Demihypercube

In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγ_n for being half of the hypercube family, γ_n. Half of the vertices are deleted and new facets are formed. The 2n facets become 2n (n−1)-demicubes, and 2ⁿ (n−1)-simplex facets are formed in place of the deleted vertices.^[1]

Alternation of the n-cube yields one of two n-demicubes, as in this 3-dimensional illustration of the two tetrahedra that arise as the 3-demicubes of the 3-cube.

They have been named with a demi- prefix to each hypercube name: demicube, demitesseract, etc. The demicube is identical to the regular tetrahedron, and the demitesseract is identical to the regular 16-cell. The demipenteract is considered semiregular for having only regular facets. Higher forms do not have all regular facets but are all uniform polytopes.

The vertex-edge graph of the demihypercube is the halved cube graph.

An n-demicube has inversion symmetry if n is even.