100 equidistant vertices dome -- is it possible?

[mireazma]

Hello everybody!

If at the beginning I thought it was a child's play after ~2 weeks it seems like I'm trying to build a time machine or something.
I'm a programmer (barely) and I'm working on this P.O.S. (S for software) which has a big nasty issue that I can't get past no matter what path I choose to follow.
I accidentally ended up here and noticed the subject is just a sibling of mine so I was like finding a soda vending machine in the middle of a desert.

OK, the matter is fully described in the title, as well: I need a dome (preferably a little under 1/2 but not over 1/2) that has 100 equidistant vertices.
I don't even know whether it's geometrically possible.

Does anybody have an idea?

[Hector Alfredo Hernández Hdez.]

do you are talking about 100 vertices equidist of one center or equidistant betwen neibor pairs?

2013/10/10 mireazma <mire...@gmail.com>

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[Hector Alfredo Hernández Hdez.]

or both at same time?

2013/10/10 Hector Alfredo Hernández Hdez. <hecto...@gmail.com>

[Gerry in Quebec]

Good question by Hector.

 

Mireazma, how about half a honeycomb sphere with 105 vertices? It has 150 struts and 46 flat faces, of which 6 are regular pentagons and the rest irregular hexagons of two types. Every vertex is equidistant to the three other vertices immediately surrounding it, but not all vertices are the same distance from the polyhedron's centre. All struts are the same length.

 

This is half an icosahedral Goldberg polyheron, I[3,0]. Taff Goch did a SketchUp model of the full sphere many moons ago:

 

http://sketchup.google.com/3dwarehouse/details?mid=73df5cebd07d1f182137e32e8ebf2b1&prevstart=0

 

The base of the half-sphere doesn't lie flat, though. It zig-zags like half a stylized Easter egg.

 

- Gerry in Quebec

[mireazma]

Thanks for replying.

I'm talking about both equidistant from the center and from neighboring hubs.
I searched on the web in the mean time and I came up with... nothing. I was looking for next figures in the pattern: tetrahedron, octahedron, icosahedron, ... meaning they are regular, equilateral triangle faceted. I'm thinking there must be a continuation maybe not 16 as in "4, 8, 12, 16" but 24 or some other higher number, much like prime numbers which are as rare as they grow larger but there exist; so the figure, even if it has much more vertices, what's the closest number to 200 for a spherical (100 for 1/2).

[Hector Alfredo Hernández Hdez.]

what kind of faces are you searching?

2013/10/10 mireazma <mire...@gmail.com>

Thanks for replying.

I'm talking about both equidistant from the center and from neighboring hubs.
I searched on the web in the mean time and I came up with... nothing. I was looking for next figures in the pattern: tetrahedron, octahedron, icosahedron, ... meaning they are regular, equilateral triangle faceted. I'm thinking there must be a continuation maybe not 16 as in "4, 8, 12, 16" but 24 or some other higher number, much like prime numbers which are as rare as they grow larger but there exist; so the figure, even if it has much more vertices, what's the closest number to 200 for a spherical (100 for 1/2).

--

[Hector Alfredo Hernández Hdez.]

sorry,

what kind of faces are you lookingfor?

2013/10/10 Hector Alfredo Hernández Hdez. <hecto...@gmail.com>

what kind of faces are you searching?

[Gerry in Quebec]

The icosahedron, with its 20 equilateral faces, 12 identical vertices, and 30 edges of equal length, is the end of the line. There are no other polyhedra that meet your criteria. Requiring that all faces be equilateral triangles and that all vertices be equidistant from the spherical centre stops you dead in your tracks. Lifting either requirement gives you options.

[Blair Wolfram]

The next level of uniformity may be along the lines of Joe Clinton's Equal Central Angle Conjecture.

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Blair F. Wolfram
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thedo...@domeincorporated.com

http://www.hurricanedomes.com

888-DOME-INC or 612-333-3663

[Gerry in Quebec]

But if we're looking at a dome with around 100 vertices, most of the "faces" of a Goldberg-Clinton "equal central angle" thingie would not be flat. From the point of view of construction, it's already bad enough that these structures aren't triangulated (like building with wet noodles). The twisted surfaces would be really annoying.

[Dick Fischbeck]

What is the goal? A simple dome?

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[Dick Fischbeck]

An edge-built hex-pent dome, ok. Wet noodle.  A face built dome, not bad. Slight variation from flat doesn't count for much, I'm guessing. How far off from flat we talking.  Then again, some domes are built with identical parts, if that helps. Mass production can be valuable.

[Ken G. Brown]

Here are some links I've saved over the years rearding the topic and related things. I see a lot of them are dead now and I don't have time to devote to fixing them. If anyone can find updated links, please post something better than this even if from Internet archives. Maybe these links will lead to something interesting. Hope you find one or two worthwhile!

http://www.ics.uci.edu/~eppstein/projects/pairs/

http://local.wasp.uwa.edu.au/~pbourke/geometry/spherepoints/

http://www.chiark.greenend.org.uk/~sgtatham/polyhedra/

http://www.mathpages.com/home/kmath005.htm

http://astronomy.swin.edu.au/~pbourke/

http://memeticdrift.net/tau/ptdistro/13nodes.htm

http://www.ogre.nu/sphere.htm

http://space.mit.edu/home/tegmark/icosahedron.html

Sphere Tilings/Tesselations

Tammes Problem http://en.wikipedia.org/wiki/Tammes_problem

Tessellation 

http://mathworld.wolfram.com/Tessellation.html

TesselSphere

http://myweb.tiscali.co.uk/oaktree/nshea/index.html

http://www.geocities.com/nicholasshea1/index.html 

http://spaces.live.com/editorial/en-us/render/msngroupsclosure.htm

*** sphere distribution problems http://www.ogre.nu/sphere.htm

3D Molecular Fractals http://ourworld.compuserve.com/homepages/robert_conroy/3dmolecu.htm

3D Packing of Sphere-Polyhedra 2.2 http://smartimtech.com/packingmink/appletpackingmink.html

Amazon.com: The Pursuit of Perfect Packing, Second Edition (9781420068177): Denis Weaire, Tomaso Aste: Books http://www.amazon.com/Pursuit-Perfect-Packing-Second/dp/1420068172/

CAGE - www.mathemati…contents.html http://www.mathematik.uni-bielefeld.de/~senkel/CAGE/contents.html

Career Networking - Companies http://www.lycomm.com.sg/career/company.html

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) http://www.dimacs.rutgers.edu/

Daniel H. Huson http://www.mathematik.uni-bielefeld.de/~huson/

Daniel Huson: Papers and Software http://www.mathematik.uni-bielefeld.de/~huson/papers.html

Geometry Dodecahedron tiling - Rafiki http://www.codefun.com/Geometry_tile1.htm

Gunnar Brinkmann http://www.mathematik.uni-bielefeld.de/~gunnar/

index.html for Gosset http://www.research.att.com/~njas/gosset/index.html

Math Forum: Macintosh Software http://mathforum.org/software.mac.html

Neil Sloane (home page) http://www.research.att.com/~njas/

Olaf Delgado's home page http://www.mathematik.uni-bielefeld.de/~delgado/

Programming with OpenGL: Advanced Rendering http://www.sgi.com:80/products/software/opengl/advanced97/notes/notes.html

Rigidity of Sphere Packings A Linear Programming Approach http://atom.princeton.edu/donev/Packing/LPRigidity/

Some Spherical Tilings - http://cs.smu.ca/~dawson/images4.html http://cs.smu.ca/~dawson/images4.html

Spacelike tessellations of tetrahedrons http://www.pimeson.com/

Tuesday afternoon, 19 March 1996 http://flux.aps.org/meetings/BAPSMAR96/tocG.html#SG23.003

www.cs.hmc.ed...noframes.html http://www.cs.hmc.edu:80/~ivl/art-main-noframes.html

Zonotiles http://www.mathematik.uni-bielefeld.de/~delgado/zono.html

Ken G. Brown

.

[TaffGoch]

(Nice list, Ken. I'll enjoy following those URLs.)

___________________

mireazma,

The ONLY tessellations ("above" the icosahedron) that... 

(1) cover a convex surface,

 - AND - 

(2) employ ONLY equilateral triangles...

... are Lobel Frames,...

... BUT, but they don't produce spheres:

http://www.equilatere.net/frame.php?lang=en&page=en/fondamentals/home.php

http://sketchup.google.com/3dwarehouse/details?mid=d707c5785e8945e15267ed5eeb5c0a1c

-Taff

[Hector Alfredo Hernández Hdez.]

The ball done with 20 hexaedrons and 12 petagones is somethig like asking.

dont forget that the faces can be no regular poligones...

2013/10/10 TaffGoch <taff...@gmail.com>

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[Gerry in Quebec]

 

Do they really cover a convex surface? Or do you mean a surface that is mostly convex?

- Gerry

[TaffGoch]

Gerry,

Hmm, well, it's not flat. I can say that much.

(Yes, indeed, I too can see concave dihedrals between some of the triangles.)

-Taff

[Ken G. Brown]

Don't get lost! But we expect a full report. :)

I've found it pretty interesting to follow up on any bibliographies found in papers I come across. 

And for sure, please keep track of what you find so I can update the links and add to. 

Ken,

from my iPhone

On 2013-10-10, at 17:54, TaffGoch <taff...@gmail.com> wrote:

(Nice list, Ken. I'll enjoy following those URLs.)

___________________

mireazma,

The ONLY tessellations ("above" the icosahedron) that... 

(1) cover a convex surface,

 - AND - 

(2) employ ONLY equilateral triangles...

... are Lobel Frames,...

... BUT, but they don't produce spheres:

<2-2-2-3.jpg>

http://www.equilatere.net/frame.php?lang=en&page=en/fondamentals/home.php

http://sketchup.google.com/3dwarehouse/details?mid=d707c5785e8945e15267ed5eeb5c0a1c

-Taff

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[mireazma]

This looks like the most friendly community I've ever been into, hands down.

Nifty stuff in the thread, I'll make it all a bookmark for future reference.
I don't need to physically build a dome (which I was hoping could help a bit) so if it comes to drop some constraints, all I'm allowed by my project to loosen is the number of hubs and the cut plane, say 100 - 200 hubs and 1/4 - 1/2 but I suspect Gerry crossed the t's -- I'm stuck.

I'll do some more research with the pencil and paper to eliminate the last left overs in my mind and then I'll call it a day.

Big Thanks to all of you.

[Hector Alfredo Hernández Hdez.]

Same length struth is a big restrictión, instead of we are going  suggest

to use Gerry designs.

See you.

2013/10/11 mireazma <mire...@gmail.com>

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