10-demicube

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Demidekeract
(10-demicube)
Demidekeract ortho petrie.svg
Petrie polygon projection
Type Uniform 10-polytope
Family demihypercube
Coxeter symbol 171
Schläfli symbol {31,7,1}
h{4,38}
s{29}
Coxeter-Dynkin diagram CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.png
9-faces 532 20 {31,6,1}Demienneract ortho petrie.svg
512 {38}9-simplex t0.svg
8-faces 5300 180 {31,5,1}Demiocteract ortho petrie.svg
5120 {37}8-simplex t0.svg
7-faces 24000 960 {31,4,1}Demihepteract ortho petrie.svg
23040 {36}7-simplex t0.svg
6-faces 64800 3360 {31,3,1}Demihexeract ortho petrie.svg
61440 {35}6-simplex t0.svg
5-faces 115584 8064 {31,2,1}Demipenteract graph ortho.svg
107520 {34}5-simplex t0.svg
4-faces 142464 13440 {31,1,1}Cross graph 4.svg
129024 {33}4-simplex t0.svg
Cells 122880 15360 {31,0,1}3-simplex t0.svg
107520 {3,3}3-simplex t0.svg
Faces 61440 {3}2-simplex t0.svg
Edges 11520
Vertices 512
Vertex figure Rectified 9-simplex
Rectified 9-simplex.png
Symmetry group D10, [37,1,1] = [1+,4,38]
[29]+
Dual ?
Properties convex

In geometry, a demidekeract or 10-demicube is a uniform 10-polytope, constructed from the 10-cube with alternated vertices truncated. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

Coxeter named this polytope as 171 from its Coxeter-Dynkin diagram, with a ring on one of the 1-length Coxeter-Dynkin diagram branches.

Cartesian coordinates[edit]

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[edit]

10-demicube graph.png
B10 coxeter plane
10-demicube.svg
D10 coxeter plane
(Vertices are colored by multiplicity: red, orange, yellow, green = 1,2,4,8)

References[edit]

  • 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-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
  • Richard Klitzing, 10D uniform polytopes (polyxenna), x3o3o *b3o3o3o3o3o3o3o - hede

External links[edit]