In geometry, a 6-demicube, demihexeract or hemihexeract 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. Acronym: hax.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM<sub>6</sub> for a 6-dimensional half measure polytope.
Coxeter named this polytope as 1<sub>31</sub> from its Coxeter diagram, with a ring on one of the 1-length branches, . It can named similarly by a 3-dimensional exponential Schläfli symbol or {3,3<sup>3,1</sup>}.
Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:
with an odd number of plus signs.
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.
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.
There are 47 uniform polytopes with D<sub>6</sub> symmetry, 31 are shared by the B<sub>6</sub> symmetry, and 16 are unique:
The 6-demicube, 1<sub>31</sub> is third in a dimensional series of uniform polytopes, expressed by Coxeter as k<sub>31</sub> series. The fifth figure is a Euclidean honeycomb, 3<sub>31</sub>, and the final is a noncompact hyperbolic honeycomb, 4<sub>31</sub>. 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<sub>3k</sub> series. The fourth figure is the Euclidean honeycomb 1<sub>33</sub> and the final is a noncompact hyperbolic honeycomb, 1<sub>34</sub>.
Coxeter identified a subset of 12 vertices that form a regular skew icosahedron {3, 5} with the same symmetries as the icosahedron itself, but at different angles. He dubbed this the regular skew icosahedron.