# 6-demicube

In geometry, a **6-demicube** or **demihexteract** is a uniform 6-polytope, constructed from a *6-cube* (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM_{6} for a 6-dimensional *half measure* polytope.

Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:

This configuration matrix represents the 6-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[1]}^{[2]}

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.^{[3]}

There are 47 uniform polytopes with D_{6} symmetry, 31 are shared by the B_{6} symmetry, and 16 are unique:

The 6-demicube, 1_{31} is third in a dimensional series of uniform polytopes, expressed by Coxeter as k_{31} series. The fifth figure is a Euclidean honeycomb, 3_{31}, and the final is a noncompact hyperbolic honeycomb, 4_{31}. Each progressive uniform polytope is constructed from the previous as its vertex figure.

It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_{3k} series. The fourth figure is the Euclidean honeycomb 1_{33} and the final is a noncompact hyperbolic honeycomb, 1_{34}.