This group has been pretty good about not posting off topic over the
last 6 years (wow Polytopia for 6 years!)-  but still a few slip
through the cracks.

I used to give warnings-  especially if I knew that the member has
made legitimate contributions to discussion here.
But now I'm just going to remove and ban OT posters at will.

I also deny membership to anyone who can't say anything interesting
when requesting membership.  So to anyone who refers a friend to this
group:
make sure to tell them to explain a little bit about their interests
and how it relates to the interests of Polytopia.

Thanks to all members who keep Polytopia alive and interesting-
I know I havn't been one of them for a while.  Maybe I should get it
going...

-Dan

Anyone familiar with Theodorus of Cyrene's construction
of square roots with a spiraling manifold of right triangles?

http://courses.wcupa.edu/jkerriga/Lessons/A%20Lesson%20on%20Spirals.html

http://en.wikipedia.org/wiki/Theodorus_of_Cyrene

He stopped at sqrt(17).
Did he know that the next triangle would overlap or
did the "school bell ring" ?

Could you prove weather it does or doesn't overlap by adding the next
triangle either by a precise compass and ruler construction or with a
more advanced algebraic or trigonometric argument?



-Dan

--- In Polytopia@yahoogroups.com, "Dan" <polychoron@...> wrote:

> Anyone familiar with Theodorus of Cyrene's construction
> of square roots with a spiraling manifold of right triangles?

> http://courses.wcupa.edu/jkerriga/Lessons/A%20Lesson%20on%20Spirals.html

> http://en.wikipedia.org/wiki/Theodorus_of_Cyrene

> He stopped at sqrt(17).
> Did he know that the next triangle would overlap or
> did the "school bell ring" ?

> Could you prove weather it does or doesn't overlap by adding the next
> triangle either by a precise compass and ruler construction or with a
> more advanced algebraic or trigonometric argument?

> -Dan

 n 
 ATAN(1/(SQRT(n))) 
 Degrees 
 Cumulative Angles 
 Degrees 
 1 
 0.7853981634 radians 
 45° 
 0.7853981634 radians 
 45° 
 2 
 0.61547970867 radians 
 35.264389683° 
 1.4008778721 radians 
 80.264389683° 
 3 
 0.5235987756 radians 
 30° 
 1.9244766477 radians 
 110.26438968° 
 4 
 0.463647609 radians 
 26.565051177° 
 2.3881242567 radians 
 136.82944086° 
 5 
 0.42053433528 radians 
 24.094842552° 
 2.808658592  radians 
 160.92428341° 
 6 
 0.38759668666 radians 
 22.207654299° 
 3.1962552786 radians 
 183.13193771° 
 7 
 0.36136712391 radians 
 20.704811055° 
 3.5576224025 radians 
 203.83674877° 
 8 
 0.33983690945 radians 
 19.471220634° 
 3.897459312  radians 
 223.3079694 ° 
 9 
 0.3217505544 radians 
 18.434948823° 
 4.2192098664 radians 
 241.74291822° 
10 
 0.30627736917 radians 
 17.548400614° 
 4.5254872355 radians 
 259.29131884° 
11 
 0.29284277173 radians 
 16.778654881° 
 4.8183300073 radians 
 276.06997372° 
12 
 0.2810349015 radians 
 16.102113752° 
 5.0993649088 radians 
 292.17208747° 
13 
 0.27054976298 radians 
 15.501359567° 
 5.3699146717 radians 
 307.67344704° 
14 
 0.2611574109 radians 
 14.963217433° 
 5.6310720826 radians 
 322.63666447° 
15 
 0.25268025514 radians 
 14.477512186° 
 5.8837523378 radians 
 337.11417666° 
16 
 0.24497866313 radians 
 14.036243468° 
 6.1287310009 radians 
 351.15042012° 
17 
 0.23794112483 radians 
 13.633022225° 
 6.3666721257 radians 
 364.78344235° 

You're right — it is an overlap!

Thanks for that table Alan!
My thoughts are that Theodorus first constructed it with compass and
ruler- which gave the required precision without much difficulty.
Then since he knew overlap would occur after sqrt(17) he could stop
there when making a freehand sketch for students.  They didn't know
about arctangent did they?

-Dan


--- In Polytopia@yahoogroups.com, "Alan Michelson"
<amichelson2002@...> wrote:
>
>
> --- In Polytopia@yahoogroups.com, "Dan" <polychoron@> wrote:
>
> > Anyone familiar with Theodorus of Cyrene's construction
> > of square roots with a spiraling manifold of right triangles?
>
> >
> http://courses.wcupa.edu/jkerriga/Lessons/A%20Lesson%20on%20Spirals.html
>
> > http://en.wikipedia.org/wiki/Theodorus_of_Cyrene
>
> > He stopped at sqrt(17).
> > Did he know that the next triangle would overlap or
> > did the "school bell ring" ?
>
> > Could you prove weather it does or doesn't overlap by adding the next
> > triangle either by a precise compass and ruler construction or with a
> > more advanced algebraic or trigonometric argument?
>
> > -Dan
>   n  ATAN(1/(SQRT(n)))  Degrees
> Cumulative Angles
> Degrees
>   1  0.7853981634 radians  45°  0.7853981634 radians  45°  2
> 0.61547970867 radians  35.264389683°  1.4008778721 radians
> 80.264389683°  3  0.5235987756 radians  30°  1.9244766477 radians
> 110.26438968°  4  0.463647609 radians  26.565051177°  2.3881242567
> radians  136.82944086°  5  0.42053433528 radians  24.094842552°
> 2.808658592  radians  160.92428341°  6  0.38759668666 radians
> 22.207654299°  3.1962552786 radians  183.13193771°  7
> 0.36136712391 radians  20.704811055°  3.5576224025 radians
> 203.83674877°  8  0.33983690945 radians  19.471220634°
> 3.897459312  radians  223.3079694 °  9  0.3217505544 radians
> 18.434948823°  4.2192098664 radians  241.74291822° 10
> 0.30627736917 radians  17.548400614°  4.5254872355 radians
> 259.29131884° 11  0.29284277173 radians  16.778654881°
> 4.8183300073 radians  276.06997372° 12  0.2810349015 radians
> 16.102113752°  5.0993649088 radians  292.17208747° 13
> 0.27054976298 radians  15.501359567°  5.3699146717 radians
> 307.67344704° 14  0.2611574109 radians  14.963217433°
> 5.6310720826 radians  322.63666447° 15  0.25268025514 radians
> 14.477512186°  5.8837523378 radians  337.11417666° 16
> 0.24497866313 radians  14.036243468°  6.1287310009 radians
> 351.15042012° 17  0.23794112483 radians  13.633022225°
> 6.3666721257 radians  364.78344235°
> You're right — it is an overlap!
>

--- In Polytopia@yahoogroups.com, "Dan" <polychoron@...> wrote:

> Thanks for that table Alan!
> My thoughts are that Theodorus first constructed it with compass and
> ruler- which gave the required precision without much difficulty.
> Then since he knew overlap would occur after sqrt(17) he could stop
> there when making a freehand sketch for students. They didn't know
> about arc-tangent did they?

• Trigonometry
• Inverse Trigonometric Functions

The history of trigonometry and of trigonometric functions may span about 4000 years.

I could have used the sines rather than the tangents, but I needed to start with the Isosceles Right Triangle as #1.

> -Dan

> > --- In Polytopia@yahoogroups.com, "Dan" <polychoron@> wrote:

> > > Anyone familiar with Theodorus of Cyrene's construction
> > > of square roots with a spiraling manifold of right triangles?

• > > > http://courses.wcupa.edu/jkerriga/Lessons/A%20Lesson%20on%20Spirals.html
• > > > http://en.wikipedia.org/wiki/Theodorus_of_Cyrene

> > > He stopped at sqrt(17).
> > > Did he know that the next triangle would overlap or
> > > did the "school bell ring" ?

> > > Could you prove weather it does or doesn't overlap by adding the next
> > > triangle either by a precise compass and ruler construction or with a
> > > more advanced algebraic or trigonometric argument?

> > > -Dan

I have a question--a comment to the original post stated that 17 is
the first number for which the even-odd concept used in the proof of
the irrationality of the square root of two cannot be generalized to
other primes, if I understood the statement. Can anyone elaborate on
this?

--john

--- In Polytopia@yahoogroups.com, John Chalmers <JHCHALMERS@...> wrote:

> I have a question—a comment to the original post stated that 17 is
> the first number for which the even-odd concept used in the proof of
> the irrationality of the square root of two cannot be generalized to
> other primes, if I understood the statement. Can anyone elaborate on
> this?

If it is a multiple of two,

then it is an even number,
else it is an odd number.

Obviously, that works if the prime number is two. See the Example proofs.

> —john

Thanks, I understand the example proof, but my question was whether
there is something unique about 17 and its square root as implied in
the post, other than the observation that the Root Spiral overlaps
after sqr(17).

Alan: Thanks--I was able to download McCabe's article on Theodorus's
methods and answer my question. The previous reference I had was
incorrect.

--John

--- In Polytopia@yahoogroups.com, John Chalmers wrote:

> Thanks, I understand the example proof, but my question was whether
> there is something unique about 17 and its square root as implied in
> the post, other than the observation that the Root Spiral overlaps
> after sqrt(17).

A Fermat prime is a Fermat number F_n = 2^^{(2^n)}+1 that is prime. For example, F₂ = 2^^{(2^2)}+1 = 17. Read up on constructible polygons and Geometric Construction. I believe that it was Gauss who made the connection!

I am a math/physics/electronics specialist as well as a very
spiritually in-tuned. Looks like the last discussions have been
numerology inclined.

I will do my best to participate.

I myself have been working on a special project for several years and
it is goin great!

I have also narrowed down my active groups so I can participate more
too.

I am looking for more members to help me verify my research; my
website is at:

http://www.freewebs.com/mrzeta/index.html

My project hopes to break the inter-dimensional communications
barrier to alternate dimensions, and communicate with other entities
there if they do indeed exist.

I could use group liason help, although I will try as well to keep
everyone informed. I really need people with excellent math skills
including vector differential calculus; not sure when this will come
into play if at all.

I will check out the group and thank you for lettin me in!

Mr Gray

--- In Polytopia@yahoogroups.com, "Mr Gray" <mrzeta7@...> wrote:

> I am a math/physics/electronics specialist as well as a very
> spiritually in-tuned. Looks like the last discussions have been
> numerology inclined.

> I will do my best to participate.

> I myself have been working on a special project for several years and
> it is going great!

> I have also narrowed down my active groups so I can participate more
> too.

> I am looking for more members to help me verify my research; my
> website is at:

> http://www.freewebs.com/mrzeta/

There is going to be a Great Pyramid replica in Japan. You remember Paolo Soleri, who designs cities in the form of Mega-Buildings, where you commute by elevator rather than by automobile!

> My project hopes to break the inter-dimensional communications
> barrier to alternate dimensions, and communicate with other entities
> there if they do indeed exist.

I believe that you saw the Stargate movie!

> I could use group liaison help, although I will try as well to keep
> everyone informed. I really need people with excellent math skills
> including vector differential calculus; not sure when this will come
> into play if at all.

> I will check out the group and thank you for letting me in!

> Mr Gray

Gregg Fleishman was on Life & Times yesterday. He's the guy with the polyhedra. "Shelters all made from beveled cubes." Notice how he uses language that lay-person Val Zavala "pretended to understand":

• 4-valent: "six squares related to the six spaces of a cube"
• 2-valent: "one to the square root of two, which relates to the forty-five degree slope of the roof panels"
• 3-valent: "eight triangles relate to the eight corners of the cube"

http://www.kcet.org/lifeandtimes/archives/200704/20070417.php

You might want to see his other stuff. By the way, he attended Play Mountain School, which his mom Phyllis founded!

--- In synergeo@yahoogroups.com, rybo6 <rybo6@...> wrote:

On Jun 16, 2006, at 10:32 PM, Alan Michelson wrote:
> TT] making
> > them practically useless as any kind of container.
>
> http://www.math.jmu.edu/~taal/CVstuff/simplicity.pdf

Alan, nice article and I like this quote "Simplicity is not simple."

...."Fleishman wants people to be able to build with
his panels without extensive training. "Making a
good system from only a few pieces which are not
too hard to machine or build is quite difficult," says
Fleishman, "Simplicity is not simple."..
.
Rybo

>
>

--- In Polytopia@yahoogroups.com, "Harold" <howard@...> wrote:
>
> Hey all,
>   I have been working on a 6v geodesic dome design that requires
> only six strut lengths. (Based on the icosahedron)
>
> Using this theory; any frequency dome would have the same number of
> different strut lengths as its frequency. 3v=3 different lengths,
> 4v=4 different lengths, 6v=6 different lengths, and so on.
>
> Has anyone ever heard of a design like this?

http://www.domeclimber.com/freqs.php

> The model I have built is beautiful and gives clear, contiguous,
> color coded patterns around the circumference of the dome. Once
> assembly is started; the builder just connects the colors to like
> colors most of the way around. It makes a six frequency as easy to
> build as a four frequency.
>
> One small drawback to this design is that the hemisphere (and each
> other cut off point) does not sit flat. This would not be a problem
> if you planned to build a dome of less than half sphere. Less than
> half sphere domes never sit flat anyway. This also would not be a
> problem for construction of complete spheres.
>
> I look forward to the feedback. If the layout already exists I would
> like to read up on it. If the layout does not exist, I would like to
> perfect this layout, and then share it with all of you!
>
> Thank you for your time,
> Harold
>

Hello Dome People,
I have a 22.68 Foot diameter 4-v Geodesic dome made of 1
inch conduit. This dome uses Hubs that were designed to speed and
simplify assembly. The new hubs allow easy construction, from the
bottom up, and require only one person to assemble. Detent pins
connect each strut to the hub. This dome was designed to fit a 35'
parachute. (I don't know why they call it "35 foot" it is only 22.68
feet diameter at the rim.)
I partially assembled this dome in a Redwood City California
park. After assembling the base ring and one row above that, the
Redwood City parks department manager came and asked me to remove it
from the park.
In order to further develop my Hub design, and mark my
parachute for "tent modifications" (doors), I need a place in
Redwood City to set up this dome tent. The tent fits into my 2007
Honda Accord. (Currently do not know weight of tent) The tent
required 840 feet of one inch conduit.
If anyone owns land that I could use, near Redwood City
California, and would like to see a 22.68 foot 4v geodesic dome
assembly and dis-assembly, please contact me at
Harold@ vwbiofuels.com  I would require eight hours, or less, from
arrival to departure (Time estimate includes lunch break and "Dome
enjoyment" time).

Thank you for your time, I look forward to all replies or comments.

Harold Westrich
P.S. I will be setting this tent up at American Cancer Society
functions near the end of May and must set it up at least once
before then, to determine assembly time and verify how they will
arrange tables.

Hi Alan

From what I see, there are 9 different triangles for the 5f triacon at
the domeclimber.com link. How many are there for Pete's 6f dome? There
must be a formula.

Dick

--- In Polytopia@yahoogroups.com, "Alan Michelson"
<amichelson2002@...> wrote:
>
> --- In Polytopia@yahoogroups.com, "Harold" <howard@> wrote:
> >
> > Hey all,
> >   I have been working on a 6v geodesic dome design that requires
> > only six strut lengths. (Based on the icosahedron)
> >
> > Using this theory; any frequency dome would have the same number of
> > different strut lengths as its frequency. 3v=3 different lengths,
> > 4v=4 different lengths, 6v=6 different lengths, and so on.
> >
> > Has anyone ever heard of a design like this?
>
> http://www.domeclimber.com/freqs.php
>
> > The model I have built is beautiful and gives clear, contiguous,
> > color coded patterns around the circumference of the dome. Once
> > assembly is started; the builder just connects the colors to like
> > colors most of the way around. It makes a six frequency as easy to
> > build as a four frequency.
> >
> > One small drawback to this design is that the hemisphere (and each
> > other cut off point) does not sit flat. This would not be a problem
> > if you planned to build a dome of less than half sphere. Less than
> > half sphere domes never sit flat anyway. This also would not be a
> > problem for construction of complete spheres.
> >
> > I look forward to the feedback. If the layout already exists I would
> > like to read up on it. If the layout does not exist, I would like to
> > perfect this layout, and then share it with all of you!
> >
> > Thank you for your time,
> > Harold
> >
>

--- In Polytopia@yahoogroups.com, "Dick Fischbeck" <dick_fischbeck@...> wrote:

> Hi Alan

> From what I see, there are 9 different triangles for the 5f triacon at
> the domeclimber.com link. How many are there for Pete's 6f dome? There
> must be a formula.

I went to id:A097108 - OEIS Search Results and I found "FORMULA: Satisfies a linear recurrence with characteristic polynomial (1+x^3)(1-x^3)^3."

In Desert Domes - Dome Formulas:

• dome radius = strut length/strut factor
• strut length = dome radius * strut factor

> Dick

> --- In Polytopia@yahoogroups.com, "Alan Michelson"
> amichelson2002@ wrote:

> > --- In Polytopia@yahoogroups.com, "Harold" <howard@> wrote:

> > > Hey all,
> > > I have been working on a 6v geodesic dome design that requires
> > > only six strut lengths. (Based on the icosahedron)

> > > Using this theory; any frequency dome would have the same number of
> > > different strut lengths as its frequency. 3v=3 different lengths,
> > > 4v=4 different lengths, 6v=6 different lengths, and so on.

> > > Has anyone ever heard of a design like this?

> > http://www.domeclimber.com/freqs.php

> > > The model I have built is beautiful and gives clear, contiguous,
> > > color coded patterns around the circumference of the dome. Once
> > > assembly is started; the builder just connects the colors to like
> > > colors most of the way around. It makes a six frequency as easy to
> > > build as a four frequency.

> > > One small drawback to this design is that the hemisphere (and each
> > > other cut off point) does not sit flat. This would not be a problem
> > > if you planned to build a dome of less than half sphere. Less than
> > > half sphere domes never sit flat anyway. This also would not be a
> > > problem for construction of complete spheres.

> > > I look forward to the feedback. If the layout already exists I would
> > > like to read up on it. If the layout does not exist, I would like to
> > > perfect this layout, and then share it with all of you!

> > > Thank you for your time,
> > > Harold

Hi all:

I'm interested in figuring out a general algorithm for mapping
2D images onto adjacent triangles of geodesic surface (for different
frequency geodesics); rather than reinvent that wheel, is there an
algorithm (or better yet POV-Ray code) and/or a pre-processor to map a
2D image onto a portion of a geodesic surface? I'm guessing that the
general approach is to use a good 2D to spherical map with minimum
distortion, and then apply sphere to geodesic triangle mapping... Any
help with part or all of this would be appreciated! Ideally, I'd like
to be able to define the orientation, and scaling for a 2D
(rectangular) bitmap (e.g. jpg) file and then apply it to a geodesic
surface (specifying the frequency (using spherical subdivision of an
icosahedron) and render with POVray.

Thanks in advance! :-)

Hey guys,

I had made a whole bunch of animations of as many polychora as I could
get net data (or figure it out) for back in 2004.  I've finally found
them, and I'm uploading as many of them as I can to youtube.  It's not
the best resolution - but it's all I can do at the moment. :)  So,
you'll see the 5-Cell, 8-Cell, 16-Cell, 24-Cell, 120-Cell, 600-Cell,
and a few bitruncates, cantellated, and cantitruncated stuff.  More to
come including a few segmentochora, a glome, and some duoprisms.  Hope
you like 'em!

- Burt (aka Tuvel)

And then, like an idiot, i forgot to include the site addy!

http://www.youtube.com/user/WildStar2002

  - Burt (aka Tuvel)

Burt~

Thank you so much for posting this!!!  I have a 7 year old son (just
turned 7 on Sunday).  He is the reason I belong to this group...his
favorite things in the world are the Platonic and Archimedean solids--
all things geometric fascinate him.

He will spend HOURS watching!!!!  Thank you!

Peace~ Jenn


--- In Polytopia@yahoogroups.com, "wildstar2002" <wildstar@...> wrote:
>
> And then, like an idiot, i forgot to include the site addy!
>
> http://www.youtube.com/user/WildStar2002
>
>  - Burt (aka Tuvel)
>

Hey, Rybo: This looks like the 6 Gr C's of the Icosahedron. Notice the six colors that correspond to the six colors of the Vector Flexor!

Hey, Tim Tyler: That looks like tri-axial weaving!

--- In Polytopia@yahoogroups.com, "Mitch Powell" <mitch@...> wrote:

I don't think this answers your question, but it sounded like something
close enough for me to show off my icosahedral creation.

Does that give you any ideas? My idea involves stitches on canvas. Each
"equilateral" triangle is made up of five stitches (2-4-6-4-2). The
resulting needlework piece is then constructed into its icosahedron and
given a string to hang by.

A slightly heavy Christmas ornament.

OR, do whatever you want with the template. Make little ceramic tiles
and create a large sculpture out of it.

I have other models with different surface patterns. I'll find them and
post them to the album I just made if anyone is interested.

Mitch

--- In Polytopia@yahoogroups.com, "bruce1618r" Bruce@ wrote:

> Hi all:

> I'm interested in figuring out a general algorithm for mapping
> 2D images onto adjacent triangles of geodesic surface (for different
> frequency geodesics); rather than reinvent that wheel, is there an
> algorithm (or better yet POV-Ray code) and/or a pre-processor to map a
> 2D image onto a portion of a geodesic surface? I'm guessing that the
> general approach is to use a good 2D to spherical map with minimum
> distortion, and then apply sphere to geodesic triangle mapping... Any
> help with part or all of this would be appreciated! Ideally, I'd like
> to be able to define the orientation, and scaling for a 2D
> (rectangular) bitmap (e.g. jpg) file and then apply it to a geodesic
> surface (specifying the frequency (using spherical subdivision of an
> icosahedron) and render with POV-ray.

> Thanks in advance! :-)

--- End forwarded message ---

Jenn,

I can't tell you how much it pleases me that your son will like these
animations. :-)  Thank you so much for saying so - and I think we can
all look forward to some amazing things from him in the future!

- Burt

--- In Polytopia@yahoogroups.com, "jenniyoungkurtz"
<jenniyoungkurtz@...> wrote:
>
> Burt~
>
> Thank you so much for posting this!!!  I have a 7 year old son (just
> turned 7 on Sunday).  He is the reason I belong to this group...his
> favorite things in the world are the Platonic and Archimedean solids--
> all things geometric fascinate him.
>
> He will spend HOURS watching!!!!  Thank you!
>
> Peace~ Jenn
>
>

I uploaded two new photo albums, Geodesic Sphere and Zome Constructions, to the
Polytopia
photos.  The Geodesic sphere is six frequency made from coffee stirrers and
twist ties.  I was
trying to find out how large a sphere I could make with coffee stirrers.  This
is about the
limit.  Any larger and the weight of the sphere itself causes the sphere to
dimple as the
surface gets flatter.  My next attempt will be a double hulled version for more
strength.

The Zome constructions I think of as quick sketches.

Allen

Allen~   Many thanks for these stunning pics!  My little guy (he just turned 7!) wants to know if he can print them out so he can try to build the Zome creations.    He is so thrilled each time there is a new email from this group--esp. when there are pictures!   Peace~  Jenn

__________________________________________________
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The result generalizes to any n-dimensional simplex in the obvious way. If the set of vertices of a simplex is v₀,...,v_n, then considering the vertices as vectors, the centroid is C = \frac{1}{n+1}\sum_{i=0}^n v_i.

--- In Polytopia@yahoogroups.com, Adrian Rossiter <adrian_r@...> wrote:

> Alan is right about this number being related to dimension. In
> an n-dimensional pyramid the centroid is at height 1/(n+1)

> https://nrich.maths.org/discus/messages/67613/70082.html

> This means that the dual of a regular symplex making vertex
> to cell contact will have 1 - 1/(n+1) of its height equal
> to 1/(n+1) of the larger simplex's height, giving a large
> to small ratio of
> scale ratio = (1 - 1/(n+1)) / (1/(n+1))
> = ((n+1) - 1)
> = n

> Adrian.
> --
> Adrian Rossiter Adrian@... Home:
> http://antiprism.com/adrian