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Demidekeract (10-demicube)
Petrie polygon projection
Type	Uniform 10-polytope
Family	demihypercube
Coxeter symbol	171
Schläfli symbol	{3} h{4,3} s{2}
Coxeter diagram	=
9-faces	532	20 {3} 512 {3}
8-faces	5300	180 {3} 5120 {3}
7-faces	24000	960 {3} 23040 {3}
6-faces	64800	3360 {3} 61440 {3}
5-faces	115584	8064 {3} 107520 {3}
4-faces	142464	13440 {3} 129024 {3}
Cells	122880	15360 {3} 107520 {3,3}
Faces	61440	{3}
Edges	11520
Vertices	512
Vertex figure	Rectified 9-simplex
Symmetry group	D10, =
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.

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}.

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.

References

1. Deza, Michael; Shtogrin, Mikhael (1998). "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: 77. doi:10.2969/aspm/02710073. ISBN 978-4-931469-77-8. Retrieved 4 April 2020.

• H.S.M. Coxeter:
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 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, ISBN 978-0-471-01003-6
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I,
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (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)
• Klitzing, Richard. "10D uniform polytopes (polyxenna) x3o3o *b3o3o3o3o3o3o3o - hede".

External links

• Olshevsky, George. "Demienneract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Multi-dimensional Glossary

• 10-polytopes