Steric 7-cubes

7-demicube	160px Steric 7-cube	Stericantic 7-cube
	Steriruncic 7-cube	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

Steric 7-cube
Type	uniform 7-polytope
Schläfli symbol	t0,3{3,34,1} h4{4,35}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	20160
Vertices	2240
Vertex figure
Coxeter groups	D7, [34,1,1]
Properties	convex

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	150px	150px	150px
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D5	D4	D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Related polytopes

Dimensional family of steric n-cubes
n	5	6	7	8
[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	120px		120px
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	150px		150px
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D5	D4	D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Steriruncic 7-cube

Images

orthographic projections

Coxeter plane	B7	D7	D6
Graph	150px		150px
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D5	D4	D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Steriruncicantic 7-cube

Images

orthographic projections

Coxeter plane	B7	D7	D6
Graph	150px		150px
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D5	D4	D3
Graph	150px	150px	150px
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
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:

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

External links

• Weisstein, Eric W., "Hypercube", MathWorld.
• Olshevsky, George, Measure polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary