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Hexic 7-cubes

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7-demicube
Hexic 7-cube
Hexicantic 7-cube
Hexiruncic 7-cube
Hexiruncicantic 7-cube


Hexisteric 7-cube
Hexistericantic 7-cube
Hexisteriruncic 7-cube
Hexisteriruncicantic 7-cube
Hexipentic 7-cube


Hexipenticantic 7-cube
Hexipentiruncic 7-cube
Hexipentiruncicantic 7-cube
Hexipentisteric 7-cube
Hexipentistericantic 7-cube


Hexipentisteriruncic 7-cube
Hexipentisteriruncicantic 7-cube
Orthogonal projections in D7 Coxeter plane

In seven-dimensional geometry, a hexic 7-cube is a convex uniform 7-polytope, constructed from the uniform 7-demicube. There are 16 unique forms.

Hexic 7-cube

Hexic 7-cube
Type uniform 7-polytope
Schläfli symbol t0,5{3,34,1}
h6{4,35}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges 4704
Vertices 448
Vertex figure
Coxeter groups D7, [34,1,1]
Properties convex

Cartesian coordinates

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

(±1,±1,±1,±1,±1,±1,±3)

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]

Hexicantic 7-cube

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]

Hexiruncic 7-cube

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]

Hexisteric 7-cube

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]

Hexipentic 7-cube

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]

Hexiruncicantic 7-cube

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]

Hexistericantic 7-cube

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]

Hexipenticantic 7-cube

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]

Hexisteriruncic 7-cube

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]

Hexipentiruncic 7-cube

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]

Hexipentisteric 7-cube

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]

Hexisteriruncicantic 7-cube

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]

Hexipentiruncicantic 7-cube

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]

Hexipentisteriruncic 7-cube

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]

Hexipentistericantic 7-cube

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]

Hexipentisteriruncicantic 7-cube

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]

This polytope is based on the 7-demicube, a part 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 BC7 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)

Notes

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 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]
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
  • Klitzing, Richard. "7D uniform polytopes (polyexa)".
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 polychoron Pentachoron 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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