10-demicube

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

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 [math]\displaystyle{ \left\{3 \begin{array}{l}3, 3, 3, 3, 3, 3, 3\\3\end{array}\right\} }[/math] 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)

References

• 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 [1]
    • (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, 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". https://bendwavy.org/klitzing/dimensions/polyxenna.htm. 

External links

• Olshevsky, George. "Demienneract". Glossary for Hyperspace. Archived from the original on 4 February 2007. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#half. 
• Multi-dimensional Glossary