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Polytopia

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• Members: 410
• Category: Geometry
• Founded: Dec 10, 2001
• Language: English

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#1543 From: "Dan" <polychoron@...>
Date: Fri Nov 9, 2007 11:10 pm
Subject: OT posters will be banned immediately

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 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

#1544 From: "Dan" <polychoron@...>
Date: Fri Nov 9, 2007 11:21 pm
Subject: Theodorus and Square Roots

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 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

#1545 From: "Alan Michelson" <amichelson2002@...>
Date: Sat Nov 10, 2007 1:34 am
Subject: Re: Theodorus and Square Roots

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 --- In Polytopia@yahoogroups.com, "Dan" 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!

#1546 From: "Dan" <polychoron@...>
Date: Tue Nov 13, 2007 3:33 pm
Subject: Re: Theodorus and Square Roots

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

#1547 From: "Alan Michelson" <amichelson2002@...>
Date: Tue Nov 13, 2007 6:40 pm
Subject: Re: Theodorus and Square Roots

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 --- In Polytopia@yahoogroups.com, "Dan" 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? 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" wrote: > > > Anyone familiar with Theodorus of Cyrene's construction> > > of square roots with a spiraling manifold of right triangles? > > > 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

#1548 From: John Chalmers <JHCHALMERS@...>
Date: Tue Nov 13, 2007 7:48 pm
Subject: Re: Re: Theodorus and Square Roots

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 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

#1549 From: "Alan Michelson" <amichelson2002@...>
Date: Tue Nov 13, 2007 8:28 pm
Subject: Re: Theodorus and Square Roots

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

#1550 From: John Chalmers <JHCHALMERS@...>
Date: Tue Nov 13, 2007 8:36 pm
Subject: Re: Re: Theodorus and Square Roots

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 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).

#1551 From: John Chalmers <JHCHALMERS@...>
Date: Tue Nov 13, 2007 8:47 pm
Subject: Re: Re: Theodorus and Square Roots 2

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 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

#1552 From: "Alan Michelson" <amichelson2002@...>
Date: Wed Nov 14, 2007 1:14 am
Subject: Fermat Prime

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 --- 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 Fn = 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!

#1553 From: "Mr Gray" <mrzeta7@...>
Date: Sun Nov 25, 2007 4:44 am
Subject: Greetings

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 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

#1554 From: "Alan Michelson" <amichelson2002@...>
Date: Mon Nov 26, 2007 1:03 am
Subject: Re: Greetings, this is StarGate-1!

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 --- In Polytopia@yahoogroups.com, "Mr Gray" 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

#1555 From: Alan Michelson <amichelson2002@...>
Date: Fri Feb 1, 2008 6:49 pm
Subject: Re: Re: [synergeo] Re: ART YOU CAN SIT ON

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 COMING today @ 12 noon90.7 KPFK Radio Interview the Sound Exchange with Jay KugelmanAlan Michelson wrote: 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 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 withhis panels without extensive training. "Making agood system from only a few pieces which are nottoo hard to machine or build is quite difficult," saysFleishman, "Simplicity is not simple."...Rybo>> - XXX - Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now.

Date: Tue Feb 5, 2008 10:42 am
Subject: A General Theory of Particles and Forces

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 For those who are interested...   Visit this website: http://home.att.net/~bob.rutkiewicz/physics.htm   However, no mathematical equations found on this site... Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now.

#1557 From: "Alan Michelson" <amichelson2002@...>
Date: Fri Apr 4, 2008 3:20 am
Subject: Re: 6V Dome With only 6 strut lengths!?!? Possible? don't know?

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 --- In Polytopia@yahoogroups.com, "Harold" 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 >

#1558 From: "Harold" <howard@...>
Date: Fri May 2, 2008 10:40 pm
Subject: Please Help ! ! I need a space. Would you like to see a 4v geodesic dome?

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 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.

#1559 From: "Dick Fischbeck" <dick_fischbeck@...>
Date: Wed Aug 6, 2008 9:30 pm
Subject: Re: 6V Dome With only 6 strut lengths!?!? Possible? don't know?

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

#1560 From: "Alan Michelson" <amichelson2002@...>
Date: Thu Aug 7, 2008 1:14 am
Subject: Re: 6V Dome With only 6 strut lengths!?!? Possible? don't know?

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 --- In Polytopia@yahoogroups.com, "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." dome radius = strut length/strut factorstrut length = dome radius * strut factor> Dick> --- In Polytopia@yahoogroups.com, "Alan Michelson"> amichelson2002@ wrote:> > --- In Polytopia@yahoogroups.com, "Harold" 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

#1561 From: "bruce1618r" <bruce@...>
Date: Mon Oct 13, 2008 5:28 pm
Subject: POVray algorithm to map 2D images onto adjacent triangles of a geodesic surface

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 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! :-)

#1562 From: "Mitch Powell" <mitch@...>
Date: Mon Oct 13, 2008 6:23 pm
Subject: Re: POVray algorithm to map 2D images onto adjacent triangles of a geodesic surface

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 I don't think this answers your question, but it sounded like something close enough for me to show off my icosahedral creation:http://tech.ph.groups.yahoo.com/group/Polytopia/photos/view/4133?b=1 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" 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 POVray.> > Thanks in advance! :-)>

#1563 From: "wildstar2002" <wildstar@...>
Date: Mon Oct 13, 2008 8:16 pm

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 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)

#1564 From: "wildstar2002" <wildstar@...>
Date: Mon Oct 13, 2008 8:26 pm
Subject: Re: 4D Polychora on youtube

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 And then, like an idiot, i forgot to include the site addy! http://www.youtube.com/user/WildStar2002 - Burt (aka Tuvel)

#1565 From: "jenniyoungkurtz" <jenniyoungkurtz@...>
Date: Mon Oct 13, 2008 10:37 pm
Subject: Re: 4D Polychora on youtube

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

#1566 From: "Alan Michelson" <a.michelson@...>
Date: Tue Oct 14, 2008 4:12 am
Subject: Hey, Rybo & Tim Tyler!

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 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" wrote:I don't think this answers your question, but it sounded like somethingclose 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). Theresulting needlework piece is then constructed into its icosahedron andgiven a string to hang by.A slightly heavy Christmas ornament.OR, do whatever you want with the template. Make little ceramic tilesand create a large sculpture out of it.I have other models with different surface patterns. I'll find them andpost 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 ---

#1567 From: ron nor <macroron@...>
Date: Tue Oct 14, 2008 3:37 pm
Subject: OT: My list of Yahoo Groups :)

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 Hi everyone, Hope this is ok to post here... You can see a list of my groups on Grouply at the link below. Maybe you'll find some you want to join. ron Here's the link: http://www.grouply.com/register.php?tmg=1091148&vt=10061170 ==================== This message was posted by a fellow group member who uses Grouply instead of email to access this group. Grouply blocks additional invitations from being sent to this group by anyone for 30 days. Group owners can permanently block future invitations. For more on how Grouply maintains privacy and protects you, see http://blog.grouply.com/protect/ .

#1568 From: "wildstar2002" <wildstar@...>
Date: Tue Oct 14, 2008 4:57 pm
Subject: Re: 4D Polychora on youtube

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 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" 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 > >

#1569 From: "phi16" <allenlubow@...>
Date: Sun Oct 19, 2008 4:22 pm
Subject: Geodesic Sphere & Zome Constructions

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 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

#1570 From: jennifer kurtz <jenniyoungkurtz@...>
Date: Mon Oct 20, 2008 11:13 pm
Subject: Re: Geodesic Sphere & Zome Constructions

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 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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#1571 From: Allen Lubow <allenlubow@...>
Date: Wed Oct 22, 2008 12:49 am
Subject: Re: Geodesic Sphere & Zome Constructions

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Jenn,

I'm glad your son enjoyed them. Of course you can print them out. They were put there to share. :)

Allen

On Oct 20, 2008, at 7:13 PM, jennifer kurtz <jenniyoungkurtz@...> wrote:

 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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#1572 From: "Alan Michelson" <a.michelson@...>
Date: Fri Dec 5, 2008 12:52 am
Subject: Re: simplex

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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,...,vn, 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 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

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