Edge figures of uniform polytera

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

Edge figures of uniform polytera

Postby Mecejide » Tue May 07, 2019 4:25 pm

Lately, I've been trying to figure out what the edge figures of (isotoxal) uniform polytera. The isotoxal uniform polytera are numbers 1-4, 7-12, 25-43, 48-240, 682-684, and 687-690 on Hedrondude's website. Here's what I have so far:
hexateron-tetrahedron
penteract-tetrahedron
triacontiditeron-octahedron
hexadecahemidecateron-tetrahemihexahedron
demipenteract-uniform triangular prism
dishexadecateron-crossed triangular antiprism
hexadecapenteract-triangular crate
retrodishexadecateron-crossed triangular bifrustum
retrointercepted hexadecateron-hollow triangular prism
hexadecateron-triangular biodontoid
rectified hexateron-triangular pyramid
dodecateron-disphenoid
rectified penteract-triangular pyramid
rectified triacontiditeron-square pyramid
triacontidihemitriacontiditeron-square retrobipyramid
retropenteractihemitriacontiditeron-square pyramid
rectified hexadecahemidecateron-crossed square pyramid
penteractidishexadecahemidecateron-square hemiretrobipyramid
penteractitriacontiditeron-digonal disphenoid
dodecahemidodecateron-triangular pyramid
Does anyone know what the others are?
Mecejide
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Re: Edge figures of uniform polytera

Postby Klitzing » Wed May 08, 2019 9:12 am

I don't know what your terms "crate", "biodontoid", "retrobipyramid", "hemiretrobipyramid" would describe …
But your quest clearly is fair and easy to solve.

Remember that edge figures are nothing but vertex figures of vertex figures.
Starting with isotoxal uniform polytopes, you'd been in the realm of quasiregulars. I.e in their Dynkin diagrams just a single node is being ringed.

The vertex figure for quasiregular polytopes is being encoded by just that very Dynkin diagram, when the former ringed node is being omitted, and the directly linked nodes get ringed instead. In fact the there to be used metrically correct edge length ought be the according shortchords, that is, for an omitted link marked 3 you'd use a unit edge, for one marked 4 you'd use an edge of size sqrt(2), for one marked 5 you'd use one of size tau etc. - Generally this results in a Dynkin diagram with various marked nodes with various applied edge sizes.

In order to derive the vertex figure of a polytope with such a more complex encoding you'd have to follow W. Krieger's advise, as being described in detail e.g. here.

--- rk
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Re: Edge figures of uniform polytera

Postby Mecejide » Mon Nov 25, 2019 6:57 pm

I meant how they look in 3d.
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