7-demicube

Summary

Demihepteract
(7-demicube)

Petrie polygon projection
Type Uniform 7-polytope
Family demihypercube
Coxeter symbol 141
Schläfli symbol {3,34,1} = h{4,35}
s{21,1,1,1,1,1}
Coxeter diagrams =






6-faces 78 14 {31,3,1}
64 {35}
5-faces 532 84 {31,2,1}
448 {34}
4-faces 1624 280 {31,1,1}
1344 {33}
Cells 2800 560 {31,0,1}
2240 {3,3}
Faces 2240 {3}
Edges 672
Vertices 64
Vertex figure Rectified 6-simplex
Symmetry group D7, [34,1,1] = [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 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 HM7 for a 7-dimensional half measure polytope.

Coxeter named this polytope as 141 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol or {3,34,1}.

Cartesian coordinates

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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

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

As a configuration

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This configuration matrix represents the 7-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.[3]

D7             k-face fk f0 f1 f2 f3 f4 f5 f6 k-figures notes
A6             ( ) f0 64 21 105 35 140 35 105 21 42 7 7 041 D7/A6 = 64*7!/7! = 64
A4A1A1             { } f1 2 672 10 5 20 10 20 10 10 5 2 { }×{3,3,3} D7/A4A1A1 = 64*7!/5!/2/2 = 672
A3A2             100 f2 3 3 2240 1 4 4 6 6 4 4 1 {3,3}v( ) D7/A3A2 = 64*7!/4!/3! = 2240
A3A3             101 f3 4 6 4 560 * 4 0 6 0 4 0 {3,3} D7/A3A3 = 64*7!/4!/4! = 560
A3A2             110 4 6 4 * 2240 1 3 3 3 3 1 {3}v( ) D7/A3A2 = 64*7!/4!/3! = 2240
D4A2             111 f4 8 24 32 8 8 280 * 3 0 3 0 {3} D7/D4A2 = 64*7!/8/4!/2 = 280
A4A1             120 5 10 10 0 5 * 1344 1 2 2 1 { }v( ) D7/A4A1 = 64*7!/5!/2 = 1344
D5A1             121 f5 16 80 160 40 80 10 16 84 * 2 0 { } D7/D5A1 = 64*7!/16/5!/2 = 84
A5             130 6 15 20 0 15 0 6 * 448 1 1 D7/A5 = 64*7!/6! = 448
D6             131 f6 32 240 640 160 480 60 192 12 32 14 * ( ) D7/D6 = 64*7!/32/6! = 14
A6             140 7 21 35 0 35 0 21 0 7 * 64 D7/A6 = 64*7!/7! = 64
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There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, and 32 are unique:

D7 polytopes
 
t0(141)  
t0,1(141)
 
t0,2(141)
 
t0,3(141)
 
t0,4(141)
 
t0,5(141)
 
t0,1,2(141)
 
t0,1,3(141)
 
t0,1,4(141)
 
t0,1,5(141)
 
t0,2,3(141)
 
t0,2,4(141)
 
t0,2,5(141)
 
t0,3,4(141)
 
t0,3,5(141)
 
t0,4,5(141)
 
t0,1,2,3(141)
 
t0,1,2,4(141)
 
t0,1,2,5(141)
 
t0,1,3,4(141)
 
t0,1,3,5(141)
 
t0,1,4,5(141)
 
t0,2,3,4(141)
 
t0,2,3,5(141)
 
t0,2,4,5(141)
 
t0,3,4,5(141)
 
t0,1,2,3,4(141)
 
t0,1,2,3,5(141)
 
t0,1,2,4,5(141)
 
t0,1,3,4,5(141)
 
t0,2,3,4,5(141)
 
t0,1,2,3,4,5(141)

References

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  1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. ^ Coxeter, Complex Regular Polytopes, p.117
  3. ^ Klitzing, Richard. "x3o3o *b3o3o3o - hax".
  • 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-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
  • Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o *b3o3o3o3o - hesa".
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  • Olshevsky, George. "Demihepteract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
  • Multi-dimensional Glossary
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds