# Polytope product

The polytope products are a set of four closely related operations that take two polytopes as input and produce a new polytope: the pyramid product, prism product, direct sum or tegum product, and comb product. The products are defined on abstract polytopes, which may be optionally accompanied with realizations in Euclidean space. Their differences are summarized in the following table, where the input polytopes have rank n and m.

Names Symbol Rank Identity Annihilator Example Regular polytope family
pyramid product (join, join product) ${\displaystyle P\bowtie Q}$ ${\displaystyle n+m+1}$ Nullitope None pentagon ${\displaystyle \bowtie }$ point = pentagonal pyramid Simplices
prism product (Cartesian product) ${\displaystyle P\times Q}$ ${\displaystyle n+m}$ Point Nullitope pentagon ${\displaystyle \times }$ line segment = pentagonal prism Hypercubes
direct sum (free sum, tegum product) ${\displaystyle P\oplus Q}$ ${\displaystyle n+m}$ Point Nullitope pentagon ${\displaystyle \oplus }$ line segment = pentagonal bipyramid Orthoplices
comb product (honeycomb product, topological product) ${\displaystyle P\,\square \,Q}$ ${\displaystyle n+m-1}$ None Point, Nullitope pentagon ${\displaystyle \square }$ pentagon = pentagonal duocomb Hypercubic honeycombs

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. On abstract polytopes, the products are commutative, associative, and have prime factorization theorems applicable to all polytopes except the annihilator.

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.

The direct sum and prism product on abstract polytopes are related by ${\displaystyle P\oplus Q=(P^{*}\times Q^{*})^{*}}$ where ${\displaystyle ^{*}}$ denotes the dual. In other words, the direct sum and prism product are "dual operations." In this manner, the pyramid product and comb product are self-dual.

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]

Each product is related to an infinite family of regular polytopes. The corresponding product of two members of the family will result in another member of the family. For example the prism product of a square and a cube is a 5-cube.