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Polyapeira

Polyapeira

Polyapeira (singular polyapeiron) are polytopes with a rank of (countable) infinity. In other words, they are infinite dimensional shapes. There are a lot of problems with these shapes (like how they are disconnected, Coxeter-Dynkin diagrams break down, etc), but just let's ignore these, or at least when they don't interfere...

This page aims to describe some of these shapes.

Platonic polyapeira

omix: omegasimplex, the simplest polyapeiron, the ∞-simplex. It has a countably infinite amount of omegasimplices as facets, three around an apeiron. It is self-dual. Its diapeiral angle is 90°.

omact: omegeract, the ∞-cube. It has a countably infinite amount of omegeracts as facets, but an uncountably infinite amount of vertices. It's dual to the omegacrux. Its diapeiral angle is 90°.

omux: omegacrux, the ∞-orthoplex. It has an uncountably infinite amount of omegasimplices as facets, but a countably infinite amount of vertices. It's dual to the omegeract. Its diapeiral angle is 180°.

Uniform

gapomix: great apeirated omegasimplex, the omnitruncation of the omegasimplex.

gapomact: great apeirated omegeract, the omnitruncation of the omegeract and omegacrux.

Codytized tilings

A codytization is when you take a polytope's skeleton (its edges and vertices) and make it into a Coxeter-Dynkin diagram. For example, a codytized square is x3o3o3o3*a, which is the tetrahedral-octahedral honeycomb. You can also apply this operation to tilings, in which they become infinite dimensional polyapeiric apeirocombs.

codsquat: codytized square tiling, the codytization of the square tiling.

gapcodsquat: great apeirated codytized square tiling, the omnitruncated codytized square tiling.

codazazat: codytized order-∞ apeirogonal tiling, the codytization of the order-∞ apeirogonal tiling.

gapcodazazat: great apeirated codytized order-∞ apeirogonal tiling, the omnitruncated codytized order-∞ apeirogonal tiling.