7-demicube

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Demihepteract
(7-demicube)
Demihepteract ortho petrie.svg
Petrie polygon projection
Type Uniform 7-polytope
Family demihypercube
Coxeter symbol 141
Schläfli symbol {3,34,1} = h{4,35}
s{26}
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.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.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.png
6-faces 78 14 {31,3,1}Demihexeract ortho petrie.svg
64 {35}6-simplex t0.svg
5-faces 532 84 {31,2,1}Demipenteract graph ortho.svg
448 {34}5-simplex t0.svg
4-faces 1624 280 {31,1,1}4-orthoplex.svg
1344 {33}4-simplex t0.svg
Cells 2800 560 {31,0,1}3-simplex t0.svg
2240 {3,3}3-simplex t0.svg
Faces 2240 {3}2-simplex t0.svg
Edges 672
Vertices 64
Vertex figure Rectified 6-simplex
6-simplex t1.svg
Symmetry group D7, [36,1,1] = [1+,4,35]
[26]+
Dual ?
Properties convex

In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices truncated. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

Coxeter named this polytope as 141 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 demihepteract centered at the origin are alternate halves of the hepteract:

(±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.


Images[edit]

orthographic projections
Coxeter plane B7 D7 D6
Graph 7-demicube t0 B7.svg 7-demicube t0 D7.svg 7-demicube t0 D6.svg
Dihedral symmetry [14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph 7-demicube t0 D5.svg 7-demicube t0 D4.svg 7-demicube t0 D3.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-demicube t0 A5.svg 7-demicube t0 A3.svg
Dihedral symmetry [6] [4]

Related polytopes[edit]

There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, and 32 are unique:

7-demicube t0 D7.svg
t0(141)
7-demicube t01 D7.svg
t0,1(141)
7-demicube t02 D7.svg
t0,2(141)
7-demicube t03 D7.svg
t0,3(141)
7-demicube t04 D7.svg
t0,4(141)
7-demicube t05 D7.svg
t0,5(141)
7-demicube t012 D7.svg
t0,1,2(141)
7-demicube t013 D7.svg
t0,1,3(141)
7-demicube t014 D7.svg
t0,1,4(141)
7-demicube t015 D7.svg
t0,1,5(141)
7-demicube t023 D7.svg
t0,2,3(141)
7-demicube t024 D7.svg
t0,2,4(141)
7-demicube t025 D7.svg
t0,2,5(141)
7-demicube t034 D7.svg
t0,3,4(141)
7-demicube t035 D7.svg
t0,3,5(141)
7-demicube t045 D7.svg
t0,4,5(141)
7-demicube t0123 D7.svg
t0,1,2,3(141)
7-demicube t0124 D7.svg
t0,1,2,4(141)
7-demicube t0125 D7.svg
t0,1,2,5(141)
7-demicube t0134 D7.svg
t0,1,3,4(141)
7-demicube t0135 D7.svg
t0,1,3,5(141)
7-demicube t0145 D7.svg
t0,1,4,5(141)
7-demicube t0234 D7.svg
t0,2,3,4(141)
7-demicube t0235 D7.svg
t0,2,3,5(141)
7-demicube t0245 D7.svg
t0,2,4,5(141)
7-demicube t0345 D7.svg
t0,3,4,5(141)
7-demicube t01234 D7.svg
t0,1,2,3,4(141)
7-demicube t01235 D7.svg
t0,1,2,3,5(141)
7-demicube t01245 D7.svg
t0,1,2,4,5(141)
7-demicube t01345 D7.svg
t0,1,3,4,5(141)
7-demicube t02345 D7.svg
t0,2,3,4,5(141)
7-demicube t012345 D7.svg
t0,1,2,3,4,5(141)

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)
  • Richard Klitzing, 7D uniform polytopes (polyexa), x3o3o *b3o3o3o3o - hesa


External links[edit]