Convex hulls of uniform polychora

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

Convex hulls of uniform polychora

Postby username5243 » Wed Feb 12, 2020 3:07 am

This came o=up in another thread: How do you find the convex hull of a general uniform polychoron?

Let's start with the basics: THe pennics are easy - since the only pennic regiments are those of the conex pen truncates, and all polytopes in the same regiment have the same convex hull, those are easy.

THe tessics I've actually thought about. It's not that hard to determine vertex coordinates for the non-convex tessics, based on the cells they have in the 8 positions corresponding to the cells of tes. From there, determining the general shape of the convex hull is easy by just looking at the coordinates - the general rule is, for instance the uniform polychoron prit has vertex coordinates ((1+2sqrt(2))/2, (1+sqrt(2))/2, 1/2, 1/2) all permutations and changes of sign - any thing with al permutations/sign changes of (a,b,c,c) (provided a > b > c > 0) will be a variation of prit. I'll list the coordinates of the non-convex tessic colonels below, assume all permutations/sign changes:

Quitit: ((sqrt(2)-1)/2, (sqrt(2)-1)/2, (sqrt(2)-1)/2, 1/2) - hull is 1/w scaled sidpith

Gittith: ((sqrt(2)-1)/2, 1/2, 1/2, 1/2) - hull is 1/w-scaled tat

Wavitoth: ((sqrt(2)-1)/2, (sqrt(2)-1)/2, 1/2, 1/2) - hull is 1/w-scaled srit

Gaqrit: ((2sqrt(2)-1)/2, (2sqrt(2)-1)/2, (sqrt(2)-1)/2, 1/2) - hull is grit variant

Thatoth: ((1+sqrt(2))/2, (1+sqrt(2))/2, 1/2, (sqrt(2)-1)/2) - hull is grit variant

Thaquitoth: ((1+sqrt(2))/2, 1/2, (sqrt(2)-1)/2, (sqrt(2)-1)/2) - hull is prit variant

Quiproh: ((2sqrt(2)-1)/2, (sqrt(2)-1)/2, (sqrt(2)-1)/2, 1/2) - hull is prit variant

Gichado: ((2sqrt(2)-1)/2 (sqrt(2)-1)/2, 1/2, 1/2) - hull is proh variant

Skiviphado: ((1+sqrt(2))/2, 1/2, 1/2, (sqrt(2)-1)/2) - hull is proh variant

Gaquidpoth: ((3sqrt(2)-1)/2, (2sqrt(2)-1)/2, (sqrt(2)-1)/2, 1/2) - hull is gidpith variant

Thatpath: ((1+2sqrt(2))/2, (1+sqrt(2))/2, 1/2, (sqrt(2)-1)/2) - hull is gidpith variant

Thaquitpath: ((1+sqrt(2))/2, 1/2, (sqrt(2)-1)/2, (2sqrt(2)-1)/2) - hull is gidpith variant

I'm not sure what the exact edge ratios of these hulls are, but there's probably some way to determine them fromt he vertex coordinates.

I don't know how to apply this method to the rest of the uniforms - the icoics and hyics can't be represented like this. And then there's the really strange ones (Bowers's "miscellaneous" category, plus the big snub regiments in categories 27-29 on his website) that I have no clue what their hulls woul look like - at least for the Wythoffian ones I can assume they'll all have Wythoffian hulls. Anyone know how to figure out these convex hulls?
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Re: Convex hulls of uniform polychora

Postby URL » Mon Apr 27, 2020 5:31 am

This is a very general question. But, if you already have the numerical coordinates of a polychoron, the easiest thing you can do is to simply plug them into Qhull and analyze whatever comes out.
I'm building a library of polyhedra and polychora:
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