Demidekeract
(10-demicube)

Petrie polygon projection
TypeUniform 10-polytope
Familydemihypercube
Coxeter symbol171
Schläfli symbol{31,7,1}
h{4,38}
s{21,1,1,1,1,1,1,1,1}
Coxeter diagram=
9-faces53220 {31,6,1}
512 {38}
8-faces5300180 {31,5,1}
5120 {37}
7-faces24000960 {31,4,1}
23040 {36}
6-faces648003360 {31,3,1}
61440 {35}
5-faces1155848064 {31,2,1}
107520 {34}
4-faces14246413440 {31,1,1}
129024 {33}
Cells12288015360 {31,0,1}
107520 {3,3}
Faces61440{3}
Edges11520
Vertices512
Vertex figureRectified 9-simplex
Symmetry groupD10, [37,1,1] = [1+,4,38]
[29]+
Dual?
Propertiesconvex

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


B10 coxeter plane
D10 coxeter plane
(Vertices are colored by multiplicity: red, orange, yellow, green = 1,2,4,8)

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]

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. 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, p. 296, 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 Ivic 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: 1n1, 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)