I don't think of these relationships as having any special meaning aside from being nice numbers.
 
John Berglund
 


Michael Donovan <michael1@...> wrote:

Crop Circle Theorems
 
Their Proofs and Relationship to Musical Notes
 

   This research began with a simple and rather limited objective: to prove the crop circle theorems of Dr. Gerald Hawkins.  In fact if I could have found the proofs in the literature of the field, this research would never have taken place at all.  Fortunately, I couldn’t find them because once I started, I could see that further work that needed to be done.
 
   As I proved Dr. Hawkins theorems, I discovered five new ones and proved them as well.  I then took the diatonic ratios of all the theorems and related them to the frequencies of the musical scale.  With some rather startling results I might add.
 
   Beginning with Theorem IA I need to make some observations that apply to all of the theorems.  In Euclidean Geometry one almost always has to see the end before making a beginning.   Also, since we are looking for diatonic ratios, we need to find an equation or equations which will let us divide one diameter or radius by the other.  Remember too, that because we are working with ratios, the constants divide out leaving diameter ratios equal to radius ratios.  And if we square them they are equal to each other and to the ratio of the areas.
 
   Applying this to Theorem IA the equation we need to write is for the diameter of the circumscribing circle.  It contains both the radii of the initial and the circumscribed circle.  So from the equation we are able to divide it and find the diatonic ratio of 4 to 3. 
 
   Although, I have proved three more Theorem I’s, I believe this is the one Dr Hawkins meant when he said Theorem I.  See Circular Relationships for The Theorems in Appendix A.  I base this belief on the 4 to 3 diatonic relationship which is related directly to Note F above Middle C.  See Frequencies In The Fields in Appendix B.
 
   Theorem IB is like Theorem IA except that the equilateral triangle is inscribed rather than being circumscribed.  It can be proved by Theorem IA and Theorem II.  The equations already exist so just divide them for the proof.  This gives a new diatonic ratio which is also the Note F, one octave lower than the previous.
This theorem is such a simple and logical extension of the first two that I am puzzled as to why Dr Hawkins did not discover and publish it.
 
   Theorem IC is also often referred to as Theorem I although it is quite different.  Sometimes both Theorem IA and Theorem IC appear in the same article as if they were identical.  They aren’t.  The proof of Theorem IC shows that it contains no diatonic ratio that can be related to a musical frequency.  I believe that this was not the theorem Dr. Hawkins was referring to when he said Theorem I.  In my mind the origin of Theorem IC is rather murky.
 
      Theorem ID would have never been discovered if I had read the instructions for constructing Theorem IA a little closer.  Instead of circumscribing the equilateral triangle, I circumscribed the three circles and then proved the theorem before realizing my mistake.  It has a nice 7 to 3 relationship but it would need to be 8 to 3 to be Note F in the next higher octave.
 
   Theorem II is easy to prove by constructing the appropriate similar triangles and remembering their relationships.  It may be proved a number of different ways I have shown just one of them.  It has the nice diatonic ratio of 4 to 1 which relates directly to the Note C which is two octaves above Middle C.
 
   Theorem III is the simplest of all proofs.  Just remember the Pythagorean Theorem. It also has a nice diatonic ratio of 2 to 1 which relates directly to the Note C which is an octave above Middle C.
 
   Again using the Pythagorean Theorem, Theorem IVA is shown to have a nice 4 to 3 relationship.  We have previously related this to Note F using Theorem IA.
 
   While proving Theorem II, it occurred to me that there should be a similar theorem related to the hexagon. There was and that led me to discover Theorem IVB by connecting the diameters at the hexagon corners.  Again by using similar triangles and writing and dividing the proper equations it is shown to have a diatonic ratio of 1 to 3 which relates directly to the Note F.  This Note F is yet another octave lower.
 
   I have included Theorem IVC mostly for completeness as it does not have a diatonic ratio which can be related to a specific note.  If I hadn’t included it you might have wondered why since it can be proved by simply dividing Theorem IVB by Theorem IVA. 
 
   There is a Theorem V which can be used for deriving (not proving) the other theorems.  However it does not of its self have diatonic ratios and therefore was not a part of this research.
 
      Appendix A Circular Relationships for The Theorems shows a summary of all the results.  Note that to go from one column to the other, you simply square or take the square root.  But how do you know which column to use?  I have followed the lead of Dr. Hawkins in that if the circles are not concentric, you use the ratio of diameters, if they are concentric you use the ratio of areas.  This means diameters for Theorems IA, IB, IC, ID, IVB, and IVC and areas for Theorems II, III, and IVA.  Why did he pick this convention?  Certainly I don’t know, perhaps he was a practical man and he did it because it works.
 
   Frequencies In The Fields in Appendix B gives four octaves: two above and two below Middle C.   This does not encompass the full 27.5 to 4,186 Hz of a piano but does include all the frequencies found so far.  Notice that all the notes are either F or C.  Coincidence or a message?  Perhaps as we discover more notes, this will become clear.
 
   Theorem T in Appendix C is not really a part of this research, but is included as help for anyone wanting to compute circle and regular polygon ratios.  It includes all cases and relies on trigonometry rather than Euclidian Geometry.
 
   Finally, if you’re wondering about me, I have a Short Bio in Appendix D.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Copyright 2004 by C. D. Gragg. All rights reserved
 
Theorem IA
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is circumscribed about the triangle, the ratio of the diameters is 4 to 3.
              
 
 
 
 
 
Theorem IB
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is inscribed within the triangle, the ratio of the diameters is  2:3.
 
              
 

Theorem IC
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is constructed using the single circle as a center and drawing the circle through the other two centers, the ratio of the diameters is 4 to Sqrt 3.
              
 

Theorem ID
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is constructed circumscribing the three circles, the ratio of the diameters is 7 to 3.
              
 
 
 
 
 
Theorem II
   If an equilateral triangle is inscribed and circumscribed the ratio of the circles’ areas is 4:1.
              
 
 
 
 
 
 
 

Theorem III
   If a square is inscribed and circumscribed the ratio of the circles’ areas is 2:1.
               
 
 
 
 
 
 
 
 
 
Theorem IVA
   If a hexagon is inscribed and circumscribed the ratio of the circles’ areas is 4:3.
              
 
 
 
 
 
 
 
 
 
Theorem IVB
      If a hexagon is inscribed and circumscribed and the corners connected by diameters, the inscribed circles of the created equilateral triangles have a diameter ratio to the inscribed circle of 1:3.
              
 
 
 
 
 
 
 
Theorem IVC
      If a hexagon is inscribed and circumscribed and the corners connected by diameters, the inscribed circles of the created equilateral triangles have a diameter ratio to the circumscribed circle of 1:2Sqrt3.
              
 
 
 
 
 
 
 
Appendix A
 
                          Circular Relationships for
                                     The Theorems
 Theorem Ratio of Diameters and
Radii Ratio of Areas, Diameters Squared, and Radii Squared
 Theorem IA 4:3 16:9
 Theorem IB  2:3  4:9
 Theorem IC 4:Sqrt3 16:3
 Theorem ID 7:3 49:9
 Theorem II 2:1 4:1
 Theorem III Sqrt2:1 2:1
 Theorem IVA 2:Sqrt3 4:3
 Theorem IVB 1:3 1:9
 Theorem IVC 1:2 Sqrt3 1:12
  
  
  
  
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Copyright 2004 by C. D. Gragg, All rights reserved
 
 
 
 
 
Appendix B
 
Frequencies In The Fields
 
Note Name C D E F G A B C
Diatonic Ratio 1/4 9/32 5/16 1/3 3/8 5/12 15/32 1/2
Frequency (Hz) 66 74.25 82.5 88 99 110 123.75 132
 
Note Name C D E F G A B C
Diatonic Ratio 1/2 9/16 5/8 2/3 3/4 5/6 15/16 1
Frequency (Hz) 132 148.5 165 176 198 220 247.5 264
 
Note Name C* D E F G A B C
Diatonic Ratio 1 9/8 5/4 4/3 3/2 5/3 15/8 2
Frequency (Hz) 264 297 330 352 396 440 495 528
 
Note Name C D E F G A B C
Diatonic Ratio 2 9/4 5/2 8/3 3 10/3 15/4 4
Frequency (Hz) 528 594 660 704 792 880 990 1056
* Middle C
 Denotes  found in the fields
 
Theorem Summary
 
Frequency (Hz) Theorem Used For Proof
88 Theorem IVB, Gragg
176 Theorem IB, Gragg
352 Theorem IA, Theorem IVA, Hawkins
528 Theorem III, Hawkins
1056 Theorem II, Hawkins
 

 
 
 
 
 
 
 
 
 
 
                    Copyright 2004 by C. D. Gragg, All rights reserved
Appendix C
Theorem T
 
   Trigonometry can be used to solve circular relationships for inscribed and circumscribed regular  polygons for polygons of any number of sides from 3 to infinity.
 
 Proof:                                      
                                                  Where:  α = 3600          and n = number of sides    
                                                  2n
                                                          
                                                                 cos α  = R1 
                                                              R2
 
                                                                    R2 =   _1     Proving the Theorem
Figure 1. Regular polygon with                  R1       cos α
               any number of sides                              
                                                           Further:   ( R2)2 =  (1)2    
                                                                                                  ( R1)2      ( cos α)2
Table of Some Common Polygons
Figure (All are equiangular) Number of Equal Sides  Ratio of Diameters and
Radii Ratio of Areas, Diameters Squared, and Radii Squared
Triangle (1)(4)         3       2.000 4      4.000
Square (2)         4       1.414 2      2.000
Pentagon         5       1.236          1.527
Hexagon (3)          6       1.155 4/3   1.333
Heptagon         7       1.110         1.232
Octagon (4)         8       1.082         1.172
Nonagon         9       1.064         1.132
Decagon       10       1.051         1.106
       15       1.022         1.045
       20       1.012         1.025
       50       1.002         1.004
     100       1.000         1.001
     200       1.000         1.000
       ∞       1.000         1.000
(1) Theorem II, by Dr. Hawkins using Euclidian Geometry
(2) Theorem III, by Dr. Hawkins using Euclidian Geometry 
(3) Theorem IV, by Dr. Hawkins using Euclidian Geometry
(4) Found in the Kekoskee/Mayville, Wisconsin Crop Circle
             Formation July 9, 2003    
                                                                           Copyright 2004 by C. D. Gragg, All rights reserved
 
 
 
 
 
 
 

Appendix D
 
Short Bio
 
   My name is Dee Gragg.  I am a retired, mechanical engineer.  My career was spent in research, testing and evaluation. My main areas of research were automotive air bags, jet aircraft ejection seats and high speed rocket sleds.  I have published 33 technical papers as either the principal author or a co-author.  They form a part of the body of literature in their respective fields.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 


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#745 From: "Michael Donovan" <michael1@...>
Date: Sat Nov 6, 2004 3:29 am
Subject: Re: back to the weird 2
jayennseth
Send Email Send Email
 
John,
    I had both a feeling and a deep hope you would appear out of the cybermist for this.  And I am very grateful that you have.  If you remember the general question of if Professor Gerald Hawkins did or did not come up with new theorems had come up on this list. At that time you pointed out that Hawkins theorem was covered by a very easily provable attribute to all regular polygons.  Perhaps I should dig that discussion up out of the files.  I acknowledged that without more information it would seem that your argument was the better.  However, I could not get further information from the web site who has stated that they have all of Hawkins work.  To some extent the issue is therefore still open.
    The work below is by Dee Gragg.  I am still separating the issues which I feel are unfortunately mixed together.  One issue is if or not Hawkins has found previously unknown Euclidean theorems.  And the other issue is if or not there are diatonic ratios in the crop circles.  And the issue of if or not there are diatonic issues in the crop circles is completely separate from the controversy of if or not the crop circles are 'real'.  In this case 'real' meaning that they are made by some unknown force, not 'hoaxers'.  In fact, when Hawkins was first asked to investigate the attitude that he took was purely mathematical, to look at the 'mind' of the circle makers whoever they were.  It was from that attitude that the observations came that whoever was doing this, either 'hoaxers' or unknown minds, they had the odd character of using not just diatonic scales, but diatonic scales that were more known to classic, not modern, history.
    Now Dee Gragg states that the issues are mixed.  Let me quote Dee...
     (quote) "  Now as to the Musical notes encoded in the circles.
They are not unrelated to the theorems.  In fact, all
four of Dr. Hawkins theorems are related to musical
notes.  Alas, I found that only two of mine were; F
below middle C and F two octaves below Middle C. ..." (unquote)
    Dee is right.  They are not unrelated.  But in this case they can still be separated and treated separate.  I advised that because of your observation, John.  But perhaps, with your astute help, the issue of if or not there are new Euclidean theorems can be resolved.
    I am passing this to Dee Gragg.  I am suggesting that she join the Polytopia list so that I am not so much in the middle of this.
    Michael Donovan
    Camden, ME.
----- Original Message -----
Sent: Friday, November 05, 2004 11:33 AM
Subject: Re: [Polytopia] back to the weird 2

There are some interesting relationships in shapes. Theorems 1A to 1D can be seen in the following picture. (Theorem 1C requires that you know that a 30-60-90 degree triangle has the edge ratios of 1:2:sqrt(3)... the others can be seen just by counting distances.) Were we to place another circle inside the three given circles, just tangent to them, its radius would be in a ratio of 1 to 3 with the original circles. There are numerous other relationships that we could point out. I don't think of these relationships as having any special meaning aside from being nice numbers.
 
John Berglund
 


Michael Donovan <michael1@...> wrote:

Crop Circle Theorems
 
Their Proofs and Relationship to Musical Notes
 

   This research began with a simple and rather limited objective: to prove the crop circle theorems of Dr. Gerald Hawkins.  In fact if I could have found the proofs in the literature of the field, this research would never have taken place at all.  Fortunately, I couldn’t find them because once I started, I could see that further work that needed to be done.
 
   As I proved Dr. Hawkins theorems, I discovered five new ones and proved them as well.  I then took the diatonic ratios of all the theorems and related them to the frequencies of the musical scale.  With some rather startling results I might add.
 
   Beginning with Theorem IA I need to make some observations that apply to all of the theorems.  In Euclidean Geometry one almost always has to see the end before making a beginning.   Also, since we are looking for diatonic ratios, we need to find an equation or equations which will let us divide one diameter or radius by the other.  Remember too, that because we are working with ratios, the constants divide out leaving diameter ratios equal to radius ratios.  And if we square them they are equal to each other and to the ratio of the areas.
 
   Applying this to Theorem IA the equation we need to write is for the diameter of the circumscribing circle.  It contains both the radii of the initial and the circumscribed circle.  So from the equation we are able to divide it and find the diatonic ratio of 4 to 3. 
 
   Although, I have proved three more Theorem I’s, I believe this is the one Dr Hawkins meant when he said Theorem I.  See Circular Relationships for The Theorems in Appendix A.  I base this belief on the 4 to 3 diatonic relationship which is related directly to Note F above Middle C.  See Frequencies In The Fields in Appendix B.
 
   Theorem IB is like Theorem IA except that the equilateral triangle is inscribed rather than being circumscribed.  It can be proved by Theorem IA and Theorem II.  The equations already exist so just divide them for the proof.  This gives a new diatonic ratio which is also the Note F, one octave lower than the previous.
This theorem is such a simple and logical extension of the first two that I am puzzled as to why Dr Hawkins did not discover and publish it.
 
   Theorem IC is also often referred to as Theorem I although it is quite different.  Sometimes both Theorem IA and Theorem IC appear in the same article as if they were identical.  They aren’t.  The proof of Theorem IC shows that it contains no diatonic ratio that can be related to a musical frequency.  I believe that this was not the theorem Dr. Hawkins was referring to when he said Theorem I.  In my mind the origin of Theorem IC is rather murky.
 
      Theorem ID would have never been discovered if I had read the instructions for constructing Theorem IA a little closer.  Instead of circumscribing the equilateral triangle, I circumscribed the three circles and then proved the theorem before realizing my mistake.  It has a nice 7 to 3 relationship but it would need to be 8 to 3 to be Note F in the next higher octave.
 
   Theorem II is easy to prove by constructing the appropriate similar triangles and remembering their relationships.  It may be proved a number of different ways I have shown just one of them.  It has the nice diatonic ratio of 4 to 1 which relates directly to the Note C which is two octaves above Middle C.
 
   Theorem III is the simplest of all proofs.  Just remember the Pythagorean Theorem. It also has a nice diatonic ratio of 2 to 1 which relates directly to the Note C which is an octave above Middle C.
 
   Again using the Pythagorean Theorem, Theorem IVA is shown to have a nice 4 to 3 relationship.  We have previously related this to Note F using Theorem IA.
 
   While proving Theorem II, it occurred to me that there should be a similar theorem related to the hexagon. There was and that led me to discover Theorem IVB by connecting the diameters at the hexagon corners.  Again by using similar triangles and writing and dividing the proper equations it is shown to have a diatonic ratio of 1 to 3 which relates directly to the Note F.  This Note F is yet another octave lower.
 
   I have included Theorem IVC mostly for completeness as it does not have a diatonic ratio which can be related to a specific note.  If I hadn’t included it you might have wondered why since it can be proved by simply dividing Theorem IVB by Theorem IVA. 
 
   There is a Theorem V which can be used for deriving (not proving) the other theorems.  However it does not of its self have diatonic ratios and therefore was not a part of this research.
 
      Appendix A Circular Relationships for The Theorems shows a summary of all the results.  Note that to go from one column to the other, you simply square or take the square root.  But how do you know which column to use?  I have followed the lead of Dr. Hawkins in that if the circles are not concentric, you use the ratio of diameters, if they are concentric you use the ratio of areas.  This means diameters for Theorems IA, IB, IC, ID, IVB, and IVC and areas for Theorems II, III, and IVA.  Why did he pick this convention?  Certainly I don’t know, perhaps he was a practical man and he did it because it works.
 
   Frequencies In The Fields in Appendix B gives four octaves: two above and two below Middle C.   This does not encompass the full 27.5 to 4,186 Hz of a piano but does include all the frequencies found so far.  Notice that all the notes are either F or C.  Coincidence or a message?  Perhaps as we discover more notes, this will become clear.
 
   Theorem T in Appendix C is not really a part of this research, but is included as help for anyone wanting to compute circle and regular polygon ratios.  It includes all cases and relies on trigonometry rather than Euclidian Geometry.
 
   Finally, if you’re wondering about me, I have a Short Bio in Appendix D.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Copyright 2004 by C. D. Gragg. All rights reserved
 
Theorem IA
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is circumscribed about the triangle, the ratio of the diameters is 4 to 3.
              
 
 
 
 
 
Theorem IB
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is inscribed within the triangle, the ratio of the diameters is  2:3.
 
              
 

Theorem IC
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is constructed using the single circle as a center and drawing the circle through the other two centers, the ratio of the diameters is 4 to Sqrt 3.
              
 

Theorem ID
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is constructed circumscribing the three circles, the ratio of the diameters is 7 to 3.
              
 
 
 
 
 
Theorem II
   If an equilateral triangle is inscribed and circumscribed the ratio of the circles’ areas is 4:1.
              
 
 
 
 
 
 
 

Theorem III
   If a square is inscribed and circumscribed the ratio of the circles’ areas is 2:1.
               
 
 
 
 
 
 
 
 
 
Theorem IVA
   If a hexagon is inscribed and circumscribed the ratio of the circles’ areas is 4:3.
              
 
 
 
 
 
 
 
 
 
Theorem IVB
      If a hexagon is inscribed and circumscribed and the corners connected by diameters, the inscribed circles of the created equilateral triangles have a diameter ratio to the inscribed circle of 1:3.
              
 
 
 
 
 
 
 
Theorem IVC
      If a hexagon is inscribed and circumscribed and the corners connected by diameters, the inscribed circles of the created equilateral triangles have a diameter ratio to the circumscribed circle of 1:2Sqrt3.
              
 
 
 
 
 
 
 
Appendix A
 
                          Circular Relationships for
                                     The Theorems
 Theorem Ratio of Diameters and
Radii Ratio of Areas, Diameters Squared, and Radii Squared
 Theorem IA 4:3 16:9
 Theorem IB  2:3  4:9
 Theorem IC 4:Sqrt3 16:3
 Theorem ID 7:3 49:9
 Theorem II 2:1 4:1
 Theorem III Sqrt2:1 2:1
 Theorem IVA 2:Sqrt3 4:3
 Theorem IVB 1:3 1:9
 Theorem IVC 1:2 Sqrt3 1:12
  
  
  
  
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Copyright 2004 by C. D. Gragg, All rights reserved
 
 
 
 
 
Appendix B
 
Frequencies In The Fields
 
Note Name C D E F G A B C
Diatonic Ratio 1/4 9/32 5/16 1/3 3/8 5/12 15/32 1/2
Frequency (Hz) 66 74.25 82.5 88 99 110 123.75 132
 
Note Name C D E F G A B C
Diatonic Ratio 1/2 9/16 5/8 2/3 3/4 5/6 15/16 1
Frequency (Hz) 132 148.5 165 176 198 220 247.5 264
 
Note Name C* D E F G A B C
Diatonic Ratio 1 9/8 5/4 4/3 3/2 5/3 15/8 2
Frequency (Hz) 264 297 330 352 396 440 495 528
 
Note Name C D E F G A B C
Diatonic Ratio 2 9/4 5/2 8/3 3 10/3 15/4 4
Frequency (Hz) 528 594 660 704 792 880 990 1056
* Middle C
 Denotes  found in the fields
 
Theorem Summary
 
Frequency (Hz) Theorem Used For Proof
88 Theorem IVB, Gragg
176 Theorem IB, Gragg
352 Theorem IA, Theorem IVA, Hawkins
528 Theorem III, Hawkins
1056 Theorem II, Hawkins
 

 
 
 
 
 
 
 
 
 
 
                    Copyright 2004 by C. D. Gragg, All rights reserved
Appendix C
Theorem T
 
   Trigonometry can be used to solve circular relationships for inscribed and circumscribed regular  polygons for polygons of any number of sides from 3 to infinity.
 
 Proof:                                      
                                                  Where:  α = 3600          and n = number of sides    
                                                  2n
                                                          
                                                                 cos α  = R1 
                                                              R2
 
                                                                    R2 =   _1     Proving the Theorem
Figure 1. Regular polygon with                  R1       cos α
               any number of sides                              
                                                           Further:   ( R2)2 =  (1)2    
                                                                                                  ( R1)2      ( cos α)2
Table of Some Common Polygons
Figure (All are equiangular) Number of Equal Sides  Ratio of Diameters and
Radii Ratio of Areas, Diameters Squared, and Radii Squared
Triangle (1)(4)         3       2.000 4      4.000
Square (2)         4       1.414 2      2.000
Pentagon         5       1.236          1.527
Hexagon (3)          6       1.155 4/3   1.333
Heptagon         7       1.110         1.232
Octagon (4)         8       1.082         1.172
Nonagon         9       1.064         1.132
Decagon       10     & nbsp; 1.051         1.106
       15       1.022         1.045
       20       1.012         1.025
       50       1.002         1.004
     100       1.000         1.001
     200       1.000         1.000
       ∞       1.000         1.000
(1) Theorem II, by Dr. Hawkins using Euclidian Geometry
(2) Theorem III, by Dr. Hawkins using Euclidian Geometry 
(3) Theorem IV, by Dr. Hawkins using Euclidian Geometry
(4) Found in the Kekoskee/Mayville, Wisconsin Crop Circle
             Formation July 9, 2003    
                                                                           Copyright 2004 by C. D. Gragg, All rights reserved
 
 
 
 
 
 
 

Appendix D
 
Short Bio
 
   My name is Dee Gragg.  I am a retired, mechanical engineer.  My career was spent in research, testing and evaluation. My main areas of research were automotive air bags, jet aircraft ejection seats and high speed rocket sleds.  I have published 33 technical papers as either the principal author or a co-author.  They form a part of the body of literature in their respective fields.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 


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#746 From: John Berglund <anisohedral@...>
Date: Sun Nov 7, 2004 3:53 am
Subject: Re: back to the weird 2
anisohedral
Send Email Send Email
 
Hey Michael,
 
I recall our discussion. I would be happy to say that Professor Hawkins has come up with new theorems, but there are thousands of new theorems found every year. I could write down 20 such theorems with no difficulty. The problem is that these particular theorems don't seem to have much significance to mathematics.
 
There is a bit of interest in the design of the crop circles, as in any designs. Were you to collect logos of companies, you could also find many mathematical relationships like symmetry in them. You could study them and find similar "theorems."
 
Concerning the question of being "diatonic:" Pythagorus noticed that notes which have frequencies in a ratio of small natural numbers sound good together. "Diatonic" refers to a musical scale - which in the old days had all the notes in ratios of small natural numbers. (Nowadays we use the equitempered scale based on irrational numbers... we lose the exact ratios, but gain the ability to play in all the different keys.) The diatonic scale uses 2, 3, and 5 and their multiples. One will note that 7 is the first number which is left out in this scheme. It would not be surprising if there are "diatonic" ratios in crop circles, company logos, or other created designs - these are the most common small numbers. If there are any crop circles that have ratios of 1:7 or 1:11 or 1:13 then we can say that these are not diatonic. Those interested could investigate.
 
Of course the claim that crop circles go with music doesn't have to stop with diatonic scales - just because western music is based on this scale, there are many other scales. The Yahoo "Tuning" group goes into loving detail about all the different ways to include 7, 11, 13 and more in scales ranging from 2 to thousands of notes per octave. By this method whatever ratios showed up anywhere - it could be matched to music.
 
The idea that since you can find certain ratios in a design, that some musical connection can be made seems silly to me. Take a yin-yang symbol. The ratio of the inner curve radius to the outer curve radius is 1:2 - a diatonic ratio. A honeycomb tiling by bees contains all of the diatonic ratios (as well as the 7, 11, 13...) The height and width of a TV screen are in a diatonic ratio. The size of an inch to a foot is a diatonic ratio. If you ask people to pick out two random numbers from 1 to 10, most of the time they will pick a diatonic ratio. Hold up some fingers on each hand - the ratio between the hands is diatonic. The ratio of how many teeth I have to how many leg bones I have is diatonic. Hopefully these examples show that diatonic ratios are everywhere - no need to get excited if they show up somewhere.
 
John Berglund

Michael Donovan <michael1@...> wrote:
John,
    I had both a feeling and a deep hope you would appear out of the cybermist for this.  And I am very grateful that you have.  If you remember the general question of if Professor Gerald Hawkins did or did not come up with new theorems had come up on this list. At that time you pointed out that Hawkins theorem was covered by a very easily provable attribute to all regular polygons.  Perhaps I should dig that discussion up out of the files.  I acknowledged that without more information it would seem that your argument was the better.  However, I could not get further information from the web site who has stated that they have all of Hawkins work.  To some extent the issue is therefore still open.
    The work below is by Dee Gragg.  I am still separating the issues which I feel are unfortunately mixed together.  One issue is if or not Hawkins has found previously unknown Euclidean theorems.  And the other issue is if or not there are diatonic ratios in the crop circles.  And the issue of if or not there are diatonic issues in the crop circles is completely separate from the controversy of if or not the crop circles are 'real'.  In this case 'real' meaning that they are made by some unknown force, not 'hoaxers'.  In fact, when Hawkins was first asked to investigate the attitude that he took was purely mathematical, to look at the 'mind' of the circle makers whoever they were.  It was from that attitude that the observations came that whoever was doing this, either 'hoaxers' or unknown minds, they had the odd character of using not just diatonic scales, but diatonic scales that were more known to classic, not modern, history.
    Now Dee Gragg states that the issues are mixed.  Let me quote Dee...
     (quote) "  Now as to the Musical notes encoded in the circles.
They are not unrelated to the theorems.  In fact, all
four of Dr. Hawkins theorems are related to musical
notes.  Alas, I found that only two of mine were; F
below middle C and F two octaves below Middle C. ..." (unquote)
    Dee is right.  They are not unrelated.  But in this case they can still be separated and treated separate.  I advised that because of your observation, John.  But perhaps, with your astute help, the issue of if or not there are new Euclidean theorems can be resolved.
    I am passing this to Dee Gragg.  I am suggesting that she join the Polytopia list so that I am not so much in the middle of this.
    Michael Donovan
    Camden, ME.
----- Original Message -----
Sent: Friday, November 05, 2004 11:33 AM
Subject: Re: [Polytopia] back to the weird 2

There are some interesting relationships in shapes. Theorems 1A to 1D can be seen in the following picture. (Theorem 1C requires that you know that a 30-60-90 degree triangle has the edge ratios of 1:2:sqrt(3)... the others can be seen just by counting distances.) Were we to place another circle inside the three given circles, just tangent to them, its radius would be in a ratio of 1 to 3 with the original circles. There are numerous other relationships that we could point out. I don't think of these relationships as having any special meaning aside from being nice numbers.
 
John Berglund
 


Michael Donovan <michael1@...> wrote:

Crop Circle Theorems
 
Their Proofs and Relationship to Musical Notes
 

   This research began with a simple and rather limited objective: to prove the crop circle theorems of Dr. Gerald Hawkins.  In fact if I could have found the proofs in the literature of the field, this research would never have taken place at all.  Fortunately, I couldn’t find them because once I started, I could see that further work that needed to be done.
 
   As I proved Dr. Hawkins theorems, I discovered five new ones and proved them as well.  I then took the diatonic ratios of all the theorems and related them to the frequencies of the musical scale.  With some rather startling results I might add.
 
   Beginning with Theorem IA I need to make some observations that apply to all of the theorems.  In Euclidean Geometry one almost always has to see the end before making a beginning.   Also, since we are looking for diatonic ratios, we need to find an equation or equations which will let us divide one diameter or radius by the other.  Remember too, that because we are working with ratios, the constants divide out leaving diameter ratios equal to radius ratios.  And if we square them they are equal to each other and to the ratio of the areas.
 
   Applying this to Theorem IA the equation we need to write is for the diameter of the circumscribing circle.  It contains both the radii of the initial and the circumscribed circle.  So from the equation we are able to divide it and find the diatonic ratio of 4 to 3. 
 
   Although, I have proved three more Theorem I’s, I believe this is the one Dr Hawkins meant when he said Theorem I.  See Circular Relationships for The Theorems in Appendix A.  I base this belief on the 4 to 3 diatonic relationship which is related directly to Note F above Middle C.  See Frequencies In The Fields in Appendix B.
 
   Theorem IB is like Theorem IA except that the equilateral triangle is inscribed rather than being circumscribed.  It can be proved by Theorem IA and Theorem II.  The equations already exist so just divide them for the proof.  This gives a new diatonic ratio which is also the Note F, one octave lower than the previous.
This theorem is such a simple and logical extension of the first two that I am puzzled as to why Dr Hawkins did not discover and publish it.
 
   Theorem IC is also often referred to as Theorem I although it is quite different.  Sometimes both Theorem IA and Theorem IC appear in the same article as if they were identical.  They aren’t.  The proof of Theorem IC shows that it contains no diatonic ratio that can be related to a musical frequency.  I believe that this was not the theorem Dr. Hawkins was referring to when he said Theorem I.  In my mind the origin of Theorem IC is rather murky.
 
      Theorem ID would have never been discovered if I had read the instructions for constructing Theorem IA a little closer.  Instead of circumscribing the equilateral triangle, I circumscribed the three circles and then proved the theorem before realizing my mistake.  It has a nice 7 to 3 relationship but it would need to be 8 to 3 to be Note F in the next higher octave.
 
   Theorem II is easy to prove by constructing the appropriate similar triangles and remembering their relationships.  It may be proved a number of different ways I have shown just one of them.  It has the nice diatonic ratio of 4 to 1 which relates directly to the Note C which is two octaves above Middle C.
 
   Theorem III is the simplest of all proofs.  Just remember the Pythagorean Theorem. It also has a nice diatonic ratio of 2 to 1 which relates directly to the Note C which is an octave above Middle C.
 
   Again using the Pythagorean Theorem, Theorem IVA is shown to have a nice 4 to 3 relationship.  We have previously related this to Note F using Theorem IA.
 
   While proving Theorem II, it occurred to me that there should be a similar theorem related to the hexagon. There was and that led me to discover Theorem IVB by connecting the diameters at the hexagon corners.  Again by using similar triangles and writing and dividing the proper equations it is shown to have a diatonic ratio of 1 to 3 which relates directly to the Note F.  This Note F is yet another octave lower.
 
   I have included Theorem IVC mostly for completeness as it does not have a diatonic ratio which can be related to a specific note.  If I hadn’t included it you might have wondered why since it can be proved by simply dividing Theorem IVB by Theorem IVA. 
 
   There is a Theorem V which can be used for deriving (not proving) the other theorems.  However it does not of its self have diatonic ratios and therefore was not a part of this research.
 
      Appendix A Circular Relationships for The Theorems shows a summary of all the results.  Note that to go from one column to the other, you simply square or take the square root.  But how do you know which column to use?  I have followed the lead of Dr. Hawkins in that if the circles are not concentric, you use the ratio of diameters, if they are concentric you use the ratio of areas.  This means diameters for Theorems IA, IB, IC, ID, IVB, and IVC and areas for Theorems II, III, and IVA.  Why did he pick this convention?  Certainly I don’t know, perhaps he was a practical man and he did it because it works.
 
   Frequencies In The Fields in Appendix B gives four octaves: two above and two below Middle C.   This does not encompass the full 27.5 to 4,186 Hz of a piano but does include all the frequencies found so far.  Notice that all the notes are either F or C.  Coincidence or a message?  Perhaps as we discover more notes, this will become clear.
 
   Theorem T in Appendix C is not really a part of this research, but is included as help for anyone wanting to compute circle and regular polygon ratios.  It includes all cases and relies on trigonometry rather than Euclidian Geometry.
 
   Finally, if you’re wondering about me, I have a Short Bio in Appendix D.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Copyright 2004 by C. D. Gragg. All rights reserved
 
Theorem IA
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is circumscribed about the triangle, the ratio of the diameters is 4 to 3.
              
 
 
 
 
 
Theorem IB
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is inscribed within the triangle, the ratio of the diameters is  2:3.
 
              
 

Theorem IC
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is constructed using the single circle as a center and drawing the circle through the other two centers, the ratio of the diameters is 4 to Sqrt 3.
              
 

Theorem ID
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is constructed circumscribing the three circles, the ratio of the diameters is 7 to 3.
              
 
 
 
 
 
Theorem II
   If an equilateral triangle is inscribed and circumscribed the ratio of the circles’ areas is 4:1.
              
 
 
 
 
 
 
 

Theorem III
   If a square is inscribed and circumscribed the ratio of the circles’ areas is 2:1.
               
 
 
 
 
 
 
 
 
 
Theorem IVA
   If a hexagon is inscribed and circumscribed the ratio of the circles’ areas is 4:3.
              
 
 
 
 
 
 
 
 
 
Theorem IVB
      If a hexagon is inscribed and circumscribed and the corners connected by diameters, the inscribed circles of the created equilateral triangles have a diameter ratio to the inscribed circle of 1:3.
              
 
 
 
 
 
 
 
Theorem IVC
      If a hexagon is inscribed and circumscribed and the corners connected by diameters, the inscribed circles of the created equilateral triangles have a diameter ratio to the circumscribed circle of 1:2Sqrt3.
              
 
 
 
 
 
 
 
Appendix A
 
                          Circular Relationships for
                                     The Theorems
 Theorem Ratio of Diameters and
Radii Ratio of Areas, Diameters Squared, and Radii Squared
 Theorem IA 4:3 16:9
 Theorem IB  2:3  4:9
 Theorem IC 4:Sqrt3 16:3
 Theorem ID 7:3 49:9
 Theorem II 2:1 4:1
 Theorem III Sqrt2:1 2:1
 Theorem IVA 2:Sqrt3 4:3
 Theorem IVB 1:3 1:9
 Theorem IVC 1:2 Sqrt3 1:12
  
  
  
  
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Copyright 2004 by C. D. Gragg, All rights reserved
 
 
 
 
 
Appendix B
 
Frequencies In The Fields
 
Note Name C D E F G A B C
Diatonic Ratio 1/4 9/32 5/16 1/3 3/8 5/12 15/32 1/2
Frequency (Hz) 66 74.25 82.5 88 99 110 123.75 132
 
Note Name C D E F G A B C
Diatonic Ratio 1/2 9/16 5/8 2/3 3/4 5/6 15/16 1
Frequency (Hz) 132 148.5 165 176 198 220 247.5 264
 
Note Name C* D E F G A B C
Diatonic Ratio 1 9/8 5/4 4/3 3/2 5/3 15/8 2
Frequency (Hz) 264 297 330 352 396 440 495 528
 
Note Name C D E F G A B C
Diatonic Ratio 2 9/4 5/2 8/3 3 10/3 15/4 4
Frequency (Hz) 528 594 660 704 792 880 990 1056
* Middle C
 Denotes  found in the fields
 
Theorem Summary
 
Frequency (Hz) Theorem Used For Proof
88 Theorem IVB, Gragg
176 Theorem IB, Gragg
352 Theorem IA, Theorem IVA, Hawkins
528 Theorem III, Hawkins
1056 Theorem II, Hawkins
 

 
 
 
 
 
 
 
 
 
 
                    Copyright 2004 by C. D. Gragg, All rights reserved
Appendix C
Theorem T
 
   Trigonometry can be used to solve circular relationships for inscribed and circumscribed regular  polygons for polygons of any number of sides from 3 to infinity.
 
 Proof:                                      
                                                  Where:  α = 3600          and n = number of sides    
                                                  2n
                                                          
                                                                 cos α  = R1 
                                                              R2
 
                                                                    R2 =   _1     Proving the Theorem
Figure 1. Regular polygon with                  R1       cos α
               any number of sides                              
                                                           Further:   ( R2)2 =  (1)2    
                                                                                                  ( R1)2      ( cos α)2
Table of Some Common Polygons
Figure (All are equiangular) Number of Equal Sides  Ratio of Diameters and
Radii Ratio of Areas, Diameters Squared, and Radii Squared
Triangle (1)(4)         3       2.000 4      4.000
Square (2)         4       1.414 2      2.000
Pentagon         5       1.236          1.527
Hexagon (3)          6       1.155 4/3   1.333
Heptagon         7       1.110         1.232
Octagon (4)         8       1.082         1.172
Nonagon         9       1.064         1.132
Decagon       10     & nbsp; 1.051         1.106
       15       1.022         1.045
       20       1.012         1.025
       50       1.002         1.004
     100       1.000         1.001
     200       1.000         1.000
       ∞       1.000         1.000
(1) Theorem II, by Dr. Hawkins using Euclidian Geometry
(2) Theorem III, by Dr. Hawkins using Euclidian Geometry 
(3) Theorem IV, by Dr. Hawkins using Euclidian Geometry
(4) Found in the Kekoskee/Mayville, Wisconsin Crop Circle
             Formation July 9, 2003    
                                                                           Copyright 2004 by C. D. Gragg, All rights reserved
 
 
 
 
 
 
 

Appendix D
 
Short Bio
 
   My name is Dee Gragg.  I am a retired, mechanical engineer.  My career was spent in research, testing and evaluation. My main areas of research were automotive air bags, jet aircraft ejection seats and high speed rocket sleds.  I have published 33 technical papers as either the principal author or a co-author.  They form a part of the body of literature in their respective fields.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 


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#747 From: "Larry Rafey" <rafey@...>
Date: Sun Nov 7, 2004 6:54 pm
Subject: Re: back to the weird 2
lrafey
Send Email Send Email
 
John & Michael,
   May I quote Stephen Jay Gould here in his Questioning the Millenium in which he states the following:
           "My argument for the origin of our fascination with numerical regularity closely parallels my claims about our affinity for dichotomous classification. In part, we latch on to numerical regularity, and seek deep meaning therein, because such order does underlie much of nature's patterning. The periodic table, after all, is not an arbitray human mnemonic, and Newtonian gravity does work by a law of inverse squares. But our search for numberical order, and our over interpretations, run so far beyond what nature could possibly exemplify, that we can only posit some inherent mental bias as a driving force ... Our searches for numerical order lead as often to terminal nuttiness as to profound insight."
    Perhaps this might shed some sobering illumination on your argument. It is in the very nature of number that patterns will be discovered no matter where you look as number is encoded within the very fabric of reality and serves as the very essence of our knowledge of it.
LD Rafey
 
 
 
 
 
 


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#748 From: rybo6 <rybo6@...>
Date: Sun Nov 7, 2004 7:30 pm
Subject: Rybonic symbolisms as textual formatting
os_jbug
Send Email Send Email
 
A) __  __  = generalized-concept and constant-eternal-truth
     ...a) e.g. sphere, Isotropic-Edge-Matrix(IEM), One-Diameter.

     B) *   *   = special-case vector-associated-values
     ....b) e.g. 1.1, 3, 5, 6.7....infinitum, vector-"edge"-values, of
any,
   non-Synergetics-based Isotropic Vector Matrix(IVM).

     C) **   ** = special-case Synergetics.
     ....c) e.g. Isotropic-Vector-Matrix(IVM) vector-"edge"-value of 
     **two(2)**

     D) ***  *** = energetic-motion, over-time, in-space.
     ...d) e.g. axis-spun polyhedral great-circles(GrCs) a.k.a as
     ultra-high-frequency polyhedra.

Rybo
































Anti-bush campagin 2004. Bush must go!

#749 From: rybo6 <rybo6@...>
Date: Sun Nov 7, 2004 7:41 pm
Subject: (No subject)
os_jbug
Send Email Send Email
 
NOTE: The following, is in reverse-order, of my initial writing of
   these comments in a "reply-email" in another group.
   ============================

     .......This may be the Great-Circle reprsentations of, sub-atomic,
     high-frequecny-polyhedra as, the phenomena we call "spin", at, our
     macro-level-existence, that, we call the 3-4D sensorial environment.

   Last I read it had confirmed some more of Generalizd Relativity
mathematical/theories, i.e. that there exists something called
"gravitaitonal-frame-dragging.

I wonder how "gravity-probe-B" is doing?

     This instrument will measure the rate-of-time arrival, between
two(2) or more variant-rates of, EM-frequenices.  There conjecture is
that one  will arrive at sooner than the other one from some celestial
source.  Which reminds me.

   This is one to watch.  In 2006-7 there will be instrument in space to
confirm or dispel Loop Quatum Gravitys conjecture of the spin-foam
fabric-of-space, quantization-abilities.

He only knows this is what his black hole mathematics had led too him
in 1/2/2002.  So of course the natural inclination is to say, "his
mathematics msut be in error"

"we humans are 2D somethings having the illusion of 3D"

Which in turn, lead to one of Jacob B's  making one(1), of the
top-ten, most-"radical"-crazy statements, known on Earth.---
Similarly, I think Holography, as expressed on 2D surface, is involved.

     This latter, is intimatly involved too Jacob Bekenstiens "Black
Hole" mathematics, whom himself, and S. Hawking discovered, the
"Bekenstien Law" regarding entropy, inside a black-hole, being
expressed at its  2D-surface.

obyR(Rybo)

#750 From: "Larry Rafey" <rafey@...>
Date: Sun Nov 7, 2004 8:59 pm
Subject: Re:GP-B Mission
lrafey
Send Email Send Email
 
FYI:
Rybo.......and anyone interested in this little piece of magic.........
   GP-B is, indeed, performing better than expected, having recently terminated its IOC (initialization and orbit calibration) phase and having at last begun transition into its scientific phase bringing it one step closer to shedding new light on the fundamental properties of the universe.
   GP-B, decades in the planning and executing, is testing whether something spinning (like our Earth) can have an effect on the spin of another object (in this case, the quartz ping-pong balls, perfectly spherical to within a few atoms and eternally undisturbed in pristine, Zen-like space-time; their axis of spin adjusted precisely to point at a selected guide star..IM Pegasi or HR 8703) and all this in the absence of any signal path (neither radio wave, static electricity, magnetism nor friction) between them!
   The answer might help to ultimately fuse the contradictory principles og Quantum Mechanics and Relativity into a more unified theory. We might even directly observe an entirely new, unanticipated force of nature !
    According to theory, a rotating massive body drags space and time around with it. A gyroscope orbiting Earth can be made to tilt away from the plane of its orbit because the Earth is dragging it. This spin axis drift referred to as the frame-dragging (or geodetic) effect of an anticipated 40.9 milliarc seconds can be measured, it is hoped, to within 1% precision by the BP-B Probe.
  All systems so far are meeting or exceeding anticipated requirements. We should have some positive info in about six months. However, no info will be forthcoming until after the science phase has concluded and further analysis is performed.
LD Rafey
----- Original Message -----
From: rybo6
Sent: Sunday, November 07, 2004 1:41 PM
Subject: [Polytopia]

NOTE: The following, is in reverse-order, of my initial writing of
  these comments in a "reply-email" in another group.
  ============================

    .......This may be the Great-Circle reprsentations of, sub-atomic,
    high-frequecny-polyhedra as, the phenomena we call "spin", at, our
    macro-level-existence, that, we call the 3-4D sensorial environment.

  Last I read it had confirmed some more of Generalizd Relativity
mathematical/theories, i.e. that there exists something called
"gravitaitonal-frame-dragging.

I wonder how "gravity-probe-B" is doing?

    This instrument will measure the rate-of-time arrival, between
two(2) or more variant-rates of, EM-frequenices.  There conjecture is
that one  will arrive at sooner than the other one from some celestial
source.  Which reminds me.

  This is one to watch.  In 2006-7 there will be instrument in space to
confirm or dispel Loop Quatum Gravitys conjecture of the spin-foam
fabric-of-space, quantization-abilities.

He only knows this is what his black hole mathematics had led too him
in 1/2/2002.  So of course the natural inclination is to say, "his
mathematics msut be in error"

"we humans are 2D somethings having the illusion of 3D"

Which in turn, lead to one of Jacob B's  making one(1), of the
top-ten, most-"radical"-crazy statements, known on Earth.---
Similarly, I think Holography, as expressed on 2D surface, is involved.

    This latter, is intimatly involved too Jacob Bekenstiens "Black
Hole" mathematics, whom himself, and S. Hawking discovered, the
"Bekenstien Law" regarding entropy, inside a black-hole, being
expressed at its  2D-surface.

obyR(Rybo)



#751 From: "Michael Donovan" <michael1@...>
Date: Sun Nov 7, 2004 9:47 pm
Subject: Re: back to the weird 2
jayennseth
Send Email Send Email
 
Yes, the famous argument between Renee Decartes (excuse my dyslexic spellings) and Pierre Furmet.  I am writing an answer, more comment, in regards to John's excellet post.
Michael
----- Original Message -----
Sent: Sunday, November 07, 2004 1:54 PM
Subject: Re: [Polytopia] back to the weird 2

John & Michael,
   May I quote Stephen Jay Gould here in his Questioning the Millenium in which he states the following:
           "My argument for the origin of our fascination with numerical regularity closely parallels my claims about our affinity for dichotomous classification. In part, we latch on to numerical regularity, and seek deep meaning therein, because such order does underlie much of nature's patterning. The periodic table, after all, is not an arbitray human mnemonic, and Newtonian gravity does work by a law of inverse squares. But our search for numberical order, and our over interpretations, run so far beyond what nature could possibly exemplify, that we can only posit some inherent mental bias as a driving force ... Our searches for numerical order lead as often to terminal nuttiness as to profound insight."
    Perhaps this might shed some sobering illumination on your argument. It is in the very nature of number that patterns will be discovered no matter where you look as number is encoded within the very fabric of reality and serves as the very essence of our knowledge of it.
LD Rafey
 
 
 
 
 
 


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#752 From: "Larry Rafey" <rafey@...>
Date: Sun Nov 7, 2004 10:11 pm
Subject: Descartes...
lrafey
Send Email Send Email
 
Michael,
  Yes. I believe you are referring to the correspondence between Descartes and P.de Fermat.
   Looking forward to you further commentary.
LD Rafey

#753 From: "Michael Donovan" <michael1@...>
Date: Sun Nov 7, 2004 10:26 pm
Subject: Re: Descartes...
jayennseth
Send Email Send Email
 
the guy who comes up with the dyslexic spell checker will make millions.
Thanks,
 
----- Original Message -----
Sent: Sunday, November 07, 2004 5:11 PM
Subject: [Polytopia] Descartes...

Michael,
  Yes. I believe you are referring to the correspondence between Descartes and P.de Fermat.
   Looking forward to you further commentary.
LD Rafey


#754 From: "Michael Donovan" <michael1@...>
Date: Fri Nov 12, 2004 1:48 am
Subject: Re: back to the weird 2
jayennseth
Send Email Send Email
 

Re: Dee Gragg's Crop Circle Paper and John Berglund's Reaction
Michael Donovan
 
(I wish to thank Larry Rafey of Polytopia for the apt quote from Stephen Jay Gould's Questioning The Millennium.  In fact, I will use it to introduce my reply to John Berglund, repeating it as this will be copied elsewhere. And I also apologize for repetition of some other material, again reposted because of the copying to others outside the list.)
 
           (quote)"My argument for the origin of our fascination with numerical regularity closely parallels my claims about our affinity for dichotomous classification. In part, we latch on to numerical regularity, and seek deep meaning therein, because such order does underlie much of nature's patterning. The periodic table, after all, is not an arbitrary human mnemonic, and Newtonian gravity does work by a law of inverse squares. But our search for numerical order, and our over interpretations, run so far beyond what nature could possibly exemplify, that we can only posit some inherent mental bias as a driving force ... Our searches for numerical order lead as often to terminal nuttiness as to profound insight." Stephen Jay Gould, Questioning The Millennium.
 
To John Berglund and Yahoo Polytopia list:
Copy to Dee Gragg.
John,
    In a few clear paragraphs your post lays out the full landscape in which to view the claims of Professor Hawkins and Dee Gragg.
    Dee Gragg is not a member of this group, though I have invited her..., more the ease of mailing.  I will copy this and your remarks to her.  Dee Gragg, as am I, is interested in the math behind crop circles.  John's comments on Dee Gragg's paper are copied below.  John Berglund has no 'new age' interests, and is a sound mathematical thinker.., perfect for bouncing such claims.  I am very grateful for John's interest, clarity and expertise. 
    I write this because I feel that Dee Gragg is on to some very important discoveries.  But that one aspect, in a technical sense, of the present manner of expressing these discoveries may not be the best or even backfire.  To me this is no small issue.  I see a giant 'math' message in the crop circles and am writing a book on what this mathematical message might be.
    Were you to view the headlines at Rense, www.rense.com, today you will see a cryptic quote from General Wesley Clark.  He was being asked what he thought of UFOs.  First he made a remark indicating that he had been briefed on the matter.  Then, with no surrounding explanatory context, he made this statement..., (quote) "There are things going on. But we will have to work out our own mathematics." (unquote)   I believe he meant exactly what he said, with the exact words that he chose to say it, and further that his doubled layered, pithy reply becomes clearer to the extent that you also realize that many at the top have been seeing a deep shift in our mathematical thinking prompted by some decidedly 'spooky' heart stoppers..., crop circles among others.  Changes which are deep in mathematical thought do not come often, but when they do they invariably have strategic geopolitical implications.  It is my firm belief that the exact same complex of forces now surrounding the new math implied in the crop circle phenomena and 'sea/air' correlation were also gathered around the advent of calculus in the 1720s.  The 'military' or 'security' becomes the first flag of interest.  Newton's blossoming of Descartes, with a few well marked measuring sticks, had world changing tactical application and strategic implication upon which this orb almost had, in the British, a fully global empire.  While lofty philosophical discussions were taking place, and the scope of technological applications of calculus were no more than a few un-watered germs in the public discussion, tight controls were placed upon the military application.  Only Royal Marines were allowed to be gunners on British men-o-war.
    I believe that now, as with most of the mid and late 1700s with calculus, the major governments know of a math advance, in this case a new geometry, and have delayed the emergence of this math into the mass mind.  But that they are slowly realizing that public awareness will grow no matter what measures are taken, and are therefore now curious as to how this might creep out into the public mind.
    There are two issues in Dee Gragg's paper.  One is if or not there are new Euclidean theorems.  What constitutes a 'new Euclidean theorem' is a matter of academic classification, an arbitrary choice, either now or in the future, decided by some combination of academic demagoguery, whim and convention.  All we need see here is that 1- such claims of 'new theorems' can be argued, and 2- such arguments will not be quickly resolved.  Because I feel that there is some interesting discovery, some real meat, in Dee's paper, I think it wise to simply diminish the 'new theorem' claims.  A simple, "Theorems which I have not seen published before and, who knows, in the future might even be considered new Euclidean theorems....," di da di da, or somesuch, would disarm time wasting arguments and be viewed as somewhere between understatement and caution.
    The meat of the paper consists in the list of diatonic ratio observations.  I will not belabor the point made both by John Berglund and in the introductory quote.  Are these ratios something that could be expected at random? Have Dee Graggs and Gerald Hawkins been reading far too much into something that would under many circumstances just naturally occur?  I think not.  And there are some standard tests for determining that in the statistical sense.  I will leave that issue alone for the moment.
    First I wish to back up and give another, I hope larger and better, framework for this entire gestalt of interlinking issues.  It is not so much the variety of 'scales' in the appearance of 'octave', but the octave itself which more intrigues.  A standard science encyclopedia such as Van Nostrand's
will say that although there are many mathematical relationships and ratios within music, that music itself is purely psychological.  That is to say that music is a 'figment of the mind' having no basis in physics.  A point I would argue.
    To start I am going to quote, and quote quite liberally, from an article in the September - October issue (page 49) of Nexus magazine.  The title of the article is An Introduction To Global Scaling Theory copyright Dr. Hartmut Muller, 2004.  Dr. Muller talks about the relationship between mathematical and physical science discovery, that at first, for seeming eons, mathematics was leading the way, but that recently it has not been, saying...:
    (quote) "...This paradigm brought about the end of the ancient student-master relationship between the natural sciences and mathematics.  In the academic enterprise of today the mathematician only develops the models.  It is the physicist (chemist, biologist, geologist) who decides which of the models matches the measurements and gets applied.  As a result of this division of labour, mathematics became more and more 'instrumentalized' and hence isolated form the its intellectual source - the natural sciences...." (unquote, Dr. Hartmut Muller)
    Yet Dr. Muller goes on to say that something is emerging that shows that once again something in pure math is taking the forefront.  Something strange seemed to pop out of a massive amount of pure measurements, an oddity that he then refers to as a 'scientific goldmine'.
    (quote from Dr. Muller continued...) "...The first indication of the existence of this scientific goldmine came from biology.  As a result of 12 years of research, Cislenko published his Structure of Fauna And Flora With Regard To Body Size Of Organisms (Moscow, 1980).  His work documents what is probably the most important biological discovery in the 20th century.  Cislenko was able to prove that segments of increased species representation are repeated on the logarithmic line of body sizes in equal intervals (aprox. 0.5 units of the decadic logarithm).  The phenomenon is not explicable from a biological point of view.  Why should mature individuals of amphibians, reptiles, fish, birds, and mammals of different species find it similarly advantageous to have a body size in the range of 8-12 centimeters, 33-55 centimeters or 1.5 - 2.4 meters...?" (unquote)
    I have postulated that there is an important new geometry based on the relationship of equal sized balls.  It is a geometry that is basic to three dimensions and uses the common 3-D object, the ball.  The sphere in this new geometry is impossible as it is a concept based on lines and points.  Further that one of two ways that 12 balls fit around one is central to this new geometry.  And that also in that manner balls make other balls that make other balls.  The geometric sequencing of this (logarithmic from another measurement view) is, to give an easy visualization, 12 peas around one pea making a ball about the size of a golf ball, 12 of these around one (13) making a softball sized ball, and 13 of these softball ball sized balls making a basketball sized ball and so on.  This, I postulate, is the real 'octave'.  Linear frequency measurements may appear to give regular intervals.  That is inherent in the measure system.  The experiments of Pavlita and the new geometry showed emotion to be the big factor.  And that certain basic 'planes of vibration' (base 'elements' and base 'compounds') repeat.  Going back to the original sequence of balls...., all of these balls within balls within balls first vibrate in seven 3-D planes.  And there is a subset of 12.  A little further there is a subset of 4 multiples, 8 and 16.  What is more important is that when you have the correct ball construction of 12 around one, you can easily see the seven core vibrations.  And you can also see that these seven vibrations are of two types, a subset of four and a subset of three.  And, more intriguing, they repeat....: pea, golf ball, soft ball, basketball and so on.  It is the similarity of 'feeling' about the note (any note in any scale) of one size/octave in regard to a higher/lower set which is far more the first consideration prior to the idea of specific 'scales'.
    I believe that it is this geometry that the 'circles' are giving giant hints of.  I believe, strongly, that the government knows and is monitoring the public awareness.  In fact a very curious website appeared which can be found under a simple 'spherepacking' google.  I just checked and can't find it.  It was on the West Coast, but had obvious connections to Princeton.  The guy was setting himself up as, believe it or not, a 'curriculum counselor' for teenagers who get into spherepacking and thereby get 'into the occult'.  Really !  Really?  Really.  Are you very worried that your teenagers are under their beds at night with a flashlight gluing ping pong balls together?  This site was well healed.  I wrote this guy, (a Kirby something, last name was short and began with U, Ulnar or something).  It was a funny letter in the vein of "I'm not believing this...", but indicating that I knew why he was doing what he was doing.  First those references in the site came out.  Now the entire site seems gone and I can't even find caches of it.  If you are scratching your head..., well..., I understand.  But the specific 12 around one construction does have a strong correlation to a number of basic occult cosmologies, the houses of astrology and the cabala map, and the basic organization of the Tarot deck.  I feel physics and metaphysics will merge.  And I believe that the powers-that-be know this and wish to monitor it, that being the real purpose for those strange things on the spherepacking site.  With that strange aside now aside, here are some very general observations in regard to observations of 'scale' ratios in crops circles, and the circles in general...:
    1- I am feeling that the ratios will be seen, particularly in certain sequences, as highly statistically improbable from a random comparison.
    2- This will be seen as moreso as in many cases the two types of notes of the seven, the four and the three, are separated.
    3-  When viewed with the crop pictorial, the many that are seven/12, it will become even more startling.
    4- In fact, as Dee Gragg noted in her paper, the obvious sub ratio (sharps / flats - 'black keys') are completely omitted.  And completely omitten until there is just the glimmer of what they might be.
    And all that with caveats noted.
    The 'meat' of Dee Gragg's paper, the ratio listings, need a deeper look.  This will take some time.  Interest will increase.  It will increase dramatically when it is seen that there is application for the geometry that the ratios (and general pictorial outlines) are hinting at.  And at that time a more careful look, a good deal of 'tweaking' will naturally come about in the new geometry itself.  Such, if you remember, was the case with trigonometry.   Even noting the vast possibilities for trig in navigation it took some massive amount of time to straighten out the trig tables to a point where there was enough accuracy to make it reliable.  And this came about much later, in the early 1800s, than most people realize.  How many New England towns have a street named after Samuel Bowditch?  Very many.  A great 'mathematical thinker' he was not.  But he may have been one of the most singularly anal compulsive men ever born.  He sat down with pen and paper and, God knows how long this took, straightened out the trig tables.  No small feat for those whose graveyards, replete with so many Biblical names such as Sumner and Assa and Ahab, were heavily peppered also with the simple markers of 'Lost At Sea'.  Until that time seamen were far more comfortable with pilots who were 'Portugee Rutter' trained, the trig then being more another 'check' than any sort of GPS.  From Cape Verde, dos Bijagos, Sao Tome, future navigators to be trained in the 'rutter' were culled from the human trade.  I was even told, (quaint family lore), that the native priests checked teeth, not so much for health, but to see lineage to certain tribes such as the Dogon.  Navigators in this odd admixture of art and science that the 'rutter' was began training early, as mere boys.  What seemed a hodge podge of sea colors, sky colors, fish, currents were all in some odd 'classification of twelves', perhaps an uncanny historic marker to present math development.  History omits that a large percentage of the pilots in the Age Of Sail were Black.
        The 'rutters' tones are four times three,
            The 'sun net' of the Portugee.
        He, oft times, went off 'to fish'
            And left on shore his heart's won dish.
        She, so sprightly, hedged all bets,
            Fished too, on shore
            With 'cast-a-nets'.
    The Portugee Rutter is now some historical footnote. The only major work in which I have seen references to it is the novel Shogun.  And it is almost hard to believe that trig, with such obvious application, went undeveloped so long. 1820s !!!  We usually do not see math development in such light.  The introductory quote by Stephen Jay Gould references Newton's inverse square law.  At the time, who cared?  This could not be used to navigate, or build bridges, or count money.  There were discussions, but in retrospect they seemed agonizingly slow to those involved.  Over and over Newton was asked, referencing back to Kepler, what, in his new system, would the orbit path of a planet be.  Again and again and again, in one discussion after another, Newton gave the same answer, "An ellipse".  It took years of that before the emergence of the collective, "Oh. I see. Wow."
    Einstein clearly stated that he felt the next big breakthrough would be in geometry.
    There are many good references to specific shapes demonstrating heretofore unexplained physical properties.  In the 1950s a Czech obtained a patent for the Great Pyramid shape.  Small pyramids with the Great Pyramid ratios built out of cardboard had a practical use.  Lined up correctly on the magnetic axis a razor blade placed in the center, say on a match box, would resharpen.  This would not work today except with the industrial razor blades.  The razor blades, to resharpen, need to be of the older 'high iron content' type.  To obtain this patent the inventor had to show that the mechanical device (it needed to be a mechanical device) was 'new', performed a worthwhile function, was 'non obvious', and had never been disclosed before.  The inventor did not need to show how it worked to obtain the patent.  And he stated clearly that he did not know.  But this device did work, and many, very many, Soviet soldiers, their salaries being meager and razor blades expensive, brought and used them.  That important little story is in chapter 27 of an old book, now in many libraries, Psychic Discoveries Behind The Iron Curtain.  
    Chapter 28 of that same book goes onto even more uncanny experiments with 'shape' and physics.  That is the story of Robert Pavlita.  But here the inventor, Pavlita, states clearly that he does know the 'secret' and that the secret is based on a new way of looking at geometry.  But that he would not divulge the secret.  Prior to that Pavlita had made a small fortune with patents in the textile industry.  But he never patented his devices.  At first blush odd.  He claimed that his devices were of two types.  One type worked off of what he called 'cosmic' energy.  The Great Pyramid shape, he said, was one of those.  The other type of devices, he stated, must draw energy from a human source.  One device, when one was close enough, and looked at it continually in a certain zig zag pattern etched into the device would produce a small amount of electricity, enough to light a very tiny little bulb.  Did not seem to make sense, but worked over and over.  Another device was producing some sort of mechanical ESP.  It had a needle on top.  A two playing cards could be placed around it.  The needle would move to the playing card thought of.  It would work, 'glass covered', or other interference screening ways, but only over very short distances, say ten feet.  It would, for example, sort blood type with 100 percent accuracy, trial after trial.  Another interesting device produced, at its tip, a sort of magnetic energy that seemed to magnetize any sort of material, even non magnetic.  This could be shown to be no form of static electricity as it also worked just as well under water.  In this class of devices, (the non 'cosmic'), the machine drew its energy from the person operating it.  Then, in a series of transformations, transformations wherein each step was a specific 3-D form built out of a specific material (in most cases metals)..., together, in specific sequence and arrangement these shapes drew a form of biological energy and did work.  Experiments were verified over and over, and all this beyond the present understanding of physics.  There was world wide notice at the top.  Suddenly the Soviet tanks rolled into Prague and little was heard of Pavlita's work.  However, if you google <"Robert Pavlita" and "Leonid Breshnev"> a 'contact' map will come up showing that Robert Pavlita was moved directly into the inner circle of Soviet power.  It is my feeling that although Pavlita knew the value of patents, building his fortune upon them, he knew that what he had in reality of that environment, well.., realistically it would be better and far less problematic to develop the new aspects of physics within the power structure that he found himself, or more specifically was, with heavy tanks, annexed.  That part, at least, is fairly clear.
    What is less clear is exactly what was done with the discoveries within the Soviet power structure.  The following is conjecture, things I have pieced together from some many sources and some dramatic personal experience.  From what I gathered it seemed that some of his work was experimented with lightly in some known 'spy centers' such as Rostov On Don, more to mislead, while the deeper material was worked on in Leningrad, a city used for the same reason New York was chosen for the Manhattan project, easier to cover goings and comings of key people.  Classical music was used as a cover for some of the projects.  A demonstration of sorts was hinted at in warnings that came directly from Chernenko sometime in 83.  In March of 84 there were unannounced naval maneuvers in conjunction with those demonstrations.
    A few years ago, the morning of the Christening of the new nuclear carrier, the U.S.S. Ronald Reagan, (still in trials), there was a meeting, open to the public but unadvertised, in Hampton a few hours before the afternoon Newport ceremonies.  I was told that attendance would probably be limited to some 'retired CIA types who were bored with golf'.  The speakers were Ramsey Clark, Edwin Meese, Jean Kirkpatrick, and (gee, who is that guy?)..., over in the corner...., oh, yes, 'former naval secretary' John Lehman.  All heavy weights of the Reagan administration.  It was telling to note that every single speaker, all four just mentioned including the MC, all of them mentioned Chernenko's warning of 1983 as the big turning point of modern history.  How many members of this Polytopia list would see that?  But with this there was an intriguing joke told by the master of ceremonies at the start of the talk. It was a mathematics joke. When he told the joke members of the panel snickered knowingly.  There might have been a few unknowing automatic-response half laughs from the audience.  Some blank stairs.  Perhaps a confused look or two.  I was jaw dropped.  And then the 'math' joke quickly passed through the few knowing snickers and on to other subjects.  Don't get too worried about falling off your chairs or slapping your thighs, I don't think you will.  Here is the joke..:  Coming into the Williamsburg Airport the master-of-ceremonies meets a Lithuanian.  The Lithuanian asked the master-of-ceremonies what brought him to the Tidewater area.  The master-of-ceremonies said that he came for the Christening of the U.S.S. Ronald Reagan.  The Lithuanian says, "Reagan, the greatest of Presidents !  He should be placed on Mount Rushmore".  The master-of-ceremonies says, "I agree, but there is not enough room".  The Lithuanian says, "No problem, we now have a new way of measuring."  Perplexed?  This panel was not.  They all laughed with twinkle snickering glee.  These were the men who lived through the tactical application and strategic implication of a change in geometry in 1984.  A scary change.  Far scarier in the light of the sinking of the H.M.S. Sheffield in 80.  Though it has been stated that the Argentine plane launched Exocit was at a distance of a mere three knots, (the maps indicate nine knots if an inch), it still dramatized the power of possible sub launch.  And what if communication to subs on one 'side' was suddenly a thousand times better than the other? 
    "But I am a bright and aware person", you say. "Certainly I would know about such goings on."  Would a very bright and astute person, not just in the United States or Germany but even in Sweden, know the importance of what was going on with heavy water in 1943 or 44?  Most probably not.  But in 1946 almost all would know.  And when was the theoretical work done for that?  Relativity was forty years prior, 1905.  Discoveries are not simply, "Eureka"..., bamb...thank you, history."  It is usually a slow process.
    In the very late 1970s and a tad into the early 1980s Charlie Rose, Democrat, North Carolina, on the House Intelligence Committee was screaming for knowledge of what Pavlita was into, (Knight Ridder).  The Soviets started making threats, an exquisitely planned sequence of them.   A sudden rumor that the British H.M.S. Intrepid would be sunk with Prince Andrew in it was enough to have Alexander Haig drop everything and go darting off to London.  It may have been a stretch, but the thinking was that 1- if Pavlita had control of biological energy that could perform (very short distance to be sure) bizarre 'ESP' tricks, and 2- if it was true that rabbits would react at a distance to the killing of their young, and 3- if this 'rabbit reaction' seemed to go through water, (rumored that this was performed with either mother or young submerged in a submarine...,) then was it possible that Pavlita's discovery could lead to some sort of underwater communication to land?  This is no small problem to the navies wherein a sub must surface to send a radio message.  Who knows?  But I do know that prior to the March 1984 'sea change' in the Straits of Denmark there was some intense interest at both Underwater Systems Command, Groton, and the Office of Naval Research in Arlington with Senator Moynihan, Senate Intel co-chair, directly involved.  That being so you might be a tad confused to see Konstantin Chernenko so 'off-to-the-side' in the <"Robert Pavlita" and "Leonid Breshnev"> google.  If Pavlita and Breshnev are central to this inner Soviet military/science map, why is Chernenko, the real mind behind Breshnev, and in command after Breshnev during the major action, seemingly so far (off to the top left) out of the loop?  The evidence suggests that great care was taken to guard the real 'inner circle' and this confirmed by letting (even coaxing) of part of the false inner circle to defect.  Chernenko was a true mystic, and like T.E. Lawrence, oddly placed as a commander or 'warrior'.  He was even almost an embarrassment in public, being apt to, on occasion, stare off into space and start drooling down his chin during a speech.
    From 83/84 on all discussion of or reference to Pavlita in major media very abruptly stopped.  And this with the House Intelligence Committee screaming into low hell and above only a year or so before.  Nothing.   Nada.  Zero. Ziltch. Zip.
    The public mind is trained to think of East and West navies during the Cold War as in continual dangerous 'cat and mouse' games.  The powers-that-be therefore loved such novels as Hunt For Red October that would promote that illusion.  Truth is, from the perspective of the top, those cat-and-mouse games were only allowed in certain well defined 'play pens'.  This was set up to keep that illusion on one hand and on the other help insure that war did not accidentally break out at sea, in the nuclear age, far too dangerous to even think of.  Therefore warnings of maneuvers were always given.  Not to do so was considered most dangerous, ultimately dangerous, and against the deepest, albeit secret, protocol.  Following the aforementioned warnings that is exactly what Chernenko did.  When the sleek Leningrad and consorts steamed surprisingly up the US Eastern Seaboard from Havana instead of into the Caribbean, the West was far more alarmed than, for example, any time during the Cuban Missile Crisis.  But this was never to enter the public mind.
   Many in the navy remember that March of 84 and the three battle groups that were put to sea from Mermansk and the Baltic following the Leningrad battle groups surprising move.  But few in our society would know that the underlying issue to these events were reasoned changes in mathematics.
    How history repeats !
    By keeping the calculus application well guarded within the Royal Marines the British Empire increased dramatically during that portion of the Age Of Sail.  Newton's use of Descartes was 1720.  The War of 1812 nearly 100 years later.  Had the Brits not been so greedy in their expansion, and not press-gang so many American seaman, perhaps they would have held that secret longer.  But press-gang our seamen in great numbers they did, and breaking their own 'only Royal Marine' rule, trained more than a few as gunners.  One, who escaped, was from a family who was friends of Madison.  With actions of the H.M.S. Nimrod and such the Brits still felt invincible.  It was application of calculus, not just the armor of iron-hard, good Georgian oak, by which the U.S.S. Constitution sank the H.M.S. Gerrier  in action off of Newfoundland which altered the balance of power.  (Granted that long range ordnance was not used in the engagement itself, but certainly a factor in the minds of both commanders.)  Again the point being that too few take note of the role played by a change in our mathematical thinking in that 'sea change' as well.
    My interest in crop circles came late into my interest in the new geometry.  But even in gross inspection of the pictorials I could see that one 'thread' of the message seemed to point to this 'new' or 'Pavlita' geometry. One ancient cosmological map, the cabala, now shown to be a watered down version of the new 3-D geometry, was a crop circle by itself.  That the diatonic scale was involved enough that it could not have appeared merely by chance was more than enough for me to make the connection.
    You teach, John, mathematics.  And your students are lucky to have such a fine mind instructing theirs.  For a moment, John, assume that I might have, in fantastic cybermist, visited your classroom of late.  It absolutely amazed me, but guess who I saw?  Maureen O'Tool and Kathleen O'Shaughnessy !  I have not been in a math class in more decades than I care to mention.  How seeming strange to not only see Maureen and Kathleen again, but to see them sitting in the back of your class in the same place and in the same manner as so many years ago, behind the backs of larger boys so you could not see them passing notes.  There are strong indications that this new geometry will shed some scientific clarity into what has been spoken of in common language as 'emotional chemistry'.  What if there was, based on a new geometry, a real scientific 'chemistry' of emotion?  Note that the notes too are still the same.  "Do you like him?" "Do you think he likes you?"  John, could you possibly imagine a time where Ms O'Tool and Ms O'Shaughnessy would not only sit up front in rapt attention to the study of geometry, but ask for further readings and be following you down the hall after class with further questions?  I can.
    Nothing so alters our history as science.  Nothing so alters our science as math.
    If I wax wordy on the wages of math it is that I glimpse now as much the terrible, as had been experienced by the crew of the Gerrier as shells exploded and burning men ran in screams for a preferred death in the cold waters, as the sublime.  And at our world doorstep, pounding like heaven in uncanny bent crop, a far greater change still.  My point in this preliminary note is not to go deep into the particular question of design versus chance.  But wish to hint at some broad strokes with a few of your examples.  Yes, as you say, (quote from below), "...Hold up some fingers on each hand - the ratio between the hands is diatonic..."  Absolutely.  Just note that the ratios obtained are done with areas that seem to exactly hit whole number ratios, a ratio of a large to small circle being, as example, 3, not 2.76 or 3.2, and even in that start, prior to diatonic, statistically uncanny.  And to follow what if these ratios are only the diatonic?
    John, you also note, (quote) "...A honeycomb tiling by bees contains all of the diatonic ratios...." Here you hit deeper.  A kaleidoscope of some glass pieces would give unending symmetrical circular patterns.  The honeycomb polygon is six sided.  Just as six balls fit around one on a surface.  One man, who could not get 'spheres' out of his thinking, began to approach this geometry in the past.  That was Kepler.  He began to come very close with a letter to a friend, Midnight Thoughts On The Six Cornered Snowflake.  The message aspect, some of the exactness of it, becomes clearer when you begin to understand what is attempting to be conveyed, and you place yourself in the position of the circle authors.  If you are hinting at a geometry then not known, or not well known, to the audience you are aiming toward, then a simple reproduction of geometric patterns that are deeply reflective of nature would not be near enough.  The audience would be seeing the 'same ol' nature with the 'same ol' geometry.  This is one reason why the crop circles will start a sequence of circle designs in geometric patterns, but intersperse these with insect like constructions, or other more obviously 'human contrived' forms to show this is just not 'random kaleidoscoping'.
    To quote again from Stephen Jay Gould's Questioning The Millennium above, there are vast amounts of "...terminal nuttiness as to profound insight..."  But meaning, also, that there are, now and then, a few pieces of candy in the vastness of ca ca.  Sifting the candy from the ca ca can be such a drudge that the ca ca begins to blur the eyes.  This is real candy, John.  Take another look.
    Sincerely,
    Michael Donovan
    39 Megunticook Street
    Camden, ME 03843   
    Website...:  The Geometry Of Robert Pavlita  www.midcoast.com/~michael1
****************************************************************
----- Original Message -----
Sent: Saturday, November 06, 2004 10:53 PM
Subject: Re: [Polytopia] back to the weird 2

Hey Michael,
 
I recall our discussion. I would be happy to say that Professor Hawkins has come up with new theorems, but there are thousands of new theorems found every year. I could write down 20 such theorems with no difficulty. The problem is that these particular theorems don't seem to have much significance to mathematics.
 
There is a bit of interest in the design of the crop circles, as in any designs. Were you to collect logos of companies, you could also find many mathematical relationships like symmetry in them. You could study them and find similar "theorems."
 
Concerning the question of being "diatonic:" Pythagorus noticed that notes which have frequencies in a ratio of small natural numbers sound good together. "Diatonic" refers to a musical scale - which in the old days had all the notes in ratios of small natural numbers. (Nowadays we use the equitempered scale based on irrational numbers... we lose the exact ratios, but gain the ability to play in all the different keys.) The diatonic scale uses 2, 3, and 5 and their multiples. One will note that 7 is the first number which is left out in this scheme. It would not be surprising if there are "diatonic" ratios in crop circles, company logos, or other created designs - these are the most common small numbers. If there are any crop circles that have ratios of 1:7 or 1:11 or 1:13 then we can say that these are not diatonic. Those interested could investigate.
 
Of course the claim that crop circles go with music doesn't have to stop with diatonic scales - just because western music is based on this scale, there are many other scales. The Yahoo "Tuning" group goes into loving detail about all the different ways to include 7, 11, 13 and more in scales ranging from 2 to thousands of notes per octave. By this method whatever ratios showed up anywhere - it could be matched to music.
 
The idea that since you can find certain ratios in a design, that some musical connection can be made seems silly to me. Take a yin-yang symbol. The ratio of the inner curve radius to the outer curve radius is 1:2 - a diatonic ratio. A honeycomb tiling by bees contains all of the diatonic ratios (as well as the 7, 11, 13...) The height and width of a TV screen are in a diatonic ratio. The size of an inch to a foot is a diatonic ratio. If you ask people to pick out two random numbers from 1 to 10, most of the time they will pick a diatonic ratio. Hold up some fingers on each hand - the ratio between the hands is diatonic. The ratio of how many teeth I have to how many leg bones I have is diatonic. Hopefully these examples show that diatonic ratios are everywhere - no need to get excited if they show up somewhere.
 
John Berglund

Michael Donovan <michael1@...> wrote:
John,
    I had both a feeling and a deep hope you would appear out of the cybermist for this.  And I am very grateful that you have.  If you remember the general question of if Professor Gerald Hawkins did or did not come up with new theorems had come up on this list. At that time you pointed out that Hawkins theorem was covered by a very easily provable attribute to all regular polygons.  Perhaps I should dig that discussion up out of the files.  I acknowledged that without more information it would seem that your argument was the better.  However, I could not get further information from the web site who has stated that they have all of Hawkins work.  To some extent the issue is therefore still open.
    The work below is by Dee Gragg.  I am still separating the issues which I feel are unfortunately mixed together.  One issue is if or not Hawkins has found previously unknown Euclidean theorems.  And the other issue is if or not there are diatonic ratios in the crop circles.  And the issue of if or not there are diatonic issues in the crop circles is completely separate from the controversy of if or not the crop circles are 'real'.  In this case 'real' meaning that they are made by some unknown force, not 'hoaxers'.  In fact, when Hawkins was first asked to investigate the attitude that he took was purely mathematical, to look at the 'mind' of the circle makers whoever they were.  It was from that attitude that the observations came that whoever was doing this, either 'hoaxers' or unknown minds, they had the odd character of using not just diatonic scales, but diatonic scales that were more known to classic, not modern , history.
    Now Dee Gragg states that the issues are mixed.  Let me quote Dee...
     (quote) "  Now as to the Musical notes encoded in the circles.
They are not unrelated to the theorems.  In fact, all
four of Dr. Hawkins theorems are related to musical
notes.  Alas, I found that only two of mine were; F
below middle C and F two octaves below Middle C. ..." (unquote)
    Dee is right.  They are not unrelated.  But in this case they can still be separated and treated separate.  I advised that because of your observation, John.  But perhaps, with your astute help, the issue of if or not there are new Euclidean theorems can be resolved.
    I am passing this to Dee Gragg.  I am suggesting that she join the Polytopia list so that I am not so much in the middle of this.
    Michael Donovan
    Camden, ME.
----- Original Message -----
Sent: Friday, November 05, 2004 11:33 AM
Subject: Re: [Polytopia] back to the weird 2

There are some interesting relationships in shapes. Theorems 1A to 1D can be seen in the following picture. (Theorem 1C requires that you know that a 30-60-90 degree triangle has the edge ratios of 1:2:sqrt(3)... the others can be seen just by counting distances.) Were we to place another circle inside the three given circles, just tangent to them, its radius would be in a ratio of 1 to 3 with the original circles. There are numerous other relationships that we could point out. I don't think of these relationships as having any special meaning aside from being nice numbers.
 
John Berglund
 


Michael Donovan <michael1@...> wrote:

Crop Circle Theorems
 
Their Proofs and Relationship to Musical Notes
 

   This research began with a simple and rather limited objective: to prove the crop circle theorems of Dr. Gerald Hawkins.  In fact if I could have found the proofs in the literature of the field, this research would never have taken place at all.  Fortunately, I couldn’t find them because once I started, I could see that further work that needed to be done.
 
   As I proved Dr. Hawkins theorems, I discovered five new ones and proved them as well.  I then took the diatonic ratios of all the theorems and related them to the frequencies of the musical scale.  With some rather startling results I might add.
 
   Beginning with Theorem IA I need to make some observations that apply to all of the theorems.  In Euclidean Geometry one almost always has to see the end before making a beginning.   Also, since we are looking for diatonic ratios, we need to find an equation or equations which will let us divide one diameter or radius by the other.  Remember too, that because we are working with ratios, the constants divide out leaving diameter ratios equal to radius ratios.  And if we square them they are equal to each other and to the ratio of the areas.
 
   Applying this to Theorem IA the equation we need to write is for the diameter of the circumscribing circle.  It contains both the radii of the initial and the circumscribed circle.  So from the equation we are able to divide it and find the diatonic ratio of 4 to 3. 
 
   Although, I have proved three more Theorem I’s, I believe this is the one Dr Hawkins meant when he said Theorem I.  See Circular Relationships for The Theorems in Appendix A.  I base this belief on the 4 to 3 diatonic relationship which is related directly to Note F above Middle C.  See Frequencies In The Fields in Appendix B.
 
   Theorem IB is like Theorem IA except that the equilateral triangle is inscribed rather than being circumscribed.  It can be proved by Theorem IA and Theorem II.  The equations already exist so just divide them for the proof.  This gives a new diatonic ratio which is also the Note F, one octave lower than the previous.
This theorem is such a simple and logical extension of the first two that I am puzzled as to why Dr Hawkins did not discover and publish it.
 
   Theorem IC is also often referred to as Theorem I although it is quite different.  Sometimes both Theorem IA and Theorem IC appear in the same article as if they were identical.  They aren’t.  The proof of Theorem IC shows that it contains no diatonic ratio that can be related to a musical frequency.  I believe that this was not the theorem Dr. Hawkins was referring to when he said Theorem I.  In my mind the origin of Theorem IC is rather murky.
 
      Theorem ID would have never been discovered if I had read the instructions for constructing Theorem IA a little closer.  Instead of circumscribing the equilateral triangle, I circumscribed the three circles and then proved the theorem before realizing my mistake.  It has a nice 7 to 3 relationship but it would need to be 8 to 3 to be Note F in the next higher octave.
 
   Theorem II is easy to prove by constructing the appropriate similar triangles and remembering their relationships.  It may be proved a number of different ways I have shown just one of them.  It has the nice diatonic ratio of 4 to 1 which relates directly to the Note C which is two octaves above Middle C.
 
   Theorem III is the simplest of all proofs.  Just remember the Pythagorean Theorem. It also has a nice diatonic ratio of 2 to 1 which relates directly to the Note C which is an octave above Middle C.
 
   Again using the Pythagorean Theorem, Theorem IVA is shown to have a nice 4 to 3 relationship.  We have previously related this to Note F using Theorem IA.
 
   While proving Theorem II, it occurred to me that there should be a similar theorem related to the hexagon. There was and that led me to discover Theorem IVB by connecting the diameters at the hexagon corners.  Again by using similar triangles and writing and dividing the proper equations it is shown to have a diatonic ratio of 1 to 3 which relates directly to the Note F.  This Note F is yet another octave lower.
 
   I have included Theorem IVC mostly for completeness as it does not have a diatonic ratio which can be related to a specific note.  If I hadn’t included it you might have wondered why since it can be proved by simply dividing Theorem IVB by Theorem IVA. 
 
   There is a Theorem V which can be used for deriving (not proving) the other theorems.  However it does not of its self have diatonic ratios and therefore was not a part of this research.
 
      Appendix A Circular Relationships for The Theorems shows a summary of all the results.  Note that to go from one column to the other, you simply square or take the square root.  But how do you know which column to use?  I have followed the lead of Dr. Hawkins in that if the circles are not concentric, you use the ratio of diameters, if they are concentric you use the ratio of areas.  This means diameters for Theorems IA, IB, IC, ID, IVB, and IVC and areas for Theorems II, III, and IVA.  Why did he pick this convention?  Certainly I don’t know, perhaps he was a practical man and he did it because it works.
 
   Frequencies In The Fields in Appendix B gives four octaves: two above and two below Middle C.   This does not encompass the full 27.5 to 4,186 Hz of a piano but does include all the frequencies found so far.  Notice that all the notes are either F or C.  Coincidence or a message?  Perhaps as we discover more notes, this will become clear.
 
   Theorem T in Appendix C is not really a part of this research, but is included as help for anyone wanting to compute circle and regular polygon ratios.  It includes all cases and relies on trigonometry rather than Euclidian Geometry.
 
   Finally, if you’re wondering about me, I have a Short Bio in Appendix D.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Copyright 2004 by C. D. Gragg. All rights reserved
 
Theorem IA
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is circumscribed about the triangle, the ratio of the diameters is 4 to 3.
              
 
 
 
 
 
Theorem IB
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is inscribed within the triangle, the ratio of the diameters is  2:3.
 
              
 

Theorem IC
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is constructed using the single circle as a center and drawing the circle through the other two centers, the ratio of the diameters is 4 to Sqrt 3.
              
 

Theorem ID
   If three equal circles are tangent to a common line and their centers can be connected by an equilateral triangle and a circle is constructed circumscribing the three circles, the ratio of the diameters is 7 to 3.
              
 
 
 
 
 
Theorem II
   If an equilateral triangle is inscribed and circumscribed the ratio of the circles’ areas is 4:1.
              
 
 
 
 
 
 
 

Theorem III
   If a square is inscribed and circumscribed the ratio of the circles’ areas is 2:1.
               
 
 
 
 
 
 
 
 
 
Theorem IVA
   If a hexagon is inscribed and circumscribed the ratio of the circles’ areas is 4:3.
              
 
 
 
 
 
 
 
 
 
Theorem IVB
      If a hexagon is inscribed and circumscribed and the corners connected by diameters, the inscribed circles of the created equilateral triangles have a diameter ratio to the inscribed circle of 1:3.
              
 
 
 
 
 
 
 
Theorem IVC
      If a hexagon is inscribed and circumscribed and the corners connected by diameters, the inscribed circles of the created equilateral triangles have a diameter ratio to the circumscribed circle of 1:2Sqrt3.
              
 
 
 
 
 
 
 
Appendix A
 
                          Circular Relationships for
                                     The Theorems
 Theorem Ratio of Diameters and
Radii Ratio of Areas, Diameters Squared, and Radii Squared
 Theorem IA 4:3 16:9
 Theorem IB  2:3  4:9
 Theorem IC 4:Sqrt3 16:3
 Theorem ID 7:3 49:9
 Theorem II 2:1 4:1
 Theorem III Sqrt2:1 2:1
 Theorem IVA 2:Sqrt3 4:3
 Theorem IVB 1:3 1:9
 Theorem IVC 1:2 Sqrt3 1:12
  
  
  
  
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Copyright 2004 by C. D. Gragg, All rights reserved
 
 
 
 
 
Appendix B
 
Frequencies In The Fields
 
Note Name C D E F G A B C
Diatonic Ratio 1/4 9/32 5/16 1/3 3/8 5/12 15/32 1/2
Frequency (Hz) 66 74.25 82.5 88 99 110 123.75 132
 
Note Name C D E F G A B C
Diatonic Ratio 1/2 9/16 5/8 2/3 3/4 5/6 15/16 1
Frequency (Hz) 132 148.5 165 176 198 220 247.5 264
 
Note Name C* D E F G A B C
Diatonic Ratio 1 9/8 5/4 4/3 3/2 5/3 15/8 2
Frequency (Hz) 264 297 330 352 396 440 495 528
 
Note Name C D E F G A B C
Diatonic Ratio 2 9/4 5/2 8/3 3 10/3 15/4 4
Frequency (Hz) 528 594 660 704 792 880 990 1056
* Middle C
 Denotes  found in the fields
 
Theorem Summary
 
Frequency (Hz) Theorem Used For Proof
88 Theorem IVB, Gragg
176 Theorem IB, Gragg
352 Theorem IA, Theorem IVA, Hawkins
528 Theorem III, Hawkins
1056 Theorem II, Hawkins
 

 
 
 
 
 
 
 
 
 
 
                    Copyright 2004 by C. D. Gragg, All rights reserved
Appendix C
Theorem T
 
   Trigonometry can be used to solve circular relationships for inscribed and circumscribed regular  polygons for polygons of any number of sides from 3 to infinity.
 
 Proof:                                      
                                                  Where:  α = 3600          and n = number of sides    
                                                  2n
                                                          
                                                                 cos α  = R1 
                                                              R2
 
                                                                    R2 =   _1     Proving the Theorem
Figure 1. Regular polygon with                  R1       cos α
               any number of sides                              
                                                           Further:   ( R2)2 =  (1)2    
                                                                                                  ( R1)2      ( cos α)2
Table of Some Common Polygons
Figure (All are equiangular) Number of Equal Sides  Ratio of Diameters and
Radii Ratio of Areas, Diameters Squared, and Radii Squared
Triangle (1)(4)         3       2.000 4      4.000
Square (2)         4       1.414 2      2.000
Pentagon         5       1.236          1.527
Hexagon (3)          6       1.155 4/3   1.333
Heptagon         7       1.110         1.232
Octagon (4)         8       1.082         1.172
Nonagon         9       1.064         1.132
Decagon       10     & amp; nbsp; 1.051         1.106
       15       1.022         1.045
       20       1.012         1.025
       50       1.002         1.004
     100       1.000         1.001
     200       1.000         1.000
       ∞       1.000         1.000
(1) Theorem II, by Dr. Hawkins using Euclidian Geometry
(2) Theorem III, by Dr. Hawkins using Euclidian Geometry 
(3) Theorem IV, by Dr. Hawkins using Euclidian Geometry
(4) Found in the Kekoskee/Mayville, Wisconsin Crop Circle
             Formation July 9, 2003    
                                                                           Copyright 2004 by C. D. Gragg, All rights reserved
 
 
 
 
 
 
 

Appendix D
 
Short Bio
 
   My name is Dee Gragg.  I am a retired, mechanical engineer.  My career was spent in research, testing and evaluation. My main areas of research were automotive air bags, jet aircraft ejection seats and high speed rocket sleds.  I have published 33 technical papers as either the principal author or a co-author.  They form a part of the body of literature in their respective fields.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 


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#755 From: John Berglund <anisohedral@...>
Date: Fri Nov 12, 2004 3:37 am
Subject: Re: back to the weird 2
anisohedral
Send Email Send Email
 
There are advances being made in mathematics all the time. A few that have happened in our lifetimes are the solution of Fermat's last Theorem, and the discovery of surreal numbers. Historically, most advances in math are made by people who are well versed in the math discovered up to that point. Eg. Newton, Gauss, etc. It is very rare to have someone who will create a completely new paradigm for math who is not fluent with the previous work. (Granted some discoveries can be made by amateurs - for example some tilings found by a Mrs. Rice contrary to professional mathematicians beliefs that such tilings couldn't exist. This is the exception rather than the rule.)
 
Most of the material presented seems to fit in the current mathematical world that I have - not requiring any extras. (The sphere packing example fits in normal geometry very nicely.) The quote about body sizes of animals is interesting. I can't be certain of an explanation, but I could advance a possibility. The sizes of a carnivore and it's prey are often correlated. A wolf could not live off fleas. If a herbivore is enough bigger than a carnivore, the carnivore won't be able to kill and eat it. Elephants are safe due to their size. Thus by natural selection, you might get certain sizes left as the others were consumed.
 
I am dubious of most of the paranormal claims. It would be easy to set up a double blind test to see if pyramids really can sharpen razor blades. If you believe that this is possible, please send me the right type of razor blade and instructions on how the blades must be oriented to become sharper. I will use half the blades as controls, and place the other half in pyramids. I will label each razor blade with a letter and record what treatment each letter razor blade received - even recording the results with my digital camera. Then I will return all the blades to you. You will test the blades and sort them according to whether they are sharper or not. When you have all of your data collected, we will trade data. To ensure that neither of us is cheating, we can send the data as a password protected file, and then trade passwords after both files are received. I am confident that this test would not work. However, I have heard some people claim that my very unbelief would be the cause of the failure. I can accept that - I have enough unbelief to stop this from happening anywhere in my universe. :) If you really want to test it, let me know.
 
John Berglund
 


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#756 From: "Michael Donovan" <michael1@...>
Date: Fri Nov 12, 2004 4:24 am
Subject: Re: back to the weird 2
jayennseth
Send Email Send Email
 
John,
    The testing of the effects of the Great Pyramid shape have been done by the Czech government, Flannagan, and many others.  The razor blades must be those of the older, high iron content, type.  Again, that old classic, Psychic Discoveries Behind The Iron Curtain, is a great source.
    The real issue of the moment concerns if or not the observations in Dee Gragg's paper concerning the ratios in the crop circles are something to expect, as you say even in company logos, or is there something very non random going on.  The initial attitude that Professor Hawkins used to get into the problem was to disregard how the crop circles were made. He was to look at the geometry itself and determine just from that what might be on the mind, or in the minds, of the crop circle creators.  I would agree with you that claims of 'additional Euclidean theorems mean little.  You can create additional proofs in Euclidean geometry that cover classes of cases, and by that call it an additional theorem.  If or not to call it a 'missing theorem that should have been included is very arbitrary.  I am leaving that aspect alone.  I think it far more important that the ratios be looked at.
    Your comparison to fingers of two hands and the diatonic is a great starting point.  As you said, in any one trial you would most likely get a diatonic.  And there I agree, so what?  But areas of circles are not discrete things such as fingers.  That such numerous diatonic ratios would appear in the same formations, without non diatonic where in any one of them the areas could be off..., that is highly statistically improbable by chance.  And that is the beginning, the very begging, of an uncanny string.
    The geometry that I am describing is akin to spherepacking, but not.  Because here you must assume that the balls are always in motion and always vibrating. Because of this the central form is one that 'spherepacking' chose not to look at much.  Again there are two ways in which 12 balls will fit around one.  And the central form would not fit inside a pile of cannon balls.  It would always need be on an edge, an interface.  It can only remain central in movement.  The central shape, (and it is a shape that relates to the Great Pyramid slopes, is one that the spherepackers threw out as having less meaning.
    But again, the central thing to look at here is if or not Dee Gragg's ratios could be expected to appear that way from random design or is that deliberate.  That is the question of the moment.
Michael
----- Original Message -----
Sent: Thursday, November 11, 2004 10:37 PM
Subject: Re: [Polytopia] back to the weird 2

There are advances being made in mathematics all the time. A few that have happened in our lifetimes are the solution of Fermat's last Theorem, and the discovery of surreal numbers. Historically, most advances in math are made by people who are well versed in the math discovered up to that point. Eg. Newton, Gauss, etc. It is very rare to have someone who will create a completely new paradigm for math who is not fluent with the previous work. (Granted some discoveries can be made by amateurs - for example some tilings found by a Mrs. Rice contrary to professional mathematicians beliefs that such tilings couldn't exist. This is the exception rather than the rule.)
 
Most of the material presented seems to fit in the current mathematical world that I have - not requiring any extras. (The sphere packing example fits in normal geometry very nicely.) The quote about body sizes of animals is interesting. I can't be certain of an explanation, but I could advance a possibility. The sizes of a carnivore and it's prey are often correlated. A wolf could not live off fleas. If a herbivore is enough bigger than a carnivore, the carnivore won't be able to kill and eat it. Elephants are safe due to their size. Thus by natural selection, you might get certain sizes left as the others were consumed.
 
I am dubious of most of the paranormal claims. It would be easy to set up a double blind test to see if pyramids really can sharpen razor blades. If you believe that this is possible, please send me the right type of razor blade and instructions on how the blades must be oriented to become sharper. I will use half the blades as controls, and place the other half in pyramids. I will label each razor blade with a letter and record what treatment each letter razor blade received - even recording the results with my digital camera. Then I will return all the blades to you. You will test the blades and sort them according to whether they are sharper or not. When you have all of your data collected, we will trade data. To ensure that neither of us is cheating, we can send the data as a password protected file, and then trade passwords after both files are received. I am confident that this test would not work. However, I have heard some people claim that my very unbelief  would be the cause of the failure. I can accept that - I have enough unbelief to stop this from happening anywhere in my universe. :) If you really want to test it, let me know.
 
John Berglund
 


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#757 From: John Berglund <anisohedral@...>
Date: Fri Nov 12, 2004 11:46 am
Subject: Re: back to the weird 2
anisohedral
Send Email Send Email
 
Hi Michael,
 
>The testing of the effects of the Great Pyramid shape have been done by the Czech government, Flannagan, and many others.  The razor blades must be those of the older, high iron content, type.  Again, that old classic, Psychic Discoveries Behind The Iron Curtain, is a great source.
 
I'm not aware of any pyramid test by a group that I trust. The sources given do not present double blind tests or give their statistics and methods of research. This is such a large jump from the current beliefs of science that I would want to see many tests done independently. The tests should be announced before taking place and should be videotaped. Again, I would be happy to participate in such a test.
 
The idea that the government has been hiding knowledge of math secrets for years seems silly. The more people who know a secret, the more chance that it will leak out - especially if the people are from varied backgrounds. It is possible that some small group within the government knows some secret... It is also possible that some small group of newspaper boys knows some secret... It is also possible that my cats know some secret and have refrained from talking when around people... I can't disprove any of these statements, but I don't believe in them.
 
>But again, the central thing to look at here is if or not Dee Gragg's ratios could be expected to appear that way from random design or is that deliberate.  That is the question of the moment.

The ratios would not appear in a random design. However, I don't believe that the phrase "random design" has much meaning as design has to do with something being thought out and not random. I think that most people would agree that the crop circles were made by some thinking being. This being the case, we would expect to see small number ratios. Yesterday I went to a math conference. A group of sixth graders had created some math designs. I was not surprised that the ratios were 1:2 or 2:3 or other small number ratios. Earlier, I mentioned that a hexagon tiling as in a honeycomb contains all diatonic ratios. This is not profound. A yardstick contains them too. So does graph paper.
 
Many times we may be misled by coincidences. For example in a random group of 35 people, we find that two share a birthday. This seems like an astonishing coincidence to some. But if you study the probability behind it, it is more likely than not to happen.
 
>The geometry that I am describing is akin to spherepacking, but not.  Because here you must assume that the balls are always in motion and always vibrating. Because of this the central form is one that 'spherepacking' chose not to look at much. 
 
I don't understand this. When you say "central form" are you talking about a 3D shape? Or a sequence of 3D shapes that change over time? I am familiar with the two ways of fitting 12 spheres around one. It seems that you are talking about something else, but I am not sure. Could you explain more?
 
John Berglund
 
 


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#758 From: "Michael Donovan" <michael1@...>
Date: Fri Nov 12, 2004 8:21 pm
Subject: Re: back to the weird 2
jayennseth
Send Email Send Email
 
John,
    It is good to see that you are in such good form.  You are completely "from Missouri" on this wacko 'new age' stuff.  Both high skepticism and a crack mind.  Wonderful.  Perfect.
    I had spent some time posting yesterday more to bring back the attention.  There were great skeptic replies to Dee Graggs paper, but they were introductory.  The big question here is this... Are the diatonic ratios observed in the formation appearing by chance, much as you might find one, as you say, in any company logo, or does this look specific to the design, something that could not occur by chance.  Here again you make the point that many seeming coinsodences are very natural and to be expected.  The number of days in a year are 365, but as you noted, in an average classroom (shown when you work out the permiatations), the odds are very high that two people will have the same birthday.  That is to be expected.  And you have brought forth many fine examples showing how we can be fooled.
    But, again, the big question is this...: regarding Dee Gragg's paper, (now a Polytopia file, tell me if there are any problems accessing it), are the diatonic ratios observed int the formation appearing by chance?  Much, as you say you might even find in some company logo, say the closest one to the start of page 50 in the Yellow Pages?  Or does this look specific to the design, something placed there with deliberation and some intent?  And again I refer to the secific paper posted by Dee Gragg.
    So yes, both your replies and the wonderful apt quote from Stephen Gould, re "...terminal nuttiness...," versus "...profound insight..," give proper introduction to the question.  But the question still is, are we seeiing a series of ratios, (and again specifically to Dee Gragg's paper)
    All the other material was more to give interest to the main question. 
    Michael
----- Original Message -----
Sent: Friday, November 12, 2004 6:46 AM
Subject: Re: [Polytopia] back to the weird 2

Hi Michael,
 
>The testing of the effects of the Great Pyramid shape have been done by the Czech government, Flannagan, and many others.  The razor blades must be those of the older, high iron content, type.  Again, that old classic, Psychic Discoveries Behind The Iron Curtain, is a great source.
 
I'm not aware of any pyramid test by a group that I trust. The sources given do not present double blind tests or give their statistics and methods of research. This is such a large jump from the current beliefs of science that I would want to see many tests done independently. The tests should be announced before taking place and should be videotaped. Again, I would be happy to participate in such a test.
 
The idea that the government has been hiding knowledge of math secrets for years seems silly. The more people who know a secret, the more chance that it will leak out - especially if the people are from varied backgrounds. It is possible that some small group within the government knows some secret... It is also possible that some small group of newspaper boys knows some secret... It is also possible that my cats know some secret and have refrained from talking when around people... I can't disprove any of these statements, but I don't believe in them.
 
>But again, the central thing to look at here is if or not Dee Gragg's ratios could be expected to appear that way from random design or is that deliberate.  That is the question of the moment.

The ratios would not appear in a random design. However, I don't believe that the phrase "random design" has much meaning as design has to do with something being thought out and not random. I think that most people would agree that the crop circles were made by some thinking being. This being the case, we would expect to see small number ratios. Yesterday I went to a math conference. A group of sixth graders had created some math designs. I was not surprised that the ratios were 1:2 or 2:3 or other small number ratios. Earlier, I mentioned that a hexagon tiling as in a honeycomb contains all diatonic ratios. This is not profound. A yardstick contains them too. So does graph paper.
 
Many times we may be misled by coincidences. For example in a random group of 35 people, we find that two share a birthday. This seems like an astonishing coincidence to some. But if you study the probability behind it, it is more likely than not to happen.
 
>The geometry that I am describing is akin to spherepacking, but not.  Because here you must assume that the balls are always in motion and always vibrating. Because of this the central form is one that 'spherepacking' chose not to look at much. 
 
I don't understand this. When you say "central form" are you talking about a 3D shape? Or a sequence of 3D shapes that change over time? I am familiar with the two ways of fitting 12 spheres around one. It seems that you are talking about something else, but I am not sure. Could you explain more?
 
John Berglund
 
 


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#759 From: "Paul Erlich" <paul@...>
Date: Mon Nov 15, 2004 7:44 pm
Subject: Re: 10 or 26 dimesions?
emotionaljou...
Send Email Send Email
 
The current thinking in Physics is that the various string theories
which posited either 10 or 26 dimensions (five such theories in fact)
have all been shown to be aspects/approximations of a unifying "M-
theory", in which there are 11 dimensions (1 time dimension, 3
ordinary space dimensions, 6 curled-up space dimensions, and 1 non-
curled bounded space dimension). Brian Greene's books, taken
together, form a very readable account of this, and much more.



--- In Polytopia@yahoogroups.com, "lrafey" <rafey@g...> wrote:
>
> Indeed, and other related theories exploit (and I use that term
with
> some purpose) this and that number of dimensions. But...we must
> remember that in the middle ages, layers of additional cycles were
> continually being added to current theories in order to explain
> continually observed but otherwise unexplainable phenomena
> unitl...at long last...it all fell apart! We must, therefore, take
> all these dimensional explications with some sense of rational
> consideration.
> LD Rafey
>
> --- In Polytopia@yahoogroups.com, "Alan Michelson"
> <amichelson2002@y...> wrote:
> >
> > Modern String Theory predicts as many as 26 dimensions!
> > http://ascension2000.com/ConvergenceIII/c314.htm
> > http://mathforum.org/mam/00/master/people/kaku/
> >
> > --- In Polytopia@yahoogroups.com, "Larry Rafey" <rafey@g...>
wrote:
> > > Peter...
> >
> > >   Modern String Theory predicts as many as ten (yes, 10)
> dimensions!
> > > These are based on an assumption that six of these are tightly
> > > curled or compact dimensions (sometimes described simply as
being
> > > wrapped up on sox circles but in more complex terms they are
> wrapped
> > > up on mathematical elaborations known as Calabi-You Manifolds
and
> > > Orbifolds). and that we might not be consciously aware of their
> > > existence. These extra degrees of freedom  behave positively by
> > > extending motion into these dimensions which would help to
> explain
> > > some of the otherwise contradictory observations of modern
> Physics.
> > > You might investigate the Kaluza-Klein Theory. A more recent
> take is
> > > a combination of the five predominate string theories, referred
> to
> > > as 'M-Theory.'
> >
> > >   Concerning Black Holes, you might have noted recent
> publications
> > > per Steve Hawking suggesting the possibility that not all
> > > information is destroyed in a black hole and, in fact, may,
under
> > > some circumstances, actually leak out.
> > >
> > > Keep Drawing...I, too, am an amateur artist. Enjoyed your
> website.
> > > LD Rafey
> > >
> > >

#760 From: "Paul Erlich" <paul@...>
Date: Mon Nov 15, 2004 7:58 pm
Subject: Re: back to the weird 2
emotionaljou...
Send Email Send Email
 
The diatonic scale (and the related pentatonic scale, etc.) indeed
represents one of the scale families that proceeds most naturally
from trying to have many intervals which approximate simple ratios
involving the first three primes. There is always approximation
involved because not all the ratios one wishes to be simple can be
exact at the same time -- if you try to make most of them exact, you
end up with 40:27 instead of 3:2 in one place and 32:27 instead of
6:5 in another. Far more commonly, some sort of temperament is used
so all the approximations are a little off but none are quite this
bad.

However, there are other scales that derive about as naturally from
the first three primes. And it is indeed possible to derive the
diatonic scale from the first four primes -- it's just that one's
tolerance for inaccurate approximations has to increase.

My latest paper on the various diatonic-like scale families that
proceed from the first three, as well as first four, primes is going
to be published in the next issue of Xenharmonikon. If anyone wants a
copy of it now, e-mail me your snail-mail address and I'll send it to
you. Half the paper consists of circular diagrams of scale families
which may or may not remind you of crop circles :)







--- In Polytopia@yahoogroups.com, John Berglund <anisohedral@y...>
wrote:
> Hey Michael,
>
> I recall our discussion. I would be happy to say that Professor
Hawkins has come up with new theorems, but there are thousands of new
theorems found every year. I could write down 20 such theorems with
no difficulty. The problem is that these particular theorems don't
seem to have much significance to mathematics.
>
> There is a bit of interest in the design of the crop circles, as in
any designs. Were you to collect logos of companies, you could also
find many mathematical relationships like symmetry in them. You could
study them and find similar "theorems."
>
> Concerning the question of being "diatonic:" Pythagorus noticed
that notes which have frequencies in a ratio of small natural numbers
sound good together. "Diatonic" refers to a musical scale - which in
the old days had all the notes in ratios of small natural numbers.
(Nowadays we use the equitempered scale based on irrational
numbers... we lose the exact ratios, but gain the ability to play in
all the different keys.) The diatonic scale uses 2, 3, and 5 and
their multiples. One will note that 7 is the first number which is
left out in this scheme. It would not be surprising if there
are "diatonic" ratios in crop circles, company logos, or other
created designs - these are the most common small numbers. If there
are any crop circles that have ratios of 1:7 or 1:11 or 1:13 then we
can say that these are not diatonic. Those interested could
investigate.
>
> Of course the claim that crop circles go with music doesn't have to
stop with diatonic scales - just because western music is based on
this scale, there are many other scales. The Yahoo "Tuning" group
goes into loving detail about all the different ways to include 7,
11, 13 and more in scales ranging from 2 to thousands of notes per
octave. By this method whatever ratios showed up anywhere - it could
be matched to music.
>
> The idea that since you can find certain ratios in a design, that
some musical connection can be made seems silly to me. Take a yin-
yang symbol. The ratio of the inner curve radius to the outer curve
radius is 1:2 - a diatonic ratio. A honeycomb tiling by bees contains
all of the diatonic ratios (as well as the 7, 11, 13...) The height
and width of a TV screen are in a diatonic ratio. The size of an inch
to a foot is a diatonic ratio. If you ask people to pick out two
random numbers from 1 to 10, most of the time they will pick a
diatonic ratio. Hold up some fingers on each hand - the ratio between
the hands is diatonic. The ratio of how many teeth I have to how many
leg bones I have is diatonic. Hopefully these examples show that
diatonic ratios are everywhere - no need to get excited if they show
up somewhere.
>
> John Berglund
>
> Michael Donovan <michael1@m...> wrote:
> John,
>     I had both a feeling and a deep hope you would appear out of
the cybermist for this.  And I am very grateful that you have.  If
you remember the general question of if Professor Gerald Hawkins did
or did not come up with new theorems had come up on this list. At
that time you pointed out that Hawkins theorem was covered by a very
easily provable attribute to all regular polygons.  Perhaps I should
dig that discussion up out of the files.  I acknowledged that without
more information it would seem that your argument was the better.
However, I could not get further information from the web site who
has stated that they have all of Hawkins work.  To some extent the
issue is therefore still open.
>     The work below is by Dee Gragg.  I am still separating the
issues which I feel are unfortunately mixed together.  One issue is
if or not Hawkins has found previously unknown Euclidean theorems.
And the other issue is if or not there are diatonic ratios in the
crop circles.  And the issue of if or not there are diatonic issues
in the crop circles is completely separate from the controversy of if
or not the crop circles are 'real'.  In this case 'real' meaning that
they are made by some unknown force, not 'hoaxers'.  In fact, when
Hawkins was first asked to investigate the attitude that he took was
purely mathematical, to look at the 'mind' of the circle makers
whoever they were.  It was from that attitude that the observations
came that whoever was doing this, either 'hoaxers' or unknown minds,
they had the odd character of using not just diatonic scales, but
diatonic scales that were more known to classic, not modern, history.
>     Now Dee Gragg states that the issues are mixed.  Let me quote
Dee...
>      (quote) "  Now as to the Musical notes encoded in the circles.
> They are not unrelated to the theorems.  In fact, all
> four of Dr. Hawkins theorems are related to musical
> notes.  Alas, I found that only two of mine were; F
> below middle C and F two octaves below Middle C. ..." (unquote)
>     Dee is right.  They are not unrelated.  But in this case they
can still be separated and treated separate.  I advised that because
of your observation, John.  But perhaps, with your astute help, the
issue of if or not there are new Euclidean theorems can be resolved.
>     I am passing this to Dee Gragg.  I am suggesting that she join
the Polytopia list so that I am not so much in the middle of this.
>     Michael Donovan
>     Camden, ME.
>
> ----- Original Message -----
> From: John Berglund
> To: Polytopia@yahoogroups.com
> Sent: Friday, November 05, 2004 11:33 AM
> Subject: Re: [Polytopia] back to the weird 2
>
>
> There are some interesting relationships in shapes. Theorems 1A to
1D can be seen in the following picture. (Theorem 1C requires that
you know that a 30-60-90 degree triangle has the edge ratios of
1:2:sqrt(3)... the others can be seen just by counting distances.)
Were we to place another circle inside the three given circles, just
tangent to them, its radius would be in a ratio of 1 to 3 with the
original circles. There are numerous other relationships that we
could point out. I don't think of these relationships as having any
special meaning aside from being nice numbers.
>
> John Berglund
>
>
>
> Michael Donovan <michael1@m...> wrote:
>  Crop Circle Theorems
>
> Their Proofs and Relationship to Musical Notes
>
>
>    This research began with a simple and rather limited objective:
to prove the crop circle theorems of Dr. Gerald Hawkins.  In fact if
I could have found the proofs in the literature of the field, this
research would never have taken place at all.  Fortunately, I
couldn’t find them because once I started, I could see that further
work that needed to be done.
>
>    As I proved Dr. Hawkins theorems, I discovered five new ones and
proved them as well.  I then took the diatonic ratios of all the
theorems and related them to the frequencies of the musical scale.
With some rather startling results I might add.
>
>    Beginning with Theorem IA I need to make some observations that
apply to all of the theorems.  In Euclidean Geometry one almost
always has to see the end before making a beginning.   Also, since we
are looking for diatonic ratios, we need to find an equation or
equations which will let us divide one diameter or radius by the
other.  Remember too, that because we are working with ratios, the
constants divide out leaving diameter ratios equal to radius ratios.
And if we square them they are equal to each other and to the ratio
of the areas.
>
>    Applying this to Theorem IA the equation we need to write is for
the diameter of the circumscribing circle.  It contains both the
radii of the initial and the circumscribed circle.  So from the
equation we are able to divide it and find the diatonic ratio of 4 to
3.
>
>    Although, I have proved three more Theorem I’s, I believe this
is the one Dr Hawkins meant when he said Theorem I.  See Circular
Relationships for The Theorems in Appendix A.  I base this belief on
the 4 to 3 diatonic relationship which is related directly to Note F
above Middle C.  See Frequencies In The Fields in Appendix B.
>
>    Theorem IB is like Theorem IA except that the equilateral
triangle is inscribed rather than being circumscribed.  It can be
proved by Theorem IA and Theorem II.  The equations already exist so
just divide them for the proof.  This gives a new diatonic ratio
which is also the Note F, one octave lower than the previous.
> This theorem is such a simple and logical extension of the first
two that I am puzzled as to why Dr Hawkins did not discover and
publish it.
>
>    Theorem IC is also often referred to as Theorem I although it is
quite different.  Sometimes both Theorem IA and Theorem IC appear in
the same article as if they were identical.  They aren’t.  The
proof of Theorem IC shows that it contains no diatonic ratio that can
be related to a musical frequency.  I believe that this was not the
theorem Dr. Hawkins was referring to when he said Theorem I.  In my
mind the origin of Theorem IC is rather murky.
>
>       Theorem ID would have never been discovered if I had read the
instructions for constructing Theorem IA a little closer.  Instead of
circumscribing the equilateral triangle, I circumscribed the three
circles and then proved the theorem before realizing my mistake.  It
has a nice 7 to 3 relationship but it would need to be 8 to 3 to be
Note F in the next higher octave.
>
>    Theorem II is easy to prove by constructing the appropriate
similar triangles and remembering their relationships.  It may be
proved a number of different ways I have shown just one of them.  It
has the nice diatonic ratio of 4 to 1 which relates directly to the
Note C which is two octaves above Middle C.
>
>    Theorem III is the simplest of all proofs.  Just remember the
Pythagorean Theorem. It also has a nice diatonic ratio of 2 to 1
which relates directly to the Note C which is an octave above Middle
C.
>
>    Again using the Pythagorean Theorem, Theorem IVA is shown to
have a nice 4 to 3 relationship.  We have previously related this to
Note F using Theorem IA.
>
>    While proving Theorem II, it occurred to me that there should be
a similar theorem related to the hexagon. There was and that led me
to discover Theorem IVB by connecting the diameters at the hexagon
corners.  Again by using similar triangles and writing and dividing
the proper equations it is shown to have a diatonic ratio of 1 to 3
which relates directly to the Note F.  This Note F is yet another
octave lower.
>
>    I have included Theorem IVC mostly for completeness as it does
not have a diatonic ratio which can be related to a specific note.
If I hadn’t included it you might have wondered why since it can be
proved by simply dividing Theorem IVB by Theorem IVA.
>
>    There is a Theorem V which can be used for deriving (not
proving) the other theorems.  However it does not of its self have
diatonic ratios and therefore was not a part of this research.
>
>       Appendix A Circular Relationships for The Theorems shows a
summary of all the results.  Note that to go from one column to the
other, you simply square or take the square root.  But how do you
know which column to use?  I have followed the lead of Dr. Hawkins in
that if the circles are not concentric, you use the ratio of
diameters, if they are concentric you use the ratio of areas.  This
means diameters for Theorems IA, IB, IC, ID, IVB, and IVC and areas
for Theorems II, III, and IVA.  Why did he pick this convention?
Certainly I don’t know, perhaps he was a practical man and he did
it because it works.
>
>    Frequencies In The Fields in Appendix B gives four octaves: two
above and two below Middle C.   This does not encompass the full 27.5
to 4,186 Hz of a piano but does include all the frequencies found so
far.  Notice that all the notes are either F or C.  Coincidence or a
message?  Perhaps as we discover more notes, this will become clear.
>
>    Theorem T in Appendix C is not really a part of this research,
but is included as help for anyone wanting to compute circle and
regular polygon ratios.  It includes all cases and relies on
trigonometry rather than Euclidian Geometry.
>
>    Finally, if you’re wondering about me, I have a Short Bio in
Appendix D.
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> Copyright 2004 by C. D. Gragg. All rights reserved
>
> Theorem IA
>    If three equal circles are tangent to a common line and their
centers can be connected by an equilateral triangle and a circle is
circumscribed about the triangle, the ratio of the diameters is 4 to
3.
>
>
>
>
>
>
> Theorem IB
>    If three equal circles are tangent to a common line and their
centers can be connected by an equilateral triangle and a circle is
inscribed within the triangle, the ratio of the diameters is  2:3.
>
>
>
>
> Theorem IC
>    If three equal circles are tangent to a common line and their
centers can be connected by an equilateral triangle and a circle is
constructed using the single circle as a center and drawing the
circle through the other two centers, the ratio of the diameters is 4
to Sqrt 3.
>
>
>
> Theorem ID
>    If three equal circles are tangent to a common line and their
centers can be connected by an equilateral triangle and a circle is
constructed circumscribing the three circles, the ratio of the
diameters is 7 to 3.
>
>
>
>
>
>
> Theorem II
>    If an equilateral triangle is inscribed and circumscribed the
ratio of the circles’ areas is 4:1.
>
>
>
>
>
>
>
>
>
> Theorem III
>    If a square is inscribed and circumscribed the ratio of the
circles’ areas is 2:1.
>
>
>
>
>
>
>
>
>
>
> Theorem IVA
>    If a hexagon is inscribed and circumscribed the ratio of the
circles’ areas is 4:3.
>
>
>
>
>
>
>
>
>
>
> Theorem IVB
>       If a hexagon is inscribed and circumscribed and the corners
connected by diameters, the inscribed circles of the created
equilateral triangles have a diameter ratio to the inscribed circle
of 1:3.
>
>
>
>
>
>
>
>
> Theorem IVC
>       If a hexagon is inscribed and circumscribed and the corners
connected by diameters, the inscribed circles of the created
equilateral triangles have a diameter ratio to the circumscribed
circle of 1:2Sqrt3.
>
>
>
>
>
>
>
>
> Appendix A
>
>                           Circular Relationships for
>                                      The Theorems
>  Theorem Ratio of Diameters and
> Radii Ratio of Areas, Diameters Squared, and Radii Squared
>  Theorem IA 4:3 16:9
>  Theorem IB  2:3  4:9
>  Theorem IC 4:Sqrt3 16:3
>  Theorem ID 7:3 49:9
>  Theorem II 2:1 4:1
>  Theorem III Sqrt2:1 2:1
>  Theorem IVA 2:Sqrt3 4:3
>  Theorem IVB 1:3 1:9
>  Theorem IVC 1:2 Sqrt3 1:12
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> Copyright 2004 by C. D. Gragg, All rights reserved
>
>
>
>
>
> Appendix B
>
> Frequencies In The Fields
>
> Note Name C D E F G A B C
> Diatonic Ratio 1/4 9/32 5/16 1/3 3/8 5/12 15/32 1/2
> Frequency (Hz) 66 74.25 82.5 88 99 110 123.75 132
>
> Note Name C D E F G A B C
> Diatonic Ratio 1/2 9/16 5/8 2/3 3/4 5/6 15/16 1
> Frequency (Hz) 132 148.5 165 176 198 220 247.5 264
>
> Note Name C* D E F G A B C
> Diatonic Ratio 1 9/8 5/4 4/3 3/2 5/3 15/8 2
> Frequency (Hz) 264 297 330 352 396 440 495 528
>
> Note Name C D E F G A B C
> Diatonic Ratio 2 9/4 5/2 8/3 3 10/3 15/4 4
> Frequency (Hz) 528 594 660 704 792 880 990 1056
> * Middle C
>  Denotes  found in the fields
>
> Theorem Summary
>
> Frequency (Hz) Theorem Used For Proof
> 88 Theorem IVB, Gragg
> 176 Theorem IB, Gragg
> 352 Theorem IA, Theorem IVA, Hawkins
> 528 Theorem III, Hawkins
> 1056 Theorem II, Hawkins
>
>
>
>
>
>
>
>
>
>
>
>
>                     Copyright 2004 by C. D. Gragg, All rights
reserved
> Appendix C
> Theorem T
>
>    Trigonometry can be used to solve circular relationships for
inscribed and circumscribed regular  polygons for polygons of any
number of sides from 3 to infinity.
>
>  Proof:
>                                                   Where:  α =
3600          and n = number of sides
>                                                   2n
>
>
cos α  = R1
>                                                               R2
>
>
R2 =   _1     Proving the Theorem
> Figure 1. Regular polygon with                  R1       cos α
>                any number of sides
>
Further:   ( R2)2 =  (1)2
>
                               ( R1)2      ( cos α)2
> Table of Some Common Polygons
> Figure (All are equiangular) Number of Equal Sides  Ratio of
Diameters and
> Radii Ratio of Areas, Diameters Squared, and Radii Squared
> Triangle (1)(4)         3       2.000 4      4.000
> Square (2)         4       1.414 2      2.000
> Pentagon         5       1.236          1.527
> Hexagon (3)          6       1.155 4/3   1.333
> Heptagon         7       1.110         1.232
> Octagon (4)         8       1.082         1.172
> Nonagon         9       1.064         1.132
> Decagon       10     & nbsp; 1.051         1.106
>        15       1.022         1.045
>        20       1.012         1.025
>        50       1.002         1.004
>      100       1.000         1.001
>      200       1.000         1.000
>        ∞       1.000         1.000
> (1) Theorem II, by Dr. Hawkins using Euclidian Geometry
> (2) Theorem III, by Dr. Hawkins using Euclidian Geometry
> (3) Theorem IV, by Dr. Hawkins using Euclidian Geometry
> (4) Found in the Kekoskee/Mayville, Wisconsin Crop Circle
>              Formation July 9, 2003
>
        Copyright 2004 by C. D. Gragg, All rights reserved
>
>
>
>
>
>
>
>
> Appendix D
>
> Short Bio
>
>    My name is Dee Gragg.  I am a retired, mechanical engineer.  My
career was spent in research, testing and evaluation. My main areas
of research were automotive air bags, jet aircraft ejection seats and
high speed rocket sleds.  I have published 33 technical papers as
either the principal author or a co-author.  They form a part of the
body of literature in their respective fields.
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> ---------------------------------
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#761 From: Peter Boos <peter_boos@...>
Date: Mon Nov 15, 2004 10:40 pm
Subject: Just wondering
peter_boos
Send Email Send Email
 
Just wondering.
 
I was watching a TV program about an indian nuclearphysic proffesor.
He was talking about duality of quantum mechanics, photons being wave AND particle.
His viewpoint was indifferent with western science. We always want to have one truth as one answer. While because of their believe he thinks such anomalies just exist as two answers at the same time. He explained quantum uncertainty as a kind of the free will of nature.... and later i thought if those things are indeed particle's then would they have an outside and an inside ?, or even a fixed size of charge..
No, i cannt really think of particles. In programming i would call these thiny things objects, some must be shaped by their environment like a function, while other must be pure a single not combined property, rather like an property or attribute those would be the core fabric. Not all of those waves would be like a light beam, but some would never be able to travel away but (perhaps combined rotations) those must be waving arround eachother looking just as a particle.
 
But then in what would those waves wave?. It must be something core fabric, i think about perhaps time, perhaps since we now of particles who can (almost) reach light speed and get into time shifts (future/past) then if I reject particles as beeing a wave,.. then it might be waves who would be pushing tricks on time and space itself.
So then such waves 'into the nothing empty space' ,,, could be differences in the core of space fabrics, tiny time fields noice zones. Apperently in quantumworld small differences wouldn't be that harmfull and perhaps a kind of noice in space in which space waves will ocure and twists and rotates etc , and at higher levels i think of that this noice would just fit in, fitting in to a reality ..... an observed reality (as only observed reality is our western yes/no kind of science, and observing makes quantum physics fixed as i understand). At some point this noice must have a certain fixed, or liked, or averaged direction. .. hmm perhaps not that bad why asume that time would everywhere be exactly the same.
 
hmm no..to easy must be something different.

 


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Discover all that’s new in My Yahoo!

#762 From: rybo6 <rybo6@...>
Date: Tue Nov 16, 2004 12:25 am
Subject: Re: Re: 10 or 26 dimesions?
os_jbug
Send Email Send Email
 
Hyper-dimension is just cells within cells.  Fullers VE fequency growth
was one of the early examples of this tho, he himself, appears not to
have been aware of that.
http://www.rwgrayprojects.com/synergetics/s02/figs/f2201.html

Rybo

On Nov 15, 2004, at 1:44 PM, Paul Erlich wrote:

>
>  The current thinking in Physics is that the various string theories
>  which posited either 10 or 26 dimensions (five such theories in fact)
>  have all been shown to be aspects/approximations of a unifying "M-
>  theory", in which there are 11 dimensions (1 time dimension, 3
>  ordinary space dimensions, 6 curled-up space dimensions, and 1 non-
>  curled bounded space dimension). Brian Greene's books, taken
>  together, form a very readable account of this, and much more.
>
>
>
>  --- In Polytopia@yahoogroups.com, "lrafey" <rafey@g...> wrote:
>  >
>  > Indeed, and other related theories exploit (and I use that term
>  with
>  > some purpose) this and that number of dimensions. But...we must
>  > remember that in the middle ages, layers of additional cycles were
>  > continually being added to current theories in order to explain
>  > continually observed but otherwise unexplainable phenomena
>  > unitl...at long last...it all fell apart! We must, therefore, take
>  > all these dimensional explications with some sense of rational
>  > consideration.
>  > LD Rafey
>  >
>  > --- In Polytopia@yahoogroups.com, "Alan Michelson"
>  > <amichelson2002@y...> wrote:
>  > >
>  > > Modern String Theory predicts as many as 26 dimensions!
>  > > http://ascension2000.com/ConvergenceIII/c314.htm
>  > > http://mathforum.org/mam/00/master/people/kaku/
>  > >
>  > > --- In Polytopia@yahoogroups.com, "Larry Rafey" <rafey@g...>
>  wrote:
>  > > > Peter...
>  > >
>  > > >   Modern String Theory predicts as many as ten (yes, 10)
>  > dimensions!
>  > > > These are based on an assumption that six of these are tightly
>  > > > curled or compact dimensions (sometimes described simply as
>  being
>  > > > wrapped up on sox circles but in more complex terms they are
>  > wrapped
>  > > > up on mathematical elaborations known as Calabi-You Manifolds
>  and
>  > > > Orbifolds). and that we might not be consciously aware of their
>  > > > existence. These extra degrees of freedom  behave positively by
>  > > > extending motion into these dimensions which would help to
>  > explain
>  > > > some of the otherwise contradictory observations of modern
>  > Physics.
>  > > > You might investigate the Kaluza-Klein Theory. A more recent
>  > take is
>  > > > a combination of the five predominate string theories, referred
>  > to
>  > > > as 'M-Theory.'
>  > >
>  > > >   Concerning Black Holes, you might have noted recent
>  > publications
>  > > > per Steve Hawking suggesting the possibility that not all
>  > > > information is destroyed in a black hole and, in fact, may,
>  under
>  > > > some circumstances, actually leak out.
>  > > >
>  > > > Keep Drawing...I, too, am an amateur artist. Enjoyed your
>  > website.
>  > > > LD Rafey
>  > > >
>  > > >
>
>
>
>
>
>
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Anti-bush campagin 2004. Bush must go!

#763 From: rybo6 <rybo6@...>
Date: Tue Nov 16, 2004 12:34 am
Subject: Re: Just wondering
os_jbug
Send Email Send Email
 
Peter, you may want ot check these out.
http://www.quantummatter.com/
http://www.poams.org/

Ryb
On Nov 15, 2004, at 4:40 PM, Peter Boos wrote:

> Just wondering.
>  
> I was watching a TV program about an indian nuclearphysic proffesor.
> He was talking about duality of quantum mechanics, photons being wave
> AND particle.
> His viewpoint was indifferent with western science. We always want to
> have one truth as one answer. While because of their believe he thinks
> such anomalies just exist as two answers at the same time. He
> explained quantum uncertainty as a kind of the free will of nature....
> and later i thought if those things are indeed particle's then would
> they have an outside and an inside ?, or even a fixed size of charge..
>  No, i cannt really think of particles. In programming i would call
> these thiny things objects, some must be shaped by their environment
> like a function, while other must be pure a single not combined
> property, rather like an property or attribute those would be the core
> fabric. Not all of those waves would be like a light beam, but some
> would never be able to travel away but (perhaps combined rotations)
> those must be waving arround eachother looking just as a particle.
>  
> But then in what would those waves wave?. It must be something core
> fabric, i think about perhaps time, perhaps since we now of particles
> who can (almost) reach light speed and get into time shifts
> (future/past) then if I reject particles as beeing a wave,.. then it
> might be waves who would be pushing tricks on time and space itself.
> So then such waves 'into the nothing empty space' ,,, could
> be differences in the core of space fabrics, tiny time fields noice
> zones. Apperently in quantumworld small differences wouldn't be that
> harmfull and perhaps a kind of noice in space in which space waves
> will ocure and twists and rotates etc , and at higher levels i think
> of that this noice would just fit in, fitting in to a reality ..... an
> observed reality (as only observed reality is our western yes/no kind
> of science, and observing makes quantum physics fixed as i
> understand). At some point this noice must have a certain fixed, or
> liked, or averaged direction. .. hmm perhaps not that bad why asume
> that time would everywhere be exactly the same.
>   
> hmm no..to easy must be something different.
>
>  
>
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Anti-bush campagin 2004. Bush must go!

#764 From: "Robert Webb" <RobertW@...>
Date: Tue Nov 16, 2004 1:36 am
Subject: RE: Robt. Webb. how difficult to program.
robertcharle...
Send Email Send Email
 
Michael Donovan wrote:

> That is to say part of the 'blank screen' of a new program would
> already have the spaces to be filled across the top, as with this
> email, File, Edit, View, etc.  That there is a package already to
> adjust, rename etc.

Some of this is made easier, and some is not.  It's usually not hard
to create a menu with the items you want, but making those menu items
do what you want is what takes time, depending on what they have to
do.

> So that the program is mostly file management.

Well, not really.

> I am taking a wild stab and saying that it would be far easier than
> Stella.

Yes, it would be, although I don't know what is required in your
program :-)

> About how long did that take.

I've been working on it for a few years now and am still working on
it!

> On the open market, how much would a package of that progarming go
> far.  For example, if you just had the ideas of Stella, did not
> program.  Ballpark, how much are we looking at?

Don't know.  I have spent more time on it than anyone would want to
pay for (and more than sales will ever cover).  It's hard to tell now,
but I guess there's a few thousand hours of work in it.  If you
charged $100 per hour it would cost $100000, but for such a huge job
the rate would probably be smaller than $100 per hour.

You won't be looking at anything like that though.  Maybe $10000 for
your job?  Maybe less?  I really don't know, and am probably not a
good person to ask.

Rob.

--

Robert Webb <RobertW@...>,
Want to make polyhedra?
<http://www.software3d.com/Stella.html>

#765 From: "Laurent Leimgruber" <leimgruber@...>
Date: Tue Nov 16, 2004 6:32 am
Subject: Re: Re: back to the weird 2
lleimgr2001
Send Email Send Email
 
Good morning Paul,
 
I am doing research in stockmarkets movements relationships, and have been comparing the normal Fibonacci series and ratios with musical harmonics.
 
I would be greatly interested to receive your paper, so as to understand better the significance of/possibilities of these scales.
 
I have already found a better response--at least for me--to the octave scale(2root12).
 
Thanks for your kind offer
Laurent
----- Original Message -----
Sent: Monday, November 15, 2004 8:58 PM
Subject: [Polytopia] Re: back to the weird 2


The diatonic scale (and the related pentatonic scale, etc.) indeed
represents one of the scale families that proceeds most naturally
from trying to have many intervals which approximate simple ratios
involving the first three primes. There is always approximation
involved because not all the ratios one wishes to be simple can be
exact at the same time -- if you try to make most of them exact, you
end up with 40:27 instead of 3:2 in one place and 32:27 instead of
6:5 in another. Far more commonly, some sort of temperament is used
so all the approximations are a little off but none are quite this
bad.

However, there are other scales that derive about as naturally from
the first three primes. And it is indeed possible to derive the
diatonic scale from the first four primes -- it's just that one's
tolerance for inaccurate approximations has to increase.

My latest paper on the various diatonic-like scale families that
proceed from the first three, as well as first four, primes is going
to be published in the next issue of Xenharmonikon. If anyone wants a
copy of it now, e-mail me your snail-mail address and I'll send it to
you. Half the paper consists of circular diagrams of scale families
which may or may not remind you of crop circles :)







--- In Polytopia@yahoogroups.com, John Berglund <anisohedral@y...>
wrote:
> Hey Michael,

> I recall our discussion. I would be happy to say that Professor
Hawkins has come up with new theorems, but there are thousands of new
theorems found every year. I could write down 20 such theorems with
no difficulty. The problem is that these particular theorems don't
seem to have much significance to mathematics.

> There is a bit of interest in the design of the crop circles, as in
any designs. Were you to collect logos of companies, you could also
find many mathematical relationships like symmetry in them. You could
study them and find similar "theorems."

> Concerning the question of being "diatonic:" Pythagorus noticed
that notes which have frequencies in a ratio of small natural numbers
sound good together. "Diatonic" refers to a musical scale - which in
the old days had all the notes in ratios of small natural numbers.
(Nowadays we use the equitempered scale based on irrational
numbers... we lose the exact ratios, but gain the ability to play in
all the different keys.) The diatonic scale uses 2, 3, and 5 and
their multiples. One will note that 7 is the first number which is
left out in this scheme. It would not be surprising if there
are "diatonic" ratios in crop circles, company logos, or other
created designs - these are the most common small numbers. If there
are any crop circles that have ratios of 1:7 or 1:11 or 1:13 then we
can say that these are not diatonic. Those interested could
investigate.

> Of course the claim that crop circles go with music doesn't have to
stop with diatonic scales - just because western music is based on
this scale, there are many other scales. The Yahoo "Tuning" group
goes into loving detail about all the different ways to include 7,
11, 13 and more in scales ranging from 2 to thousands of notes per
octave. By this method whatever ratios showed up anywhere - it could
be matched to music.

> The idea that since you can find certain ratios in a design, that
some musical connection can be made seems silly to me. Take a yin-
yang symbol. The ratio of the inner curve radius to the outer curve
radius is 1:2 - a diatonic ratio. A honeycomb tiling by bees contains
all of the diatonic ratios (as well as the 7, 11, 13...) The height
and width of a TV screen are in a diatonic ratio. The size of an inch
to a foot is a diatonic ratio. If you ask people to pick out two
random numbers from 1 to 10, most of the time they will pick a
diatonic ratio. Hold up some fingers on each hand - the ratio between
the hands is diatonic. The ratio of how many teeth I have to how many
leg bones I have is diatonic. Hopefully these examples show that
diatonic ratios are everywhere - no need to get excited if they show
up somewhere.

> John Berglund
>
> Michael Donovan <michael1@m...> wrote:
> John,
>     I had both a feeling and a deep hope you would appear out of
the cybermist for this.  And I am very grateful that you have.  If
you remember the general question of if Professor Gerald Hawkins did
or did not come up with new theorems had come up on this list. At
that time you pointed out that Hawkins theorem was covered by a very
easily provable attribute to all regular polygons.  Perhaps I should
dig that discussion up out of the files.  I acknowledged that without
more information it would seem that your argument was the better. 
However, I could not get further information from the web site who
has stated that they have all of Hawkins work.  To some extent the
issue is therefore still open.
>     The work below is by Dee Gragg.  I am still separating the
issues which I feel are unfortunately mixed together.  One issue is
if or not Hawkins has found previously unknown Euclidean theorems. 
And the other issue is if or not there are diatonic ratios in the
crop circles.  And the issue of if or not there are diatonic issues
in the crop circles is completely separate from the controversy of if
or not the crop circles are 'real'.  In this case 'real' meaning that
they are made by some unknown force, not 'hoaxers'.  In fact, when
Hawkins was first asked to investigate the attitude that he took was
purely mathematical, to look at the 'mind' of the circle makers
whoever they were.  It was from that attitude that the observations
came that whoever was doing this, either 'hoaxers' or unknown minds,
they had the odd character of using not just diatonic scales, but
diatonic scales that were more known to classic, not modern, history.
>     Now Dee Gragg states that the issues are mixed.  Let me quote
Dee...
>      (quote) "  Now as to the Musical notes encoded in the circles.
> They are not unrelated to the theorems.  In fact, all
> four of Dr. Hawkins theorems are related to musical
> notes.  Alas, I found that only two of mine were; F
> below middle C and F two octaves below Middle C. ..." (unquote)
>     Dee is right.  They are not unrelated.  But in this case they
can still be separated and treated separate.  I advised that because
of your observation, John.  But perhaps, with your astute help, the
issue of if or not there are new Euclidean theorems can be resolved.
>     I am passing this to Dee Gragg.  I am suggesting that she join
the Polytopia list so that I am not so much in the middle of this.
>     Michael Donovan
>     Camden, ME.
>
> ----- Original Message -----
> From: John Berglund
> To: Polytopia@yahoogroups.com
> Sent: Friday, November 05, 2004 11:33 AM
> Subject: Re: [Polytopia] back to the weird 2
>
>
> There are some interesting relationships in shapes. Theorems 1A to
1D can be seen in the following picture. (Theorem 1C requires that
you know that a 30-60-90 degree triangle has the edge ratios of
1:2:sqrt(3)... the others can be seen just by counting distances.)
Were we to place another circle inside the three given circles, just
tangent to them, its radius would be in a ratio of 1 to 3 with the
original circles. There are numerous other relationships that we
could point out. I don't think of these relationships as having any
special meaning aside from being nice numbers.

> John Berglund

>
>
> Michael Donovan <michael1@m...> wrote:
>  Crop Circle Theorems

> Their Proofs and Relationship to Musical Notes

>
>    This research began with a simple and rather limited objective:
to prove the crop circle theorems of Dr. Gerald Hawkins.  In fact if
I could have found the proofs in the literature of the field, this
research would never have taken place at all.  Fortunately, I
couldn’t find them because once I started, I could see that further
work that needed to be done.

>    As I proved Dr. Hawkins theorems, I discovered five new ones and
proved them as well.  I then took the diatonic ratios of all the
theorems and related them to the frequencies of the musical scale. 
With some rather startling results I might add.

>    Beginning with Theorem IA I need to make some observations that
apply to all of the theorems.  In Euclidean Geometry one almost
always has to see the end before making a beginning.   Also, since we
are looking for diatonic ratios, we need to find an equation or
equations which will let us divide one diameter or radius by the
other.  Remember too, that because we are working with ratios, the
constants divide out leaving diameter ratios equal to radius ratios. 
And if we square them they are equal to each other and to the ratio
of the areas.

>    Applying this to Theorem IA the equation we need to write is for
the diameter of the circumscribing circle.  It contains both the
radii of the initial and the circumscribed circle.  So from the
equation we are able to divide it and find the diatonic ratio of 4 to
3. 

>    Although, I have proved three more Theorem I’s, I believe this
is the one Dr Hawkins meant when he said Theorem I.  See Circular
Relationships for The Theorems in Appendix A.  I base this belief on
the 4 to 3 diatonic relationship which is related directly to Note F
above Middle C.  See Frequencies In The Fields in Appendix B.

>    Theorem IB is like Theorem IA except that the equilateral
triangle is inscribed rather than being circumscribed.  It can be
proved by Theorem IA and Theorem II.  The equations already exist so
just divide them for the proof.  This gives a new diatonic ratio
which is also the Note F, one octave lower than the previous.
> This theorem is such a simple and logical extension of the first
two that I am puzzled as to why Dr Hawkins did not discover and
publish it.

>    Theorem IC is also often referred to as Theorem I although it is
quite different.  Sometimes both Theorem IA and Theorem IC appear in
the same article as if they were identical.  They aren’t.  The
proof of Theorem IC shows that it contains no diatonic ratio that can
be related to a musical frequency.  I believe that this was not the
theorem Dr. Hawkins was referring to when he said Theorem I.  In my
mind the origin of Theorem IC is rather murky.

>       Theorem ID would have never been discovered if I had read the
instructions for constructing Theorem IA a little closer.  Instead of
circumscribing the equilateral triangle, I circumscribed the three
circles and then proved the theorem before realizing my mistake.  It
has a nice 7 to 3 relationship but it would need to be 8 to 3 to be
Note F in the next higher octave.

>    Theorem II is easy to prove by constructing the appropriate
similar triangles and remembering their relationships.  It may be
proved a number of different ways I have shown just one of them.  It
has the nice diatonic ratio of 4 to 1 which relates directly to the
Note C which is two octaves above Middle C.

>    Theorem III is the simplest of all proofs.  Just remember the
Pythagorean Theorem. It also has a nice diatonic ratio of 2 to 1
which relates directly to the Note C which is an octave above Middle
C.

>    Again using the Pythagorean Theorem, Theorem IVA is shown to
have a nice 4 to 3 relationship.  We have previously related this to
Note F using Theorem IA.

>    While proving Theorem II, it occurred to me that there should be
a similar theorem related to the hexagon. There was and that led me
to discover Theorem IVB by connecting the diameters at the hexagon
corners.  Again by using similar triangles and writing and dividing
the proper equations it is shown to have a diatonic ratio of 1 to 3
which relates directly to the Note F.  This Note F is yet another
octave lower.

>    I have included Theorem IVC mostly for completeness as it does
not have a diatonic ratio which can be related to a specific note. 
If I hadn’t included it you might have wondered why since it can be
proved by simply dividing Theorem IVB by Theorem IVA. 

>    There is a Theorem V which can be used for deriving (not
proving) the other theorems.  However it does not of its self have
diatonic ratios and therefore was not a part of this research.

>       Appendix A Circular Relationships for The Theorems shows a
summary of all the results.  Note that to go from one column to the
other, you simply square or take the square root.  But how do you
know which column to use?  I have followed the lead of Dr. Hawkins in
that if the circles are not concentric, you use the ratio of
diameters, if they are concentric you use the ratio of areas.  This
means diameters for Theorems IA, IB, IC, ID, IVB, and IVC and areas
for Theorems II, III, and IVA.  Why did he pick this convention? 
Certainly I don’t know, perhaps he was a practical man and he did
it because it works.

>    Frequencies In The Fields in Appendix B gives four octaves: two
above and two below Middle C.   This does not encompass the full 27.5
to 4,186 Hz of a piano but does include all the frequencies found so
far.  Notice that all the notes are either F or C.  Coincidence or a
message?  Perhaps as we discover more notes, this will become clear.

>    Theorem T in Appendix C is not really a part of this research,
but is included as help for anyone wanting to compute circle and
regular polygon ratios.  It includes all cases and relies on
trigonometry rather than Euclidian Geometry.

>    Finally, if you’re wondering about me, I have a Short Bio in
Appendix D.



















>
> Copyright 2004 by C. D. Gragg. All rights reserved

> Theorem IA
>    If three equal circles are tangent to a common line and their
centers can be connected by an equilateral triangle and a circle is
circumscribed about the triangle, the ratio of the diameters is 4 to
3.
>               





> Theorem IB
>    If three equal circles are tangent to a common line and their
centers can be connected by an equilateral triangle and a circle is
inscribed within the triangle, the ratio of the diameters is  2:3.

>               

>
> Theorem IC
>    If three equal circles are tangent to a common line and their
centers can be connected by an equilateral triangle and a circle is
constructed using the single circle as a center and drawing the
circle through the other two centers, the ratio of the diameters is 4
to Sqrt 3.
>               

>
> Theorem ID
>    If three equal circles are tangent to a common line and their
centers can be connected by an equilateral triangle and a circle is
constructed circumscribing the three circles, the ratio of the
diameters is 7 to 3.
>               





> Theorem II
>    If an equilateral triangle is inscribed and circumscribed the
ratio of the circles’ areas is 4:1.
>               







>
> Theorem III
>    If a square is inscribed and circumscribed the ratio of the
circles’ areas is 2:1.
>                









> Theorem IVA
>    If a hexagon is inscribed and circumscribed the ratio of the
circles’ areas is 4:3.
>               









> Theorem IVB
>       If a hexagon is inscribed and circumscribed and the corners
connected by diameters, the inscribed circles of the created
equilateral triangles have a diameter ratio to the inscribed circle
of 1:3.
>               







> Theorem IVC
>       If a hexagon is inscribed and circumscribed and the corners
connected by diameters, the inscribed circles of the created
equilateral triangles have a diameter ratio to the circumscribed
circle of 1:2Sqrt3.
>               







> Appendix A

>                           Circular Relationships for
>                                      The Theorems
>  Theorem Ratio of Diameters and
> Radii Ratio of Areas, Diameters Squared, and Radii Squared
>  Theorem IA 4:3 16:9
>  Theorem IB  2:3  4:9
>  Theorem IC 4:Sqrt3 16:3
>  Theorem ID 7:3 49:9
>  Theorem II 2:1 4:1
>  Theorem III Sqrt2:1 2:1
>  Theorem IVA 2:Sqrt3 4:3
>  Theorem IVB 1:3 1:9
>  Theorem IVC 1:2 Sqrt3 1:12
>  
>  
>  
>  
>  















> Copyright 2004 by C. D. Gragg, All rights reserved





> Appendix B

> Frequencies In The Fields

> Note Name C D E F G A B C
> Diatonic Ratio 1/4 9/32 5/16 1/3 3/8 5/12 15/32 1/2
> Frequency (Hz) 66 74.25 82.5 88 99 110 123.75 132

> Note Name C D E F G A B C
> Diatonic Ratio 1/2 9/16 5/8 2/3 3/4 5/6 15/16 1
> Frequency (Hz) 132 148.5 165 176 198 220 247.5 264

> Note Name C* D E F G A B C
> Diatonic Ratio 1 9/8 5/4 4/3 3/2 5/3 15/8 2
> Frequency (Hz) 264 297 330 352 396 440 495 528

> Note Name C D E F G A B C
> Diatonic Ratio 2 9/4 5/2 8/3 3 10/3 15/4 4
> Frequency (Hz) 528 594 660 704 792 880 990 1056
> * Middle C
>  Denotes  found in the fields

> Theorem Summary

> Frequency (Hz) Theorem Used For Proof
> 88 Theorem IVB, Gragg
> 176 Theorem IB, Gragg
> 352 Theorem IA, Theorem IVA, Hawkins
> 528 Theorem III, Hawkins
> 1056 Theorem II, Hawkins

>










>                     Copyright 2004 by C. D. Gragg, All rights
reserved
> Appendix C
> Theorem T

>    Trigonometry can be used to solve circular relationships for
inscribed and circumscribed regular  polygons for polygons of any
number of sides from 3 to infinity.

>  Proof:                                      
>                                                   Where:  α =
3600          and n = number of sides    
>                                                   2n
>                                                           
>                                                                 
cos α  = R1 
>                                                               R2

>                                                                    
R2 =   _1     Proving the Theorem
> Figure 1. Regular polygon with                  R1       cos α
>                any number of sides                              
>                                                           
Further:   ( R2)2 =  (1)2    
>                                                                    
                              ( R1)2      ( cos α)2
> Table of Some Common Polygons
> Figure (All are equiangular) Number of Equal Sides  Ratio of
Diameters and
> Radii Ratio of Areas, Diameters Squared, and Radii Squared
> Triangle (1)(4)         3       2.000 4      4.000
> Square (2)         4       1.414 2      2.000
> Pentagon         5       1.236          1.527
> Hexagon (3)          6       1.155 4/3   1.333
> Heptagon         7       1.110         1.232
> Octagon (4)         8       1.082         1.172
> Nonagon         9       1.064         1.132
> Decagon       10     & nbsp; 1.051         1.106
>        15       1.022         1.045
>        20       1.012         1.025
>        50       1.002         1.004
>      100       1.000         1.001
>      200       1.000         1.000
>        ∞       1.000         1.000
> (1) Theorem II, by Dr. Hawkins using Euclidian Geometry
> (2) Theorem III, by Dr. Hawkins using Euclidian Geometry 
> (3) Theorem IV, by Dr. Hawkins using Euclidian Geometry
> (4) Found in the Kekoskee/Mayville, Wisconsin Crop Circle
>              Formation July 9, 2003    
>                                                                    
       Copyright 2004 by C. D. Gragg, All rights reserved







>
> Appendix D

> Short Bio

>    My name is Dee Gragg.  I am a retired, mechanical engineer.  My
career was spent in research, testing and evaluation. My main areas
of research were automotive air bags, jet aircraft ejection seats and
high speed rocket sleds.  I have published 33 technical papers as
either the principal author or a co-author.  They form a part of the
body of literature in their respective fields.  




















>
>
>
>
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#766 From: "Paul Erlich" <paul@...>
Date: Tue Nov 16, 2004 4:46 pm
Subject: Re: 10 or 26 dimesions?
emotionaljou...
Send Email Send Email
 
This has nothing to do with what I was talking about, but thanks.

--- In Polytopia@yahoogroups.com, rybo6 <rybo6@u...> wrote:
> Hyper-dimension is just cells within cells.  Fullers VE fequency
growth
> was one of the early examples of this tho, he himself, appears not
to
> have been aware of that.
> http://www.rwgrayprojects.com/synergetics/s02/figs/f2201.html
>
> Rybo
>
> On Nov 15, 2004, at 1:44 PM, Paul Erlich wrote:
>
> >
> >  The current thinking in Physics is that the various string
theories
> >  which posited either 10 or 26 dimensions (five such theories in
fact)
> >  have all been shown to be aspects/approximations of a
unifying "M-
> >  theory", in which there are 11 dimensions (1 time dimension, 3
> >  ordinary space dimensions, 6 curled-up space dimensions, and 1
non-
> >  curled bounded space dimension). Brian Greene's books, taken
> >  together, form a very readable account of this, and much more.
> >
> >
> >
> >  --- In Polytopia@yahoogroups.com, "lrafey" <rafey@g...> wrote:
> >  >
> >  > Indeed, and other related theories exploit (and I use that term
> >  with
> >  > some purpose) this and that number of dimensions. But...we must
> >  > remember that in the middle ages, layers of additional cycles
were
> >  > continually being added to current theories in order to explain
> >  > continually observed but otherwise unexplainable phenomena
> >  > unitl...at long last...it all fell apart! We must, therefore,
take
> >  > all these dimensional explications with some sense of rational
> >  > consideration.
> >  > LD Rafey
> >  >
> >  > --- In Polytopia@yahoogroups.com, "Alan Michelson"
> >  > <amichelson2002@y...> wrote:
> >  > >
> >  > > Modern String Theory predicts as many as 26 dimensions!
> >  > > http://ascension2000.com/ConvergenceIII/c314.htm
> >  > > http://mathforum.org/mam/00/master/people/kaku/
> >  > >
> >  > > --- In Polytopia@yahoogroups.com, "Larry Rafey" <rafey@g...>
> >  wrote:
> >  > > > Peter...
> >  > >
> >  > > >   Modern String Theory predicts as many as ten (yes, 10)
> >  > dimensions!
> >  > > > These are based on an assumption that six of these are
tightly
> >  > > > curled or compact dimensions (sometimes described simply as
> >  being
> >  > > > wrapped up on sox circles but in more complex terms they
are
> >  > wrapped
> >  > > > up on mathematical elaborations known as Calabi-You
Manifolds
> >  and
> >  > > > Orbifolds). and that we might not be consciously aware of
their
> >  > > > existence. These extra degrees of freedom  behave
positively by
> >  > > > extending motion into these dimensions which would help to
> >  > explain
> >  > > > some of the otherwise contradictory observations of modern
> >  > Physics.
> >  > > > You might investigate the Kaluza-Klein Theory. A more
recent
> >  > take is
> >  > > > a combination of the five predominate string theories,
referred
> >  > to
> >  > > > as 'M-Theory.'
> >  > >
> >  > > >   Concerning Black Holes, you might have noted recent
> >  > publications
> >  > > > per Steve Hawking suggesting the possibility that not all
> >  > > > information is destroyed in a black hole and, in fact, may,
> >  under
> >  > > > some circumstances, actually leak out.
> >  > > >
> >  > > > Keep Drawing...I, too, am an amateur artist. Enjoyed your
> >  > website.
> >  > > > LD Rafey
> >  > > >
> >  > > >
> >
> >
> >
> >
> >
> >
> > Yahoo! Groups Sponsor
> >
> > ADVERTISEMENT
> > <111004_1104_f_300250a.gif>
> > <l.gif>
> >
> > Yahoo! Groups Links
> >
> >  •  To visit your group on the web, go to:
> > http://groups.yahoo.com/group/Polytopia/
> >  
> >  • 	 To unsubscribe from this group, send an email to:
> > Polytopia-unsubscribe@yahoogroups.com
> >  
> >  • 	 Your use of Yahoo! Groups is subject to the Yahoo!
Terms of
> > Service.
> >
> >
> Anti-bush campagin 2004. Bush must go!

#767 From: "Paul Erlich" <paul@...>
Date: Tue Nov 16, 2004 4:51 pm
Subject: Re: back to the weird 2
emotionaljou...
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Hi Laurent,

If you mean the 12th-root-of-2, that is indeed the solution that
Western music has been stuck in for over a century.

Some creative and sensitive musicians feel it is time to break out of
this hegemony. It is in this spirit that my new paper is offered.

Besides playing music, I work part-time in an investment firm, doing
stock forecasting. So we have something in common!

Please e-mail me your snail-mail address so that I may send you a
copy of my new paper.

Best,
Paul





--- In Polytopia@yahoogroups.com, "Laurent Leimgruber"
<leimgruber@d...> wrote:
> Good morning Paul,
>
> I am doing research in stockmarkets movements relationships, and
have been comparing the normal Fibonacci series and ratios with
musical harmonics.
>
> I would be greatly interested to receive your paper, so as to
understand better the significance of/possibilities of these scales.
>
> I have already found a better response--at least for me--to the
octave scale(2root12).
>
> Thanks for your kind offer
> Laurent
>   ----- Original Message -----
>   From: Paul Erlich
>   To: Polytopia@yahoogroups.com
>   Sent: Monday, November 15, 2004 8:58 PM
>   Subject: [Polytopia] Re: back to the weird 2
>
>
>
>   The diatonic scale (and the related pentatonic scale, etc.)
indeed
>   represents one of the scale families that proceeds most naturally
>   from trying to have many intervals which approximate simple
ratios
>   involving the first three primes. There is always approximation
>   involved because not all the ratios one wishes to be simple can
be
>   exact at the same time -- if you try to make most of them exact,
you
>   end up with 40:27 instead of 3:2 in one place and 32:27 instead
of
>   6:5 in another. Far more commonly, some sort of temperament is
used
>   so all the approximations are a little off but none are quite
this
>   bad.
>
>   However, there are other scales that derive about as naturally
from
>   the first three primes. And it is indeed possible to derive the
>   diatonic scale from the first four primes -- it's just that one's
>   tolerance for inaccurate approximations has to increase.
>
>   My latest paper on the various diatonic-like scale families that
>   proceed from the first three, as well as first four, primes is
going
>   to be published in the next issue of Xenharmonikon. If anyone
wants a
>   copy of it now, e-mail me your snail-mail address and I'll send
it to
>   you. Half the paper consists of circular diagrams of scale
families
>   which may or may not remind you of crop circles :)
>
>
>
>
>
>
>
>   --- In Polytopia@yahoogroups.com, John Berglund
<anisohedral@y...>
>   wrote:
>   > Hey Michael,
>   >
>   > I recall our discussion. I would be happy to say that Professor
>   Hawkins has come up with new theorems, but there are thousands of
new
>   theorems found every year. I could write down 20 such theorems
with
>   no difficulty. The problem is that these particular theorems
don't
>   seem to have much significance to mathematics.
>   >
>   > There is a bit of interest in the design of the crop circles,
as in
>   any designs. Were you to collect logos of companies, you could
also
>   find many mathematical relationships like symmetry in them. You
could
>   study them and find similar "theorems."
>   >
>   > Concerning the question of being "diatonic:" Pythagorus noticed
>   that notes which have frequencies in a ratio of small natural
numbers
>   sound good together. "Diatonic" refers to a musical scale - which
in
>   the old days had all the notes in ratios of small natural
numbers.
>   (Nowadays we use the equitempered scale based on irrational
>   numbers... we lose the exact ratios, but gain the ability to play
in
>   all the different keys.) The diatonic scale uses 2, 3, and 5 and
>   their multiples. One will note that 7 is the first number which
is
>   left out in this scheme. It would not be surprising if there
>   are "diatonic" ratios in crop circles, company logos, or other
>   created designs - these are the most common small numbers. If
there
>   are any crop circles that have ratios of 1:7 or 1:11 or 1:13 then
we
>   can say that these are not diatonic. Those interested could
>   investigate.
>   >
>   > Of course the claim that crop circles go with music doesn't
have to
>   stop with diatonic scales - just because western music is based
on
>   this scale, there are many other scales. The Yahoo "Tuning" group
>   goes into loving detail about all the different ways to include
7,
>   11, 13 and more in scales ranging from 2 to thousands of notes
per
>   octave. By this method whatever ratios showed up anywhere - it
could
>   be matched to music.
>   >
>   > The idea that since you can find certain ratios in a design,
that
>   some musical connection can be made seems silly to me. Take a yin-
>   yang symbol. The ratio of the inner curve radius to the outer
curve
>   radius is 1:2 - a diatonic ratio. A honeycomb tiling by bees
contains
>   all of the diatonic ratios (as well as the 7, 11, 13...) The
height
>   and width of a TV screen are in a diatonic ratio. The size of an
inch
>   to a foot is a diatonic ratio. If you ask people to pick out two
>   random numbers from 1 to 10, most of the time they will pick a
>   diatonic ratio. Hold up some fingers on each hand - the ratio
between
>   the hands is diatonic. The ratio of how many teeth I have to how
many
>   leg bones I have is diatonic. Hopefully these examples show that
>   diatonic ratios are everywhere - no need to get excited if they
show
>   up somewhere.
>   >
>   > John Berglund
>   >
>   > Michael Donovan <michael1@m...> wrote:
>   > John,
>   >     I had both a feeling and a deep hope you would appear out
of
>   the cybermist for this.  And I am very grateful that you have.
If
>   you remember the general question of if Professor Gerald Hawkins
did
>   or did not come up with new theorems had come up on this list. At
>   that time you pointed out that Hawkins theorem was covered by a
very
>   easily provable attribute to all regular polygons.  Perhaps I
should
>   dig that discussion up out of the files.  I acknowledged that
without
>   more information it would seem that your argument was the
better.
>   However, I could not get further information from the web site
who
>   has stated that they have all of Hawkins work.  To some extent
the
>   issue is therefore still open.
>   >     The work below is by Dee Gragg.  I am still separating the
>   issues which I feel are unfortunately mixed together.  One issue
is
>   if or not Hawkins has found previously unknown Euclidean
theorems.
>   And the other issue is if or not there are diatonic ratios in the
>   crop circles.  And the issue of if or not there are diatonic
issues
>   in the crop circles is completely separate from the controversy
of if
>   or not the crop circles are 'real'.  In this case 'real' meaning
that
>   they are made by some unknown force, not 'hoaxers'.  In fact,
when
>   Hawkins was first asked to investigate the attitude that he took
was
>   purely mathematical, to look at the 'mind' of the circle makers
>   whoever they were.  It was from that attitude that the
observations
>   came that whoever was doing this, either 'hoaxers' or unknown
minds,
>   they had the odd character of using not just diatonic scales, but
>   diatonic scales that were more known to classic, not modern,
history.
>   >     Now Dee Gragg states that the issues are mixed.  Let me
quote
>   Dee...
>   >      (quote) "  Now as to the Musical notes encoded in the
circles.
>   > They are not unrelated to the theorems.  In fact, all
>   > four of Dr. Hawkins theorems are related to musical
>   > notes.  Alas, I found that only two of mine were; F
>   > below middle C and F two octaves below Middle C. ..." (unquote)
>   >     Dee is right.  They are not unrelated.  But in this case
they
>   can still be separated and treated separate.  I advised that
because
>   of your observation, John.  But perhaps, with your astute help,
the
>   issue of if or not there are new Euclidean theorems can be
resolved.
>   >     I am passing this to Dee Gragg.  I am suggesting that she
join
>   the Polytopia list so that I am not so much in the middle of this.
>   >     Michael Donovan
>   >     Camden, ME.
>   >
>   > ----- Original Message -----
>   > From: John Berglund
>   > To: Polytopia@yahoogroups.com
>   > Sent: Friday, November 05, 2004 11:33 AM
>   > Subject: Re: [Polytopia] back to the weird 2
>   >
>   >
>   > There are some interesting relationships in shapes. Theorems 1A
to
>   1D can be seen in the following picture. (Theorem 1C requires
that
>   you know that a 30-60-90 degree triangle has the edge ratios of
>   1:2:sqrt(3)... the others can be seen just by counting
distances.)
>   Were we to place another circle inside the three given circles,
just
>   tangent to them, its radius would be in a ratio of 1 to 3 with
the
>   original circles. There are numerous other relationships that we
>   could point out. I don't think of these relationships as having
any
>   special meaning aside from being nice numbers.
>   >
>   > John Berglund
>   >
>   >
>   >
>   > Michael Donovan <michael1@m...> wrote:
>   >  Crop Circle Theorems
>   >
>   > Their Proofs and Relationship to Musical Notes
>   >
>   >
>   >    This research began with a simple and rather limited
objective:
>   to prove the crop circle theorems of Dr. Gerald Hawkins.  In fact
if
>   I could have found the proofs in the literature of the field,
this
>   research would never have taken place at all.  Fortunately, I
>   couldnâ?Tt find them because once I started, I could see that
further
>   work that needed to be done.
>   >
>   >    As I proved Dr. Hawkins theorems, I discovered five new ones
and
>   proved them as well.  I then took the diatonic ratios of all the
>   theorems and related them to the frequencies of the musical
scale.
>   With some rather startling results I might add.
>   >
>   >    Beginning with Theorem IA I need to make some observations
that
>   apply to all of the theorems.  In Euclidean Geometry one almost
>   always has to see the end before making a beginning.   Also,
since we
>   are looking for diatonic ratios, we need to find an equation or
>   equations which will let us divide one diameter or radius by the
>   other.  Remember too, that because we are working with ratios,
the
>   constants divide out leaving diameter ratios equal to radius
ratios.
>   And if we square them they are equal to each other and to the
ratio
>   of the areas.
>   >
>   >    Applying this to Theorem IA the equation we need to write is
for
>   the diameter of the circumscribing circle.  It contains both the
>   radii of the initial and the circumscribed circle.  So from the
>   equation we are able to divide it and find the diatonic ratio of
4 to
>   3.
>   >
>   >    Although, I have proved three more Theorem Iâ?Ts, I believe
this
>   is the one Dr Hawkins meant when he said Theorem I.  See Circular
>   Relationships for The Theorems in Appendix A.  I base this belief
on
>   the 4 to 3 diatonic relationship which is related directly to
Note F
>   above Middle C.  See Frequencies In The Fields in Appendix B.
>   >
>   >    Theorem IB is like Theorem IA except that the equilateral
>   triangle is inscribed rather than being circumscribed.  It can be
>   proved by Theorem IA and Theorem II.  The equations already exist
so
>   just divide them for the proof.  This gives a new diatonic ratio
>   which is also the Note F, one octave lower than the previous.
>   > This theorem is such a simple and logical extension of the
first
>   two that I am puzzled as to why Dr Hawkins did not discover and
>   publish it.
>   >
>   >    Theorem IC is also often referred to as Theorem I although
it is
>   quite different.  Sometimes both Theorem IA and Theorem IC appear
in
>   the same article as if they were identical.  They arenâ?Tt.  The
>   proof of Theorem IC shows that it contains no diatonic ratio that
can
>   be related to a musical frequency.  I believe that this was not
the
>   theorem Dr. Hawkins was referring to when he said Theorem I.  In
my
>   mind the origin of Theorem IC is rather murky.
>   >
>   >       Theorem ID would have never been discovered if I had read
the
>   instructions for constructing Theorem IA a little closer.
Instead of
>   circumscribing the equilateral triangle, I circumscribed the
three
>   circles and then proved the theorem before realizing my mistake.
It
>   has a nice 7 to 3 relationship but it would need to be 8 to 3 to
be
>   Note F in the next higher octave.
>   >
>   >    Theorem II is easy to prove by constructing the appropriate
>   similar triangles and remembering their relationships.  It may be
>   proved a number of different ways I have shown just one of them.
It
>   has the nice diatonic ratio of 4 to 1 which relates directly to
the
>   Note C which is two octaves above Middle C.
>   >
>   >    Theorem III is the simplest of all proofs.  Just remember
the
>   Pythagorean Theorem. It also has a nice diatonic ratio of 2 to 1
>   which relates directly to the Note C which is an octave above
Middle
>   C.
>   >
>   >    Again using the Pythagorean Theorem, Theorem IVA is shown to
>   have a nice 4 to 3 relationship.  We have previously related this
to
>   Note F using Theorem IA.
>   >
>   >    While proving Theorem II, it occurred to me that there
should be
>   a similar theorem related to the hexagon. There was and that led
me
>   to discover Theorem IVB by connecting the diameters at the
hexagon
>   corners.  Again by using similar triangles and writing and
dividing
>   the proper equations it is shown to have a diatonic ratio of 1 to
3
>   which relates directly to the Note F.  This Note F is yet another
>   octave lower.
>   >
>   >    I have included Theorem IVC mostly for completeness as it
does
>   not have a diatonic ratio which can be related to a specific
note.
>   If I hadnâ?Tt included it you might have wondered why since it
can be
>   proved by simply dividing Theorem IVB by Theorem IVA.
>   >
>   >    There is a Theorem V which can be used for deriving (not
>   proving) the other theorems.  However it does not of its self
have
>   diatonic ratios and therefore was not a part of this research.
>   >
>   >       Appendix A Circular Relationships for The Theorems shows
a
>   summary of all the results.  Note that to go from one column to
the
>   other, you simply square or take the square root.  But how do you
>   know which column to use?  I have followed the lead of Dr.
Hawkins in
>   that if the circles are not concentric, you use the ratio of
>   diameters, if they are concentric you use the ratio of areas.
This
>   means diameters for Theorems IA, IB, IC, ID, IVB, and IVC and
areas
>   for Theorems II, III, and IVA.  Why did he pick this convention?
>   Certainly I donâ?Tt know, perhaps he was a practical man and he
did
>   it because it works.
>   >
>   >    Frequencies In The Fields in Appendix B gives four octaves:
two
>   above and two below Middle C.   This does not encompass the full
27.5
>   to 4,186 Hz of a piano but does include all the frequencies found
so
>   far.  Notice that all the notes are either F or C.  Coincidence
or a
>   message?  Perhaps as we discover more notes, this will become
clear.
>   >
>   >    Theorem T in Appendix C is not really a part of this
research,
>   but is included as help for anyone wanting to compute circle and
>   regular polygon ratios.  It includes all cases and relies on
>   trigonometry rather than Euclidian Geometry.
>   >
>   >    Finally, if youâ?Tre wondering about me, I have a Short Bio
in
>   Appendix D.
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   > Copyright 2004 by C. D. Gragg. All rights reserved
>   >
>   > Theorem IA
>   >    If three equal circles are tangent to a common line and
their
>   centers can be connected by an equilateral triangle and a circle
is
>   circumscribed about the triangle, the ratio of the diameters is 4
to
>   3.
>   >
>   >
>   >
>   >
>   >
>   >
>   > Theorem IB
>   >    If three equal circles are tangent to a common line and
their
>   centers can be connected by an equilateral triangle and a circle
is
>   inscribed within the triangle, the ratio of the diameters is  2:3.
>   >
>   >
>   >
>   >
>   > Theorem IC
>   >    If three equal circles are tangent to a common line and
their
>   centers can be connected by an equilateral triangle and a circle
is
>   constructed using the single circle as a center and drawing the
>   circle through the other two centers, the ratio of the diameters
is 4
>   to Sqrt 3.
>   >
>   >
>   >
>   > Theorem ID
>   >    If three equal circles are tangent to a common line and
their
>   centers can be connected by an equilateral triangle and a circle
is
>   constructed circumscribing the three circles, the ratio of the
>   diameters is 7 to 3.
>   >
>   >
>   >
>   >
>   >
>   >
>   > Theorem II
>   >    If an equilateral triangle is inscribed and circumscribed
the
>   ratio of the circlesâ?T areas is 4:1.
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   > Theorem III
>   >    If a square is inscribed and circumscribed the ratio of the
>   circlesâ?T areas is 2:1.
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   > Theorem IVA
>   >    If a hexagon is inscribed and circumscribed the ratio of the
>   circlesâ?T areas is 4:3.
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   > Theorem IVB
>   >       If a hexagon is inscribed and circumscribed and the
corners
>   connected by diameters, the inscribed circles of the created
>   equilateral triangles have a diameter ratio to the inscribed
circle
>   of 1:3.
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   > Theorem IVC
>   >       If a hexagon is inscribed and circumscribed and the
corners
>   connected by diameters, the inscribed circles of the created
>   equilateral triangles have a diameter ratio to the circumscribed
>   circle of 1:2Sqrt3.
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   > Appendix A
>   >
>   >                           Circular Relationships for
>   >                                      The Theorems
>   >  Theorem Ratio of Diameters and
>   > Radii Ratio of Areas, Diameters Squared, and Radii Squared
>   >  Theorem IA 4:3 16:9
>   >  Theorem IB  2:3  4:9
>   >  Theorem IC 4:Sqrt3 16:3
>   >  Theorem ID 7:3 49:9
>   >  Theorem II 2:1 4:1
>   >  Theorem III Sqrt2:1 2:1
>   >  Theorem IVA 2:Sqrt3 4:3
>   >  Theorem IVB 1:3 1:9
>   >  Theorem IVC 1:2 Sqrt3 1:12
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   > Copyright 2004 by C. D. Gragg, All rights reserved
>   >
>   >
>   >
>   >
>   >
>   > Appendix B
>   >
>   > Frequencies In The Fields
>   >
>   > Note Name C D E F G A B C
>   > Diatonic Ratio 1/4 9/32 5/16 1/3 3/8 5/12 15/32 1/2
>   > Frequency (Hz) 66 74.25 82.5 88 99 110 123.75 132
>   >
>   > Note Name C D E F G A B C
>   > Diatonic Ratio 1/2 9/16 5/8 2/3 3/4 5/6 15/16 1
>   > Frequency (Hz) 132 148.5 165 176 198 220 247.5 264
>   >
>   > Note Name C* D E F G A B C
>   > Diatonic Ratio 1 9/8 5/4 4/3 3/2 5/3 15/8 2
>   > Frequency (Hz) 264 297 330 352 396 440 495 528
>   >
>   > Note Name C D E F G A B C
>   > Diatonic Ratio 2 9/4 5/2 8/3 3 10/3 15/4 4
>   > Frequency (Hz) 528 594 660 704 792 880 990 1056
>   > * Middle C
>   >  Denotes  found in the fields
>   >
>   > Theorem Summary
>   >
>   > Frequency (Hz) Theorem Used For Proof
>   > 88 Theorem IVB, Gragg
>   > 176 Theorem IB, Gragg
>   > 352 Theorem IA, Theorem IVA, Hawkins
>   > 528 Theorem III, Hawkins
>   > 1056 Theorem II, Hawkins
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >                     Copyright 2004 by C. D. Gragg, All rights
>   reserved
>   > Appendix C
>   > Theorem T
>   >
>   >    Trigonometry can be used to solve circular relationships for
>   inscribed and circumscribed regular  polygons for polygons of any
>   number of sides from 3 to infinity.
>   >
>   >  Proof:
>   >                                                   Where:  α =
>   3600          and n = number of sides
>   >                                                   2n
>   >
>
>
>   cos α  = R1
>   >                                                               R2
>   >
>
>
>   R2 =   _1     Proving the Theorem
>   > Figure 1. Regular polygon with                  R1       cos α
>   >                any number of
sides
>   >
>   Further:   ( R2)2 =  (1)2
>
>
>                                 ( R1)2      ( cos α)2
>   > Table of Some Common Polygons
>   > Figure (All are equiangular) Number of Equal Sides  Ratio of
>   Diameters and
>   > Radii Ratio of Areas, Diameters Squared, and Radii Squared
>   > Triangle (1)(4)         3       2.000 4      4.000
>   > Square (2)         4       1.414 2      2.000
>   > Pentagon         5       1.236          1.527
>   > Hexagon (3)          6       1.155 4/3   1.333
>   > Heptagon         7       1.110         1.232
>   > Octagon (4)         8       1.082         1.172
>   > Nonagon         9       1.064         1.132
>   > Decagon       10     & nbsp; 1.051         1.106
>   >        15       1.022         1.045
>   >        20       1.012         1.025
>   >        50       1.002         1.004
>   >      100       1.000         1.001
>   >      200       1.000         1.000
>   >        â^z       1.000         1.000
>   > (1) Theorem II, by Dr. Hawkins using Euclidian Geometry
>   > (2) Theorem III, by Dr. Hawkins using Euclidian Geometry
>   > (3) Theorem IV, by Dr. Hawkins using Euclidian Geometry
>   > (4) Found in the Kekoskee/Mayville, Wisconsin Crop Circle
>   >              Formation July 9, 2003
>
>
>          Copyright 2004 by C. D. Gragg, All rights reserved
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   > Appendix D
>   >
>   > Short Bio
>   >
>   >    My name is Dee Gragg.  I am a retired, mechanical engineer.
My
>   career was spent in research, testing and evaluation. My main
areas
>   of research were automotive air bags, jet aircraft ejection seats
and
>   high speed rocket sleds.  I have published 33 technical papers as
>   either the principal author or a co-author.  They form a part of
the
>   body of literature in their respective fields.
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   >
>   > ---------------------------------
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#768 From: "Laurent Leimgruber" <leimgruber@...>
Date: Tue Nov 16, 2004 5:06 pm
Subject: Re: Re: back to the weird 2
lleimgr2001
Send Email Send Email
 
Good morning Paul,
Sorry for the silly question: what is snail-mail adress??
 
my email is the following: Leimgruber@...       in case it is OK for you.
 
Thanks Paul. I was for 30-Y chief investment strategist with a Private Geneva Bank, and fund manager, so we really have a lot to share.
 
Have a nice morning.
Laurent
----- Original Message -----
Sent: Tuesday, November 16, 2004 5:51 PM
Subject: [Polytopia] Re: back to the weird 2


Hi Laurent,

If you mean the 12th-root-of-2, that is indeed the solution that
Western music has been stuck in for over a century.

Some creative and sensitive musicians feel it is time to break out of
this hegemony. It is in this spirit that my new paper is offered.

Besides playing music, I work part-time in an investment firm, doing
stock forecasting. So we have something in common!

Please e-mail me your snail-mail address so that I may send you a
copy of my new paper.

Best,
Paul





--- In Polytopia@yahoogroups.com, "Laurent Leimgruber"
<leimgruber@d...> wrote:
> Good morning Paul,
>
> I am doing research in stockmarkets movements relationships, and
have been comparing the normal Fibonacci series and ratios with
musical harmonics.
>
> I would be greatly interested to receive your paper, so as to
understand better the significance of/possibilities of these scales.
>
> I have already found a better response--at least for me--to the
octave scale(2root12).
>
> Thanks for your kind offer
> Laurent
>   ----- Original Message -----
>   From: Paul Erlich
>   To: Polytopia@yahoogroups.com
>   Sent: Monday, November 15, 2004 8:58 PM
>   Subject: [Polytopia] Re: back to the weird 2
>
>
>
>   The diatonic scale (and the related pentatonic scale, etc.)
indeed
>   represents one of the scale families that proceeds most naturally
>   from trying to have many intervals which approximate simple
ratios
>   involving the first three primes. There is always approximation
>   involved because not all the ratios one wishes to be simple can
be
>   exact at the same time -- if you try to make most of them exact,
you
>   end up with 40:27 instead of 3:2 in one place and 32:27 instead
of
>   6:5 in another. Far more commonly, some sort of temperament is
used
>   so all the approximations are a little off but none are quite
this
>   bad.
>
>   However, there are other scales that derive about as naturally
from
>   the first three primes. And it is indeed possible to derive the
>   diatonic scale from the first four primes -- it's just that one's
>   tolerance for inaccurate approximations has to increase.
>
>   My latest paper on the various diatonic-like scale families that
>   proceed from the first three, as well as first four, primes is
going
>   to be published in the next issue of Xenharmonikon. If anyone
wants a
>   copy of it now, e-mail me your snail-mail address and I'll send
it to
>   you. Half the paper consists of circular diagrams of scale
families
>   which may or may not remind you of crop circles :)
>
>
>
>
>
>
>
>   --- In Polytopia@yahoogroups.com, John Berglund
<anisohedral@y...>
>   wrote:
>   > Hey Michael,
>   > 
>   > I recall our discussion. I would be happy to say that Professor
>   Hawkins has come up with new theorems, but there are thousands of
new
>   theorems found every year. I could write down 20 such theorems
with
>   no difficulty. The problem is that these particular theorems
don't
>   seem to have much significance to mathematics.
>   > 
>   > There is a bit of interest in the design of the crop circles,
as in
>   any designs. Were you to collect logos of companies, you could
also
>   find many mathematical relationships like symmetry in them. You
could
>   study them and find similar "theorems."
>   > 
>   > Concerning the question of being "diatonic:" Pythagorus noticed
>   that notes which have frequencies in a ratio of small natural
numbers
>   sound good together. "Diatonic" refers to a musical scale - which
in
>   the old days had all the notes in ratios of small natural
numbers.
>   (Nowadays we use the equitempered scale based on irrational
>   numbers... we lose the exact ratios, but gain the ability to play
in
>   all the different keys.) The diatonic scale uses 2, 3, and 5 and
>   their multiples. One will note that 7 is the first number which
is
>   left out in this scheme. It would not be surprising if there
>   are "diatonic" ratios in crop circles, company logos, or other
>   created designs - these are the most common small numbers. If
there
>   are any crop circles that have ratios of 1:7 or 1:11 or 1:13 then
we
>   can say that these are not diatonic. Those interested could
>   investigate.
>   > 
>   > Of course the claim that crop circles go with music doesn't
have to
>   stop with diatonic scales - just because western music is based
on
>   this scale, there are many other scales. The Yahoo "Tuning" group
>   goes into loving detail about all the different ways to include
7,
>   11, 13 and more in scales ranging from 2 to thousands of notes
per
>   octave. By this method whatever ratios showed up anywhere - it
could
>   be matched to music.
>   > 
>   > The idea that since you can find certain ratios in a design,
that
>   some musical connection can be made seems silly to me. Take a yin-
>   yang symbol. The ratio of the inner curve radius to the outer
curve
>   radius is 1:2 - a diatonic ratio. A honeycomb tiling by bees
contains
>   all of the diatonic ratios (as well as the 7, 11, 13...) The
height
>   and width of a TV screen are in a diatonic ratio. The size of an
inch
>   to a foot is a diatonic ratio. If you ask people to pick out two
>   random numbers from 1 to 10, most of the time they will pick a
>   diatonic ratio. Hold up some fingers on each hand - the ratio
between
>   the hands is diatonic. The ratio of how many teeth I have to how
many
>   leg bones I have is diatonic. Hopefully these examples show that
>   diatonic ratios are everywhere - no need to get excited if they
show
>   up somewhere.
>   > 
>   > John Berglund
>   >
>   > Michael Donovan <michael1@m...> wrote:
>   > John,
>   >     I had both a feeling and a deep hope you would appear out
of
>   the cybermist for this.  And I am very grateful that you have. 
If
>   you remember the general question of if Professor Gerald Hawkins
did
>   or did not come up with new theorems had come up on this list. At
>   that time you pointed out that Hawkins theorem was covered by a
very
>   easily provable attribute to all regular polygons.  Perhaps I
should
>   dig that discussion up out of the files.  I acknowledged that
without
>   more information it would seem that your argument was the
better. 
>   However, I could not get further information from the web site
who
>   has stated that they have all of Hawkins work.  To some extent
the
>   issue is therefore still open.
>   >     The work below is by Dee Gragg.  I am still separating the
>   issues which I feel are unfortunately mixed together.  One issue
is
>   if or not Hawkins has found previously unknown Euclidean
theorems. 
>   And the other issue is if or not there are diatonic ratios in the
>   crop circles.  And the issue of if or not there are diatonic
issues
>   in the crop circles is completely separate from the controversy
of if
>   or not the crop circles are 'real'.  In this case 'real' meaning
that
>   they are made by some unknown force, not 'hoaxers'.  In fact,
when
>   Hawkins was first asked to investigate the attitude that he took
was
>   purely mathematical, to look at the 'mind' of the circle makers
>   whoever they were.  It was from that attitude that the
observations
>   came that whoever was doing this, either 'hoaxers' or unknown
minds,
>   they had the odd character of using not just diatonic scales, but
>   diatonic scales that were more known to classic, not modern,
history.
>   >     Now Dee Gragg states that the issues are mixed.  Let me
quote
>   Dee...
>   >      (quote) "  Now as to the Musical notes encoded in the
circles.
>   > They are not unrelated to the theorems.  In fact, all
>   > four of Dr. Hawkins theorems are related to musical
>   > notes.  Alas, I found that only two of mine were; F
>   > below middle C and F two octaves below Middle C. ..." (unquote)
>   >     Dee is right.  They are not unrelated.  But in this case
they
>   can still be separated and treated separate.  I advised that
because
>   of your observation, John.  But perhaps, with your astute help,
the
>   issue of if or not there are new Euclidean theorems can be
resolved.
>   >     I am passing this to Dee Gragg.  I am suggesting that she
join
>   the Polytopia list so that I am not so much in the middle of this.
>   >     Michael Donovan
>   >     Camden, ME.
>   >
>   > ----- Original Message -----
>   > From: John Berglund
>   > To: Polytopia@yahoogroups.com
>   > Sent: Friday, November 05, 2004 11:33 AM
>   > Subject: Re: [Polytopia] back to the weird 2
>   >
>   >
>   > There are some interesting relationships in shapes. Theorems 1A
to
>   1D can be seen in the following picture. (Theorem 1C requires
that
>   you know that a 30-60-90 degree triangle has the edge ratios of
>   1:2:sqrt(3)... the others can be seen just by counting
distances.)
>   Were we to place another circle inside the three given circles,
just
>   tangent to them, its radius would be in a ratio of 1 to 3 with
the
>   original circles. There are numerous other relationships that we
>   could point out. I don't think of these relationships as having
any
>   special meaning aside from being nice numbers.
>   > 
>   > John Berglund
>   > 
>   >
>   >
>   > Michael Donovan <michael1@m...> wrote:
>   >  Crop Circle Theorems
>   > 
>   > Their Proofs and Relationship to Musical Notes
>   > 
>   >
>   >    This research began with a simple and rather limited
objective:
>   to prove the crop circle theorems of Dr. Gerald Hawkins.  In fact
if
>   I could have found the proofs in the literature of the field,
this
>   research would never have taken place at all.  Fortunately, I
>   couldnâ?Tt find them because once I started, I could see that
further
>   work that needed to be done.
>   > 
>   >    As I proved Dr. Hawkins theorems, I discovered five new ones
and
>   proved them as well.  I then took the diatonic ratios of all the
>   theorems and related them to the frequencies of the musical
scale. 
>   With some rather startling results I might add.
>   > 
>   >    Beginning with Theorem IA I need to make some observations
that
>   apply to all of the theorems.  In Euclidean Geometry one almost
>   always has to see the end before making a beginning.   Also,
since we
>   are looking for diatonic ratios, we need to find an equation or
>   equations which will let us divide one diameter or radius by the
>   other.  Remember too, that because we are working with ratios,
the
>   constants divide out leaving diameter ratios equal to radius
ratios. 
>   And if we square them they are equal to each other and to the
ratio
>   of the areas.
>   > 
>   >    Applying this to Theorem IA the equation we need to write is
for
>   the diameter of the circumscribing circle.  It contains both the
>   radii of the initial and the circumscribed circle.  So from the
>   equation we are able to divide it and find the diatonic ratio of
4 to
>   3. 
>   > 
>   >    Although, I have proved three more Theorem Iâ?Ts, I believe
this
>   is the one Dr Hawkins meant when he said Theorem I.  See Circular
>   Relationships for The Theorems in Appendix A.  I base this belief
on
>   the 4 to 3 diatonic relationship which is related directly to
Note F
>   above Middle C.  See Frequencies In The Fields in Appendix B.
>   > 
>   >    Theorem IB is like Theorem IA except that the equilateral
>   triangle is inscribed rather than being circumscribed.  It can be
>   proved by Theorem IA and Theorem II.  The equations already exist
so
>   just divide them for the proof.  This gives a new diatonic ratio
>   which is also the Note F, one octave lower than the previous.
>   > This theorem is such a simple and logical extension of the
first
>   two that I am puzzled as to why Dr Hawkins did not discover and
>   publish it.
>   > 
>   >    Theorem IC is also often referred to as Theorem I although
it is
>   quite different.  Sometimes both Theorem IA and Theorem IC appear
in
>   the same article as if they were identical.  They arenâ?Tt.  The
>   proof of Theorem IC shows that it contains no diatonic ratio that
can
>   be related to a musical frequency.  I believe that this was not
the
>   theorem Dr. Hawkins was referring to when he said Theorem I.  In
my
>   mind the origin of Theorem IC is rather murky.
>   > 
>   >       Theorem ID would have never been discovered if I had read
the
>   instructions for constructing Theorem IA a little closer. 
Instead of
>   circumscribing the equilateral triangle, I circumscribed the
three
>   circles and then proved the theorem before realizing my mistake. 
It
>   has a nice 7 to 3 relationship but it would need to be 8 to 3 to
be
>   Note F in the next higher octave.
>   > 
>   >    Theorem II is easy to prove by constructing the appropriate
>   similar triangles and remembering their relationships.  It may be
>   proved a number of different ways I have shown just one of them. 
It
>   has the nice diatonic ratio of 4 to 1 which relates directly to
the
>   Note C which is two octaves above Middle C.
>   > 
>   >    Theorem III is the simplest of all proofs.  Just remember
the
>   Pythagorean Theorem. It also has a nice diatonic ratio of 2 to 1
>   which relates directly to the Note C which is an octave above
Middle
>   C.
>   > 
>   >    Again using the Pythagorean Theorem, Theorem IVA is shown to
>   have a nice 4 to 3 relationship.  We have previously related this
to
>   Note F using Theorem IA.
>   > 
>   >    While proving Theorem II, it occurred to me that there
should be
>   a similar theorem related to the hexagon. There was and that led
me
>   to discover Theorem IVB by connecting the diameters at the
hexagon
>   corners.  Again by using similar triangles and writing and
dividing
>   the proper equations it is shown to have a diatonic ratio of 1 to
3
>   which relates directly to the Note F.  This Note F is yet another
>   octave lower.
>   > 
>   >    I have included Theorem IVC mostly for completeness as it
does
>   not have a diatonic ratio which can be related to a specific
note. 
>   If I hadnâ?Tt included it you might have wondered why since it
can be
>   proved by simply dividing Theorem IVB by Theorem IVA. 
>   > 
>   >    There is a Theorem V which can be used for deriving (not
>   proving) the other theorems.  However it does not of its self
have
>   diatonic ratios and therefore was not a part of this research.
>   > 
>   >       Appendix A Circular Relationships for The Theorems shows
a
>   summary of all the results.  Note that to go from one column to
the
>   other, you simply square or take the square root.  But how do you
>   know which column to use?  I have followed the lead of Dr.
Hawkins in
>   that if the circles are not concentric, you use the ratio of
>   diameters, if they are concentric you use the ratio of areas. 
This
>   means diameters for Theorems IA, IB, IC, ID, IVB, and IVC and
areas
>   for Theorems II, III, and IVA.  Why did he pick this convention? 
>   Certainly I donâ?Tt know, perhaps he was a practical man and he
did
>   it because it works.
>   > 
>   >    Frequencies In The Fields in Appendix B gives four octaves:
two
>   above and two below Middle C.   This does not encompass the full
27.5
>   to 4,186 Hz of a piano but does include all the frequencies found
so
>   far.  Notice that all the notes are either F or C.  Coincidence
or a
>   message?  Perhaps as we discover more notes, this will become
clear.
>   > 
>   >    Theorem T in Appendix C is not really a part of this
research,
>   but is included as help for anyone wanting to compute circle and
>   regular polygon ratios.  It includes all cases and relies on
>   trigonometry rather than Euclidian Geometry.
>   > 
>   >    Finally, if youâ?Tre wondering about me, I have a Short Bio
in
>   Appendix D.
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   >
>   > Copyright 2004 by C. D. Gragg. All rights reserved
>   > 
>   > Theorem IA
>   >    If three equal circles are tangent to a common line and
their
>   centers can be connected by an equilateral triangle and a circle
is
>   circumscribed about the triangle, the ratio of the diameters is 4
to
>   3.
>   >               
>   > 
>   > 
>   > 
>   > 
>   > 
>   > Theorem IB
>   >    If three equal circles are tangent to a common line and
their
>   centers can be connected by an equilateral triangle and a circle
is
>   inscribed within the triangle, the ratio of the diameters is  2:3.
>   > 
>   >               
>   > 
>   >
>   > Theorem IC
>   >    If three equal circles are tangent to a common line and
their
>   centers can be connected by an equilateral triangle and a circle
is
>   constructed using the single circle as a center and drawing the
>   circle through the other two centers, the ratio of the diameters
is 4
>   to Sqrt 3.
>   >               
>   > 
>   >
>   > Theorem ID
>   >    If three equal circles are tangent to a common line and
their
>   centers can be connected by an equilateral triangle and a circle
is
>   constructed circumscribing the three circles, the ratio of the
>   diameters is 7 to 3.
>   >               
>   > 
>   > 
>   > 
>   > 
>   > 
>   > Theorem II
>   >    If an equilateral triangle is inscribed and circumscribed
the
>   ratio of the circlesâ?T areas is 4:1.
>   >               
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   >
>   > Theorem III
>   >    If a square is inscribed and circumscribed the ratio of the
>   circlesâ?T areas is 2:1.
>   >                
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > Theorem IVA
>   >    If a hexagon is inscribed and circumscribed the ratio of the
>   circlesâ?T areas is 4:3.
>   >               
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > Theorem IVB
>   >       If a hexagon is inscribed and circumscribed and the
corners
>   connected by diameters, the inscribed circles of the created
>   equilateral triangles have a diameter ratio to the inscribed
circle
>   of 1:3.
>   >               
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > Theorem IVC
>   >       If a hexagon is inscribed and circumscribed and the
corners
>   connected by diameters, the inscribed circles of the created
>   equilateral triangles have a diameter ratio to the circumscribed
>   circle of 1:2Sqrt3.
>   >               
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > Appendix A
>   > 
>   >                           Circular Relationships for
>   >                                      The Theorems
>   >  Theorem Ratio of Diameters and
>   > Radii Ratio of Areas, Diameters Squared, and Radii Squared
>   >  Theorem IA 4:3 16:9
>   >  Theorem IB  2:3  4:9
>   >  Theorem IC 4:Sqrt3 16:3
>   >  Theorem ID 7:3 49:9
>   >  Theorem II 2:1 4:1
>   >  Theorem III Sqrt2:1 2:1
>   >  Theorem IVA 2:Sqrt3 4:3
>   >  Theorem IVB 1:3 1:9
>   >  Theorem IVC 1:2 Sqrt3 1:12
>   >  
>   >  
>   >  
>   >  
>   >  
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > Copyright 2004 by C. D. Gragg, All rights reserved
>   > 
>   > 
>   > 
>   > 
>   > 
>   > Appendix B
>   > 
>   > Frequencies In The Fields
>   > 
>   > Note Name C D E F G A B C
>   > Diatonic Ratio 1/4 9/32 5/16 1/3 3/8 5/12 15/32 1/2
>   > Frequency (Hz) 66 74.25 82.5 88 99 110 123.75 132
>   > 
>   > Note Name C D E F G A B C
>   > Diatonic Ratio 1/2 9/16 5/8 2/3 3/4 5/6 15/16 1
>   > Frequency (Hz) 132 148.5 165 176 198 220 247.5 264
>   > 
>   > Note Name C* D E F G A B C
>   > Diatonic Ratio 1 9/8 5/4 4/3 3/2 5/3 15/8 2
>   > Frequency (Hz) 264 297 330 352 396 440 495 528
>   > 
>   > Note Name C D E F G A B C
>   > Diatonic Ratio 2 9/4 5/2 8/3 3 10/3 15/4 4
>   > Frequency (Hz) 528 594 660 704 792 880 990 1056
>   > * Middle C
>   >  Denotes  found in the fields
>   > 
>   > Theorem Summary
>   > 
>   > Frequency (Hz) Theorem Used For Proof
>   > 88 Theorem IVB, Gragg
>   > 176 Theorem IB, Gragg
>   > 352 Theorem IA, Theorem IVA, Hawkins
>   > 528 Theorem III, Hawkins
>   > 1056 Theorem II, Hawkins
>   > 
>   >
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   >                     Copyright 2004 by C. D. Gragg, All rights
>   reserved
>   > Appendix C
>   > Theorem T
>   > 
>   >    Trigonometry can be used to solve circular relationships for
>   inscribed and circumscribed regular  polygons for polygons of any
>   number of sides from 3 to infinity.
>   > 
>   >  Proof:                                      
>   >                                                   Where:  α =
>   3600          and n = number of sides    
>   >                                                   2n
>   >                                                           
>  
>                                                                 
>   cos α  = R1 
>   >                                                               R2
>   > 
>  
>                                                                    
>   R2 =   _1     Proving the Theorem
>   > Figure 1. Regular polygon with                  R1       cos α
>   >                any number of
sides                              
>   >                                                           
>   Further:   ( R2)2 =  (1)2    
>  
>                                                                    
>                                 ( R1)2      ( cos α)2
>   > Table of Some Common Polygons
>   > Figure (All are equiangular) Number of Equal Sides  Ratio of
>   Diameters and
>   > Radii Ratio of Areas, Diameters Squared, and Radii Squared
>   > Triangle (1)(4)         3       2.000 4      4.000
>   > Square (2)         4       1.414 2      2.000
>   > Pentagon         5       1.236          1.527
>   > Hexagon (3)          6       1.155 4/3   1.333
>   > Heptagon         7       1.110         1.232
>   > Octagon (4)         8       1.082         1.172
>   > Nonagon         9       1.064         1.132
>   > Decagon       10     & nbsp; 1.051         1.106
>   >        15       1.022         1.045
>   >        20       1.012         1.025
>   >        50       1.002         1.004
>   >      100       1.000         1.001
>   >      200       1.000         1.000
>   >        â^z       1.000         1.000
>   > (1) Theorem II, by Dr. Hawkins using Euclidian Geometry
>   > (2) Theorem III, by Dr. Hawkins using Euclidian Geometry 
>   > (3) Theorem IV, by Dr. Hawkins using Euclidian Geometry
>   > (4) Found in the Kekoskee/Mayville, Wisconsin Crop Circle
>   >              Formation July 9, 2003    
>  
>                                                                    
>          Copyright 2004 by C. D. Gragg, All rights reserved
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   >
>   > Appendix D
>   > 
>   > Short Bio
>   > 
>   >    My name is Dee Gragg.  I am a retired, mechanical engineer. 
My
>   career was spent in research, testing and evaluation. My main
areas
>   of research were automotive air bags, jet aircraft ejection seats
and
>   high speed rocket sleds.  I have published 33 technical papers as
>   either the principal author or a co-author.  They form a part of
the
>   body of literature in their respective fields.  
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   > 
>   >
>   >
>   >
>   >
>   > ---------------------------------
>   > Do you Yahoo!?
>   > Check out the new Yahoo! Front Page. www.yahoo.com
>   >
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>   >
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#769 From: rybo6 <rybo6@...>
Date: Tue Nov 16, 2004 6:56 pm
Subject: Re: Re: 10 or 26 dimesions?
os_jbug
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On Nov 16, 2004, at 10:46 AM, Paul Erlich wrote:
>  This has nothing to do with what I was talking about, but thanks.

Or so you think.

Obviously I think otherwise and that is why I posted the link.
Hyper-spatial dimension is 3D polyhedral-cells within polyhedral-cells.

The cube was first and currently most commonly used example of
hyper-spatial polyhedral-cell within a  polyhedral-cell.

The Synergetics info is linked to is the same set of cells within cells
scenario nd is dfeined by Fuller as Frequency.

   The Synergtics VE info is even closely related to cube since the VE is
just a truncated cube.

Rybo

#770 From: "Paul Erlich" <paul@...>
Date: Tue Nov 16, 2004 6:59 pm
Subject: Re: 10 or 26 dimesions?
emotionaljou...
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--- In Polytopia@yahoogroups.com, rybo6 <rybo6@u...> wrote:
> On Nov 16, 2004, at 10:46 AM, Paul Erlich wrote:
> >  This has nothing to do with what I was talking about, but thanks.
>
> Or so you think.
>
> Obviously I think otherwise and that is why I posted the link.

OK. Can you fill me in on the connection?

> Hyper-spatial dimension is 3D polyhedral-cells within polyhedral-
cells.
>
> The cube was first and currently most commonly used example of
> hyper-spatial polyhedral-cell within a  polyhedral-cell.
>
> The Synergetics info is linked to is the same set of cells within
cells
> scenario nd is dfeined by Fuller as Frequency.
>
>   The Synergtics VE info is even closely related to cube since the
VE is
> just a truncated cube.
>
> Rybo

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