10-demicube

In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube 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 HM10 for a ten-dimensional half measure polytope.

Coxeter named this polytope as 171 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol $$\left\{3 \begin{array}{l}3, 3, 3, 3, 3, 3, 3\\3\end{array}\right\}$$ or {3,37,1}.

Cartesian coordinates
Cartesian coordinates for the vertices of a demidekeract centered at the origin are alternate halves of the dekeract:
 * (±1,±1,±1,±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

Related polytopes
A regular dodecahedron can be embedded as a regular skew polyhedron within the vertices in the 10-demicube, possessing the same symmetries as the 3-dimensional dodecahedron.