Steric 5-cubes

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.

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

Steric 5-cube

Steric 5-cube
Type	uniform polyteron
Schläfli symbol	t_0,3{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]^{: (x3o3o *b3o3x - siphin)}

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⁺,4,3³] = [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]
Steric figure
Coxeter	=	=	=	=
Schläfli	h₄{4,3³}	h₄{4,3⁴}	h₄{4,3⁵}	h₄{4,3⁶}

Stericantic 5-cube

Stericantic 5-cube
Type	uniform polyteron
Schläfli symbol	t_0,1,3{3,3^2,1} h_2,4{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)[1]^{: (x3x3o *b3o3x - pithin)}

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	t_0,2,3{3,3^2,1} h_3,4{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)[1]^{: (x3o3o *b3x3x - pirhin)}

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	t_0,1,2,3{3,3^2,1} h_2,3,4{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)[1]^{: (x3x3o *b3x3x - giphin)}

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}	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}

References

Klitzing, Richard. "5D uniform polytopes (polytera)".

Further reading

Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed.). New York City: Dover. Retrieved 2022-05-19.
Coxeter, H. S. M. (1995-05-17). Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivić (eds.). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons. ISBN 978-0-471-01003-6. LCCN 94047368. OCLC 632987525. OL 7598569M. Retrieved 2022-05-19.
Coxeter, H. S. M. (1940-12-01). "Regular and Semi Regular Polytopes I". Mathematische Zeitschrift. Springer Nature. 46: 380–407. doi:10.1007/BF01181449. ISSN 1432-1823. S2CID 186237114. Retrieved 2022-05-19.
Coxeter, H. S. M. (1985-12-01). "Regular and Semi-Regular Polytopes II". Mathematische Zeitschrift. Springer Nature. 188 (4): 559–591. doi:10.1007/BF01161657. ISSN 1432-1823. S2CID 120429557. Retrieved 2022-05-19.
Coxeter, H. S. M. (1988-03-01). "Regular and Semi-Regular Polytopes III". Mathematische Zeitschrift. Springer Nature. 200 (1): 3–45. doi:10.1007/BF01161745. ISSN 1432-1823. S2CID 186237142. Retrieved 2022-05-19.
Johnson, Norman W. (1991). Uniform Polytopes (Unfinished manuscript thesis).
Johnson, Norman W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD thesis). University of Toronto. Retrieved 2022-05-19.

External links

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

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