# Pure polytope

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A skew polytope is **pure** if it cannot be expressed as the blend of two polytopes in a non-trivial way.^{[1]}

## Trivial blends[edit | edit source]

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[edit | edit source]

- 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[edit | edit source]

- 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".

## Reference[edit | edit source]

## Bibliography[edit | edit source]

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