10-demicube

Demidekeract (10-demicube)
Petrie polygon projection
Type	Uniform 10-polytope
Family	demihypercube
Coxeter symbol	1₇₁
Schläfli symbol	{3^1,7,1} h{4,3⁸} s{2^{1,1,1,1,1,1,1,1,1}}
Coxeter diagram	=
9-faces	532	20 {3^1,6,1} 512 {3⁸}
8-faces	5300	180 {3^1,5,1} 5120 {3⁷}
7-faces	24000	960 {3^1,4,1} 23040 {3⁶}
6-faces	64800	3360 {3^1,3,1} 61440 {3⁵}
5-faces	115584	8064 {3^1,2,1} 107520 {3⁴}
4-faces	142464	13440 {3^1,1,1} 129024 {3³}
Cells	122880	15360 {3^1,0,1} 107520 {3,3}
Faces	61440	{3}
Edges	11520
Vertices	512
Vertex figure	Rectified 9-simplex
Symmetry group	D₁₀, [3^7,1,1] = [1⁺,4,3⁸] [2⁹]⁺
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 $\left\{3{\begin{array}{l}3,3,3,3,3,3,3\\3\end{array}}\right\}$ or {3,3^7,1}.
Acronym: hede^[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.^[2]

References

↑ Klitzing, Richard. "x3o3o *b3o3o3o3o3o3o3o - hede".
↑ Deza, Michael; Shtogrin, Mikhael (2000). "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. 27. Mathematical Society of Japan: 73–92[77]. doi:10.2969/aspm/02710073. ISBN 978-4-931469-77-8.

H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, pp. 294–295, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5), ISBN 0-486-61480-8
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6
  - (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, p. 409, Hemicubes: 1_n1, ISBN 978-1-56881-220-5
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

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

[1] Klitzing, Richard. "x3o3o *b3o3o3o3o3o3o3o - hede".

[2] Deza, Michael; Shtogrin, Mikhael (2000). "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. 27. Mathematical Society of Japan: 73–92[77]. doi:10.2969/aspm/02710073. ISBN 978-4-931469-77-8.

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