Steric 7-cubes

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7-demicube

 150px
Steric 7-cube

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Stericantic 7-cube

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Steriruncic 7-cube

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Steriruncicantic 7-cube

Orthogonal projections in D7 Coxeter plane

In seven-dimensional geometry, a stericated 7-cube (or runcinated 7-demicube) is a convex uniform 7-polytope, being a runcination of the uniform 7-demicube. There are 4 unique runcinations for the 7-demicube including truncation and cantellation.

Steric 7-cube

Steric 7-cube
Type uniform 7-polytope
Schläfli symbol t0,3{3,34,1}
h4{4,35}
Coxeter-Dynkin diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 20160
Vertices 2240
Vertex figure
Coxeter groups D7, [34,1,1]
Properties convex

Alternate names

  • Small prismated demihepteract (acronym: sphosa)[1]

Cartesian coordinates

The Cartesian coordinates for the vertices of a steric 7-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±3,±3)

with an odd number of plus signs.

Images

Template:7-demicube Coxeter plane graphs

Template:Steric cube table

Stericantic 7-cube

Alternate names

  • Prismatotruncated demihepteract (acronym: pothesa)[2]

Images

Template:7-demicube Coxeter plane graphs

Steriruncic 7-cube

Alternate names

  • Prismatorhombated demihepteract (acronym: prohesa)[3]

Images

Template:7-demicube Coxeter plane graphs

Steriruncicantic 7-cube

Alternate names

  • Great prismated demihepteract (acronym: gephosa)[4]

Images

Template:7-demicube Coxeter plane graphs

These polytopes are based on the 7-demicube, a member of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 95 uniform polytopes with D7 symmetry, 63 are shared by the BC6 symmetry, and 32 are unique: Template:Demihepteract family

Notes

  1. ↑ Klitzing, (x3o3o *b3o3x3o3o - sphosa)
  2. ↑ Klitzing, (x3x3o *b3o3x3o3o - pothesa)
  3. ↑ Klitzing, (x3o3o *b3x3x3o3o - prohesa)
  4. ↑ Klitzing, (x3x3o *b3x3x3o3o - gephosa)

References

  • H.S.M. Coxeter:
    • H.S.M. Coxeter, Regular Polytopes, 3rd edition, Dover, New York, 1973
    • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, 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]
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
  • Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms". https://bendwavy.org/klitzing/dimensions/polyexa.htm. 
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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