7-demicube

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.

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-faces7814 {31,3,1}
64 {35}
5-faces53284 {31,2,1}
448 {34}
4-faces1624280 {31,1,1}
1344 {33}
Cells2800560 {31,0,1}
2240 {3,3}
Faces2240{3}
Edges672
Vertices64
Vertex figure Rectified 6-simplex
Symmetry group D7, [34,1,1] = [1+,4,35]
[26]+
Dual ?
Properties convex

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

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
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

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]

D7k-facefkf0f1f2f3f4f5f6k-figuresnotes
A6( ) f0 64211053514035105214277041D7/A6 = 64*7!/7! = 64
A4A1A1{ } f1 2672105201020101052{ }×{3,3,3}D7/A4A1A1 = 64*7!/5!/2/2 = 672
A3A2100 f2 33224014466441{3,3}v( )D7/A3A2 = 64*7!/4!/3! = 2240
A3A3101 f3 464560*406040{3,3}D7/A3A3 = 64*7!/4!/4! = 560
A3A2110 464*2240133331{3}v( )D7/A3A2 = 64*7!/4!/3! = 2240
D4A2111 f4 8243288280*3030{3}D7/D4A2 = 64*7!/8/4!/2 = 280
A4A1120 5101005*13441221{ }v( )D7/A4A1 = 64*7!/5!/2 = 1344
D5A1121 f5 16801604080101684*20{ }D7/D5A1 = 64*7!/16/5!/2 = 84
A5130 6152001506*44811D7/A5 = 64*7!/6! = 448
D6131 f6 3224064016048060192123214*( )D7/D6 = 64*7!/32/6! = 14
A6140 7213503502107*64D7/A6 = 64*7!/7! = 64

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

References
  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
      • (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)
  • Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o *b3o3o3o3o - hesa".
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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