# 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*. Half of the vertices are deleted and new facets are formed. The 2

_{n}*n*facets become 2

*n*

**(**, and 2

*n*−1)-demicubes^{n}

**(**facets are formed in place of the deleted vertices.

*n*−1)-simplex^{[1]}

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 *k*_{21} polytope family.

They are represented by Coxeter-Dynkin diagrams of three constructive forms:

H.S.M. Coxeter also labeled the third bifurcating diagrams as **1 _{k1}** 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 C_{n,m} = *m ^{th}*-face count in

*n*-cube = 2

^{n−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.

The rhombic disphenoid is the three-dimensional example as alternated cuboid. It has three sets of edge lengths, and scalene triangle faces.