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

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

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, [3^2,1,1]
Properties	convex

Alternate names

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

Cartesian coordinates

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.

Images

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]

Related polytopes

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

Runcicantic 5-cube

Runcicantic 5-cube
Type	uniform 5-polytope
Schläfli symbol	t_0,1,2{3,3^2,1} h₃{4,3³}
Coxeter-Dynkin diagram
4-faces	42
Cells	360
Faces	1040
Edges	1200
Vertices	480
Vertex figure
Coxeter groups	D₅, [3^2,1,1]
Properties	convex

Alternate names

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

Cartesian coordinates

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.

Images

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]

Related polytopes

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

Related polytopes

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}	h_2,3{4,3,3,3}	h_2,4{4,3,3,3}	h_3,4{4,3,3,3}	h_2,3,4{4,3,3,3}

Notes

^ Klitzing, (x3o3o *b3x3o - sirhin)
^ Klitzing, (x3x3o *b3x3o - girhin)

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

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds