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.

Demidekeract (10-demicube)
Petrie polygon projection
Type	Uniform 10-polytope
Family	demihypercube
Coxeter symbol	171
Schläfli symbol	{31,7,1} h{4,38} s{21,1,1,1,1,1,1,1,1}
Coxeter diagram	=
9-faces	532	20 {31,6,1} 512 {38}
8-faces	5300	180 {31,5,1} 5120 {37}
7-faces	24000	960 {31,4,1} 23040 {36}
6-faces	64800	3360 {31,3,1} 61440 {35}
5-faces	115584	8064 {31,2,1} 107520 {34}
4-faces	142464	13440 {31,1,1} 129024 {33}
Cells	122880	15360 {31,0,1} 107520 {3,3}
Faces	61440	{3}
Edges	11520
Vertices	512
Vertex figure	Rectified 9-simplex
Symmetry group	D10, [37,1,1] = [1+,4,38] [29]+
Dual	?
Properties	convex

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 ${\displaystyle \left\{3{\begin{array}{l}3,3,3,3,3,3,3\\3\end{array}}\right\}}$ or {3,3^7,1}.

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.

Images

B10 coxeter plane

D10 coxeter plane (Vertices are colored by multiplicity: red, orange, yellow, green = 1,2,4,8)

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.^[1]