Steric 5-cubes

5-cube	Steric 5-cube	Stericantic 5-cube
Half 5-cube	Steriruncic 5-cube	Steriruncicantic 5-cube
Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a steric 5-cube or (steric 5-demicube or sterihalf 5-cube) is a convex uniform 5-polytope. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of stericated 5-cubes.

Steric 5-cube

Steric 5-cube
Type	uniform polyteron
Schläfli symbol	t0,3{3,3^2,1} h4{4,3,3,3}
Coxeter-Dynkin diagram
4-faces	82
Cells	480
Faces	720
Edges	400
Vertices	80
Vertex figure	{3,3}-t1{3,3} antiprism
Coxeter groups	D5, [3^2,1,1]
Properties	convex

Alternate names

• Steric penteract, runcinated demipenteract
• Small prismated hemipenteract (siphin) (Jonathan Bowers)^[1]

Cartesian coordinates

The Cartesian coordinates for the 80 vertices of a steric 5-cube centered at the origin are the permutations of

(±1,±1,±1,±1,±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

Dimensional family of steric n-cubes
n	5	6	7	8
[1^+,4,3^n-2] = [3,3^n-3,1]	[1^+,4,3^3] = [3,3^2,1]	[1^+,4,3^4] = [3,3^3,1]	[1^+,4,3^5] = [3,3^4,1]	[1^+,4,3^6] = [3,3^5,1]
Steric figure
Coxeter	=	=	=	=
Schläfli	h4{4,3^3}	h4{4,3^4}	h4{4,3^5}	h4{4,3^6}

Stericantic 5-cube

Stericantic 5-cube
Type	uniform polyteron
Schläfli symbol	t0,1,3{3,3^2,1} h2,4{4,3,3,3}
Coxeter-Dynkin diagram
4-faces	82
Cells	720
Faces	1840
Edges	1680
Vertices	480
Vertex figure
Coxeter groups	D5, [3^2,1,1]
Properties	convex

Alternate names

• Prismatotruncated hemipenteract (pithin) (Jonathan Bowers)^[2]

Cartesian coordinates

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

(±1,±1,±3,±3,±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]

Steriruncic 5-cube

Steriruncic 5-cube
Type	uniform polyteron
Schläfli symbol	t0,2,3{3,3^2,1} h3,4{4,3,3,3}
Coxeter-Dynkin diagram
4-faces	82
Cells	560
Faces	1280
Edges	1120
Vertices	320
Vertex figure
Coxeter groups	D5, [3^2,1,1]
Properties	convex

Alternate names

• Prismatorhombated hemipenteract (pirhin) (Jonathan Bowers)^[3]

Cartesian coordinates

The Cartesian coordinates for the 320 vertices of a steriruncic 5-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±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]

Steriruncicantic 5-cube

Steriruncicantic 5-cube
Type	uniform polyteron
Schläfli symbol	t0,1,2,3{3,3^2,1} h2,3,4{4,3,3,3}
Coxeter-Dynkin diagram
4-faces	82
Cells	720
Faces	2080
Edges	2400
Vertices	960
Vertex figure
Coxeter groups	D5, [3^2,1,1]
Properties	convex

Alternate names

• Great prismated hemipenteract (giphin) (Jonathan Bowers)^[4]

Cartesian coordinates

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

(±1,±1,±3,±5,±7)

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

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 polytera (uniform 5-polytope) 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}

Notes

1. Klitzing, (x3o3o *b3o3x - siphin)
2. Klitzing, (x3x3o *b3o3x - pithin)
3. Klitzing, (x3o3o *b3x3x - pirhin)
4. Klitzing, (x3x3o *b3x3x - giphin)

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
    • (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 *b3o3x - siphin, x3x3o *b3o3x - pithin, x3o3o *b3x3x - pirhin, x3x3o *b3x3x - giphin

External links

• Weisstein, Eric W. "Hypercube". MathWorld.

• Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / E9 / E10 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform 4-polytope	5-cell	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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds