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10-demicube

Uniform 10-polytope

Uniform 10-polytope

Demidekeract
(10-demicube)
[[File:Demidekeract ortho petrie.svg320px]]
Petrie polygon projection
Type
Family
Coxeter symbol
Schläfli symbol
Coxeter diagram
9-faces
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Symmetry group
Dual
Properties

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 {3,37,1}.

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

[[File:10-demicube graph.png240px]]B10 coxeter plane[[File:10-demicube.svg240px]]D10 coxeter plane(Vertices are colored by multiplicity: red, orange, yellow, green = 1,2,4,8)

References

  • H.S.M. Coxeter:
    • H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, 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, wiley.com,
      • (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: 1n1,
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)

References

  1. (1998). "Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices". Advanced Studies in Pure Mathematics.
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