Strange polytopes

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

Strange polytopes

Postby ubersketch » Tue Dec 25, 2018 8:52 pm

I’ve been thinking about a strange kind of polytope. Essentially these are polytopes which are incidentally identical which belong to multiple armies. Has anyone found any example of these?
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Re: Strange polytopes

Postby wendy » Wed Jan 02, 2019 10:42 am

I would imagine not, because by definition, an army means that the cloud of vertices are the same.

We did have some fun with 'sub-regiments' etc, where the cloud of one is a sub-set of another, like the pap (pentagonal antiprism) is the subset of an icosahedron.
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Re: Strange polytopes

Postby ubersketch » Sun Jan 06, 2019 2:53 am

I mean that two polytopes belong in different armies but their abstract versions are identical.
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Re: Strange polytopes

Postby Klitzing » Sun Jan 06, 2019 11:00 am

ubersketch wrote:I mean that two polytopes belong in different armies but their abstract versions are identical.

This would be easy. Take any polyhedron and strech it a bit along the z-direction. Then the set of vertices obviously would differ, and thus those 2 versions of your polyhedron would belong to different armies. But they still have the same incidence structure and so still are realisations of the same abstract polyhedron.

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Re: Strange polytopes

Postby username5243 » Sun Jan 06, 2019 3:41 pm

Would this include polytopes that only differ by (for example) having pentagrams in place of pentagons?

If so, then a rather simple example is tic (x4x3o) and quith (x4/3x3o) - the latter is in a completely different army (sirco) but has the exact same incidence matrix, only using octagrams instead of octagons.
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Re: Strange polytopes

Postby Klitzing » Sun Jan 06, 2019 5:00 pm

yes, you are right, algebraic conjugacy here indeed provides different representations of the same abstract polytopes.
In fact, you could freely vary in a Dynkin diagram

  • 4 vs. 4/3
  • 5 vs. 5/3
  • 5/2 vs. 5/4
while the incidence matrix will not change at all.
But the vertex set surely would change thereby and hence those representations do belong to different armies.

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