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 hγn for being half of the hypercube family, γn. 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.
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 don't have all regular facets but are all uniform polytopes.
The vertices and edges of a demihypercube form two copies of the halved cube graph.
Thorold Gosset described the demipenteract in his 1900 publication listing all of the regular and semiregular figures in n-dimensions above 3. He called it a 5-ic semi-regular. It also exists within the semiregular k21 polytope family.
They are represented by Coxeter-Dynkin diagrams of three constructive forms:
H.S.M. Coxeter also labeled the third bifurcating diagrams as 1k1 representing the lengths of the 3 branches and led by the ringed branch.
An n-demicube, n greater than 2, has n(n−1)/2 edges meeting at each vertex. The graphs below show less edges at each vertex due to overlapping edges in the symmetry projection.
In general, a demicube's elements can be determined from the original n-cube: (with Cn,m = mth-face count in n-cube = 2n−m n!/(m!(n−m)!))
Constructions as alternated orthotopes have the same topology, but can be stretched with different lengths in n-axes of symmetry.