Steric 7-cubes

150px 7-demicube	150px Steric 7-cube	150px Stericantic 7-cube
	150px Steriruncic 7-cube	150px 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
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

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

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.

Template:7-demicube Coxeter plane graphs

Template:Steric cube table

• Prismatotruncated demihepteract (acronym: pothesa)^[2]

Template:7-demicube Coxeter plane graphs

• Prismatorhombated demihepteract (acronym: prohesa)^[3]

Template:7-demicube Coxeter plane graphs

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

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

• 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. 

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

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