Catch-all term for any interesting polytope

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Catch-all term for any interesting polytope

Postby ubersketch » Wed Dec 06, 2017 10:46 pm

In an attempt to create a catch-all term for any interesting polytope I created the term demiuniform, now renamed symmetroform. A symmetroform polytope is one that has less vertex-transitive groups than it has vertices and its elements must be symmetroform. Symmetroform polytopes include uniform polytopes, johnson solids, and perhaps scaliform. Tell me what you think. I half think somebody has already created this term on this forum but I'm posting this anyways.
Axioms:
1 - Every symmetroform polytope's dual is symmetroform.
2 - Every symmetroform polytope's conjugate is symmetroform.
3 - Every self-dual polytope is symmetroform.
4 - Every self-conjugate polytope is symmetroform.
5 - Every polytope with congruent elements are symmetroform.
6 - Every polytope with symmetroform elements is symmetroform.
7 - Every stellation of a symmetroform polytope must be symmetroform.
8 - Every polytope circumscribable on a hypersphere/orb is symmetroform.
Last edited by ubersketch on Wed Dec 13, 2017 11:29 pm, edited 6 times in total.
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Re: Catch-all term for any interesting polytope

Postby Mercurial, the Spectre » Fri Dec 08, 2017 3:42 pm

Hmm... if you mean vertex-uniform polytopes, you can consider other subgroups that would still define it as a vertex-uniform (isogonal) polytope.
Take a hexagonal prism with symmetry D6h (order 24). It is uniform, and it has 12 vertices. Now list all the possible vertex-uniform groups: D6h (24), D6 (12), D3h (12), and D3d (12). You have four possible options, so it is demiregular.
The tetrahedron, a Platonic solid with 4 vertices, has these vertex-uniform groups: Td (24), T (12), D2d (8), D2 (4), and S4 (4) for a total of 5. Since it is greater than the number of vertices, it is not demiregular.

In fact, a regular polygon is always demiregular since there are at most 3 groups. Here is a list:
Triangle (3 vertices): D3 (6), S3 (3), demiregular
Square (4 vertices): D4 (8), S4 (4), D2 (4), demiregular
Pentagon (5 vertices): D5 (10), S5 (5), demiregular
Hexagon (6 vertices): D6 (12), S6 (6), D3 (6), demiregular
Octagon (8 vertices): D8 (16), S8 (8), D4 (8), demiregular

If you are talking about demiuniform duals, take the cube with 8 vertices. You have Oh (48), O (24), Th (24), D4h (16), D4 (16), C4h (8), and D2d (8) for a total of 7. The octahedron with 6 vertices is its dual. The cube is demiuniform, but the octahedron is not. Therefore, the postulate isn't true.

Besides, the definition of an interesting polytope is purely based on human understanding.
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Re: Catch-all term for any interesting polytope

Postby ubersketch » Fri Dec 08, 2017 6:26 pm

That's not what I mean by vertex transitive group.
https://en.wikipedia.org/wiki/Isogonal_ ... rm_figures
This article sums up what vertex transitive group means adequately.
The reason why I came up with demiuniformity was to create a group of polytopes that was very inclusive but not too inclusive, excluding an endless amount of polytopes with uninteresting properties.
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Re: Catch-all term for any interesting polytope

Postby Klitzing » Sat Dec 09, 2017 12:01 am

In order to see what is realy going up here, one has to contrast your "demi-uniformity" definition with the well-known definition of "uniformity".

Uniformity requires
  • symmetry applies transitively on all vertices (i.e. all vertices are alike)
  • the elements of uniform polytopes have to be uniform polytopes in turn (i.e. hierarchicality)
  • the 2d uniforms are just the regular polygons
Thus esp. all edges are bound to have the same size throughout, say unity.

Your demi-uniformity in contrast
  • softens the first condition (in the form as you provided)
  • still requires the second, then being applied to the wider class of objects obtained
  • nothing is said to the former third requirement, the base of dimensional induction
Thus esp. you might well have edges of arbitrary sizes being contained.
Also, by allowing for more than just a single transitivity class of vertices, nothing restricts to convex figures anymore, even when looking onto not self-intersecting ones.

I'd think that your definition is valide and indeed provides a rather huge class of polytopes. But I wouldn't call them "demi-uniform", because they not even are as close as demi, neither quarter, nor eighth. In fact, esp. the mentioned property of allowing for different vertices, allowing for different edge lengths, allowing for non-convexity makes your definition a very, very far cousin to uniformity.

BTW, I for one do not like the term "uniform" (with any adjective what-so-ever) to be applied to multiform ones (eg. calling biform figures as 2-uniform). Because "uni" just means "1". Thus said example would mean 2-1-form, a mere nonsense.

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Re: Catch-all term for any interesting polytope

Postby ubersketch » Sat Dec 09, 2017 1:10 pm

Demiuniform sounds long and not very accurate but it's the best name I could come up with and I'm planning to rename it something more original.
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Re: Catch-all term for any interesting polytope

Postby Mercurial, the Spectre » Sun Dec 10, 2017 2:27 pm

ubersketch wrote:Demiuniform sounds long and not very accurate but it's the best name I could come up with and I'm planning to rename it something more original.

Good luck then on your journey! :P
Maybe you should call them "symmetroforms".
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Re: Catch-all term for any interesting polytope

Postby ubersketch » Sun Dec 10, 2017 6:20 pm

Mercurial, the Spectre wrote:
ubersketch wrote:Demiuniform sounds long and not very accurate but it's the best name I could come up with and I'm planning to rename it something more original.

Good luck then on your journey! :P
Maybe you should call them "symmetroforms".

Thanks, I'm going to use symmetroform instead now because its much more nice sounding and original than demiuniform and also unlike, multiform, doesn't assume every edge is equally sized.
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