8-demicube

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Demiocteract
(8-demicube)
Demiocteract ortho petrie.svg
Petrie polygon projection
Type Uniform 8-polytope
Family demihypercube
Coxeter symbol 151
Schläfli symbol {3,35,1} = h{4,36}
s{27}
Coxeter 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.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.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.png
7-faces 144:
16 {31,4,1}Demihepteract ortho petrie.svg
128 {36}7-simplex t0.svg
6-faces 112 {31,3,1}Demihexeract ortho petrie.svg
1024 {35}6-simplex t0.svg
5-faces 448 {31,2,1}Demipenteract graph ortho.svg
3584 {34}5-simplex t0.svg
4-faces 1120 {31,1,1}Cross graph 4.svg
7168 {3,3,3}4-simplex t0.svg
Cells 10752:
1792 {31,0,1}3-simplex t0.svg
8960 {3,3}3-simplex t0.svg
Faces 7168 {3}2-simplex t0.svg
Edges 1792
Vertices 128
Vertex figure Rectified 7-simplex
7-simplex t1.svg
Symmetry group D8, [37,1,1] = [1+,4,36]
A18, [27]+
Dual ?
Properties convex

In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices truncated. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

Coxeter named this polytope as 151 from its Coxeter-Dynkin diagram, with a ring on one of the 1-length branches.

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of an 8-demicube centered at the origin are alternate halves of the 8-cube:

(±1,±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

Related polytopes and honeycombs[edit]

This polytope is the vertex figure for the uniform tessellation, 251 with Coxeter-Dynkin diagram:

CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Images[edit]

orthographic projections
Coxeter plane B8 D8 D7 D6 D5
Graph 8-demicube t0 B8.svg 8-demicube t0 D8.svg 8-demicube t0 D7.svg 8-demicube t0 D6.svg 8-demicube t0 D5.svg
Dihedral symmetry [16/2] [14] [12] [10] [8]
Coxeter plane D4 D3 A7 A5 A3
Graph 8-demicube t0 D4.svg 8-demicube t0 D3.svg 8-demicube t0 A7.svg 8-demicube t0 A5.svg 8-demicube t0 A3.svg
Dihedral symmetry [6] [4] [8] [6] [4]

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)

External links[edit]