Polytope product

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There are four operations on polytopes that are collectively known as the polytope products. These are:

Geometrically, these operations produce quite different results, but abstractly they're all built using a very similar procedure which mimics the direct product on posets.

The join, Cartesian product, and direct sum always produce convex polytopes if the base polytopes are convex, and these products are therefore well studied in the field of convex geometry. The comb product is not, as it generally produces skew polytopes even if the input polytopes are planar.

These products are useful tools to describe certain polytopes, or to build new ones with various desirable properties, including regularity or uniformity. They're also of theoretical interest – their automorphism groups in particular have been well-studied.[1]

External links[edit | edit source]

References[edit | edit source]

  1. Gleason, Ian; Hubard, Isabel (2018). "Products of abstract polytopes". Journal of Combinatorial Theory, Series A. 157: 287–320. doi:10.1016/j.jcta.2018.02.002.