Runcic 5-cubes

5-cube	Runcic 5-cube =
5-demicube =	Runcicantic 5-cube =
Orthogonal projections in B5 Coxeter plane

In six-dimensional geometry, a runcic 5-cube or (runcic 5-demicube, runcihalf 5-cube) is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

Runcic 5-cube
Type	uniform 5-polytope
Schläfli symbol	h3{4,3,3,3}
Coxeter-Dynkin diagram
4-faces	42
Cells	360
Faces	880
Edges	720
Vertices	160
Vertex figure
Coxeter groups	D5, [32,1,1]
Properties	convex

• Cantellated 5-demicube/demipenteract
• Small rhombated hemipenteract (sirhin) (Jonathan Bowers)^[1]

The Cartesian coordinates for the 960 vertices of a runcic 5-cubes centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3)

with an odd number of plus signs.

orthographic projections

Coxeter plane	B5
Graph
Dihedral symmetry	[10/2]
Coxeter plane	D5	D4
Graph
Dihedral symmetry	[8]	[6]
Coxeter plane	D3	A3
Graph
Dihedral symmetry	[4]	[4]

It has half the vertices of the runcinated 5-cube, as compared here in the B5 Coxeter plane projections:

Runcic 5-cube

Runcinated 5-cube

Runcic n-cubes
n	4	5	6	7	8
[1+,4,3n-2] = [3,3n-3,1]	[1+,4,32] = [3,31,1]	[1+,4,33] = [3,32,1]	[1+,4,34] = [3,33,1]	[1+,4,35] = [3,34,1]	[1+,4,36] = [3,35,1]
Runcic figure
Coxeter	=	=	=	=	=
Schläfli	h3{4,32}	h3{4,33}	h3{4,34}	h3{4,35}	h3{4,36}

Runcicantic 5-cube
Type	uniform 5-polytope
Schläfli symbol	t0,1,2{3,32,1} h3{4,33}
Coxeter-Dynkin diagram
4-faces	42
Cells	360
Faces	1040
Edges	1200
Vertices	480
Vertex figure
Coxeter groups	D5, [32,1,1]
Properties	convex

• Cantitruncated 5-demicube/demipenteract
• Great rhombated hemipenteract (girhin) (Jonathan Bowers)^[2]

The Cartesian coordinates for the 480 vertices of a runcicantic 5-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±5,±5)

with an odd number of plus signs.

orthographic projections

Coxeter plane	B5
Graph
Dihedral symmetry	[10/2]
Coxeter plane	D5	D4
Graph
Dihedral symmetry	[8]	[6]
Coxeter plane	D3	A3
Graph
Dihedral symmetry	[4]	[4]

It has half the vertices of the runcicantellated 5-cube, as compared here in the B5 Coxeter plane projections:

Runcicantic 5-cube

Runcicantellated 5-cube

This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 uniform 5-polytopes that can be constructed from the D₅ symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.

D5 polytopes
h{4,3,3,3}	h2{4,3,3,3}	h3{4,3,3,3}	h4{4,3,3,3}	h2,3{4,3,3,3}	h2,4{4,3,3,3}	h3,4{4,3,3,3}	h2,3,4{4,3,3,3}

• 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, 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. "5D uniform polytopes (polytera) with acronyms". x3o3o *b3x3o - sirhin, x3x3o *b3x3o - girhin

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