10-demicube explained

bgcolor=#e7dcc3 colspan=3	Demidekeract (10-demicube)
bgcolor=#ffffff align=center colspan=3	Petrie polygon projection
Type	Uniform 10-polytope
Family	demihypercube
Coxeter symbol	1₇₁
Schläfli symbol	h s
Coxeter diagram	=
9-faces	532
8-faces	5300
7-faces	24000
6-faces	64800
5-faces	115584
4-faces	142464
Cells	122880
Faces	61440
Edges	11520
Vertices	512
Vertex figure	Rectified 9-simplex
Symmetry group	D₁₀, [3<sup>7,1,1</sup>] = [1<sup>+</sup>,4,3<sup>8</sup>] [2<sup>9</sup>]⁺
Dual	?
Properties	convex

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.

\left\{3\begin{array}{l}3,3,3,3,3,3,3\\3\end{array}\right\}

or .

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.

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]

H.S.M. Coxeter:
- Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition,, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  - (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
  - (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
  - (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 26. pp. 409: Hemicubes: 1_n1)
Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)

Deza . Michael . Shtogrin . Mikhael . Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices . Advanced Studies in Pure Mathematics . Arrangements – Tokyo 1998 . 1998 . 77 . 10.2969/aspm/02710073 . 978-4-931469-77-8 . 4 April 2020. free .