----- 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@...> 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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Check out the new Yahoo! Front Page. www.yahoo.com

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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@...> 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.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

---

Do you Yahoo!?
Check out the new Yahoo! Front Page. www.yahoo.com

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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!

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)

----- Original Message -----

From: rybo6

To: Polytopia@yahoogroups.com

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)

----- Original Message -----

From: Larry Rafey

To: Polytopia@yahoogroups.com

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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----- Original Message -----

From: Larry Rafey

To: Polytopia@yahoogroups.com

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

From: John Berglund

To: Polytopia@yahoogroups.com

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 -----

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@...> 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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----- Original Message -----

From: John Berglund

To: Polytopia@yahoogroups.com

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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----- Original Message -----

From: John Berglund

To: Polytopia@yahoogroups.com

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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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
> > >
> > >

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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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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>
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Anti-bush campagin 2004. Bush must go!

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!

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>

----- 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
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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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!

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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Service.

----- Original Message -----

From: Paul Erlich

To: Polytopia@yahoogroups.com

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
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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

--- 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