Semi-uniform polytope

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A semi-uniform polytope is a polytope which is isogonal, along with all of its elements. In contrast to the stricter uniform polytopes, the semi-uniform polytopes are not necessarily required to have only one type of edge length. Many semi-uniform polytopes can have multiple edge lengths that can be continuously varied.

All known convex semi-uniform polytopes are equivalent to a variation of some convex uniform polytope, though it is not known for certain if this is always the case; a counterexample has not yet been discovered. By contrast, nonconvex semi-uniforms have somewhat more variety. Already in 2 dimensions, semi-uniform polygons include a variety of polygons, such as rectangles, ditrigons, and tripods, that have two alternating edge lengths. In 3 dimensions the number of types of semi-uniform polyhedra is still largely unknown, but goes well into the thousands at least. Semi-uniform polychora and higher have largely still yet to be investigated.

A notable infinite family of semi-uniform polytopes is the kaleidoscopical polytopes.

Enumeration[edit | edit source]

By geometrical similarity, there are an uncountably infinite amount of semi-uniform polytopes in each dimension as the measures are continuously variable - there is a type of rectangle for each real number greater or equal to 1. To enumerate them, they are grouped into teepees and tribes. A teepee is a set of semi-uniform polytopes that can be continuously morphed through without vertices crossing. A tribe is a set of semi-uniform polytopes that can be continuously morphed through, allowing vertices to cross.

The exact number of teepees or tribes of semi-uniform polytopes is not known, but it is known that there are (at least) 7078 teepees of "primary tame semi-uniform" polyhedra (which have no off-axis faces and are tame).