Pure polytope

This article is about the property of skew polytopes. For the property of posets, see Abstract polytope § Chains and flags.

A skew polytope is pure if it cannot be expressed as the blend of two polytopes in a non-trivial way.

Trivial blends

A blend is considered trivial if the result is one of the arguments. For example, any polytope 𝓟 is the result of the blend 𝓟#𝓟, thus this blend is trivial. Similarly any polytope is a blend of itself with the point.

A non-trivial blend is simply any blend that isn't trivial.

Examples

• The mucube is a pure apeirohedron.
• The square duocomb is pure.
• The Petrial tetrahedron is pure.
• The blended square tiling is not pure, because it is the blend of the square tiling and a digon.

Properties

• All regular polygons are either planar or a non-trivial blend. In other words, there are no regular pure skew polygons.
• All the regular pure polyhedra in 3-dimensional Euclidean space are either the Petrial of a non-skew regular polyhedron or an apeirohedron. The latter are collectively referred to as the "pure apeirohedra".

Bibliography

• *McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF). Discrete Computational Geometry (47): 449–478. doi:10.1007/PL00009304.