// Workers AI · dad joke modeWhat did the Runcic 7-cubes say? "We're cubing our options.

7-demicube	Runcic 7-cube	Runcicantic 7-cube
Orthogonal projections in D7 Coxeter plane

In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.

Runcic 7-cube

Runcic 7-cube
Type	uniform 7-polytope
Schläfli symbol	t0,2{3,34,1} h3{4,35}
Coxeter-Dynkin diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	16800
Vertices	2240
Vertex figure
Coxeter groups	D7, [34,1,1]
Properties	convex

A runcic 7-cube, h₃{4,3⁵}, has half the vertices of a runcinated 7-cube, t_0,3{4,3⁵}.

Alternate names

• Small rhombated hemihepteract (acronym: sirhesa) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the vertices of a cantellated demihepteract centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±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]

Runcicantic 7-cube

Runcicantic 7-cube
Type	uniform 7-polytope
Schläfli symbol	t0,1,2{3,34,1} h2,3{4,35}
Coxeter-Dynkin diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges	23520
Vertices	6720
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

A runcicantic 7-cube, h_2,3{4,3⁵}, has half the vertices of a runcicantellated 7-cube, t_0,1,3{4,3⁵}.

Alternate names

• Great rhombated hemihepteract (acronym: girhesa) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the vertices of a runcicantic 7-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±5,±5)

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]

Related polytopes

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:

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 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". x3o3o *b3x3o3o3o - sirhesa, x3x3o *b3x3o3o3o - girhesa

External links

• Weisstein, Eric W. "Hypercube". MathWorld.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary