Runcinated 7-demicubes

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7-demicube

Runcinated 7-demicube

Runcitruncated 7-demicube

Runcicantellated 7-demicube

Runcicantitruncated 7-demicube
Orthogonal projections in D7 Coxeter plane

In seven-dimensional geometry, a 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.


Runcinated 7-demicube

Runcinated 7-demicube
Typeuniform polyexon
Schläfli symbol t0,3{3,34,1}
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 cantellated demihepteract 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]

Runcitruncated 7-demicube

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]

Runcicantellated 7-demicube

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]

Runcicantitruncated 7-demicube

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]

Related polytopes

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:


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, editied 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.
    • Richard Klitzing, 7D, uniform polytopes (polyexa)

    External links

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