Steric 7-cubes

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.


7-demicube
Steric 7-cube
Stericantic 7-cube


Steriruncic 7-cube
Steriruncicantic 7-cube

Orthogonal projections in D7 Coxeter plane

Steric 7-cube
Steric 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,3{3,34,1}
h4{4,35}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges20160
Vertices2240
Vertex figure
Coxeter groupsD7, [34,1,1]
Propertiesconvex

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
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]
Dimensional family of steric n-cubes
n5678
[1+,4,3n-2]
= [3,3n-3,1]		 [1+,4,33]
= [3,32,1]		 [1+,4,34]
= [3,33,1]		 [1+,4,35]
= [3,34,1]		 [1+,4,36]
= [3,35,1]
Steric
figure
Coxeter
=
=
=
=
Schläfli h4{4,33} h4{4,34} h4{4,35} h4{4,36}

Stericantic 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]

Steriruncic 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]

Steriruncicantic 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 BC6 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
        • (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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