Hexic 7-cubes

120px 7-demicube	120px Hexic 7-cube	120px Hexicantic 7-cube	120px Hexiruncic 7-cube	120px Hexiruncicantic 7-cube
120px Hexisteric 7-cube	120px Hexistericantic 7-cube	120px Hexisteriruncic 7-cube	120px Hexisteriruncicantic 7-cube	120px Hexipentic 7-cube
120px Hexipenticantic 7-cube	120px Hexipentiruncic 7-cube	120px Hexipentiruncicantic 7-cube	120px Hexipentisteric 7-cube	120px Hexipentistericantic 7-cube
120px Hexipentisteriruncic 7-cube	120px 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.

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 BC₇ symmetry, and 32 are unique: Template:Demihepteract family

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)". https://bendwavy.org/klitzing/dimensions/polyexa.htm. 

External links

• Weisstein, Eric W.. "Hypercube". http://mathworld.wolfram.com/Hypercube.html. 
• Polytopes of Various Dimensions
• Multi-dimensional Glossary