Runcinated 5-cubes

In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube.

More information Orthogonal projections in B5 Coxeter plane ...

Orthogonal projections in B₅ Coxeter plane
5-cube	Runcinated 5-cube	Runcinated 5-orthoplex
Runcitruncated 5-cube	Runcicantellated 5-cube	Runcicantitruncated 5-cube
Runcitruncated 5-orthoplex	Runcicantellated 5-orthoplex	Runcicantitruncated 5-orthoplex

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There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncations and cantellations. Four are more simply constructed relative to the 5-orthoplex.

Runcinated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t_0,3{4,3,3,3}
Coxeter diagram
4-faces	202	10 80 80 32
Cells	1240	40 240 320 160 320 160
Faces	2160	240 960 640 320
Edges	1440	480+960
Vertices	320
Vertex figure
Coxeter group	B₅ [4,3,3,3]
Properties	convex

Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph
Dihedral symmetry	[4]	[4]

Runcitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t_0,1,3{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces	202	10 80 80 32
Cells	1560	40 240 320 320 160 320 160
Faces	3760	240 960 320 960 640 640
Edges	3360	480+960+1920
Vertices	960
Vertex figure
Coxeter group	B₅, [3,3,3,4]
Properties	convex

Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph
Dihedral symmetry	[4]	[4]

Runcicantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t_0,2,3{4,3,3,3}
Coxeter-Dynkin diagram
4-faces	202	10 80 80 32
Cells	1240	40 240 320 320 160 160
Faces	2960	240 480 960 320 640 320
Edges	2880	960+960+960
Vertices	960
Vertex figure
Coxeter group	B₅ [4,3,3,3]
Properties	convex

Runcinated 5-cubes

Runcinated 5-cube

Alternate names

Coordinates

Images

Runcitruncated 5-cube

Alternate names

Construction and coordinates

Images

Runcicantellated 5-cube

Alternate names

Coordinates

Images

Runcicantitruncated 5-cube

Alternate names

Coordinates

Images

Related polytopes

Notes

References

External links

Related Articles

B5 polytopes
β₅	t₁β₅	t₂γ₅	t₁γ₅	γ₅	t_0,1β₅	t_0,2β₅	t_1,2β₅
t_0,3β₅	t_1,3γ₅	t_1,2γ₅	t_0,4γ₅	t_0,3γ₅	t_0,2γ₅	t_0,1γ₅	t_0,1,2β₅
t_0,1,3β₅	t_0,2,3β₅	t_1,2,3γ₅	t_0,1,4β₅	t_0,2,4γ₅	t_0,2,3γ₅	t_0,1,4γ₅	t_0,1,3γ₅
t_0,1,2γ₅	t_0,1,2,3β₅	t_0,1,2,4β₅	t_0,1,3,4γ₅	t_0,1,2,4γ₅	t_0,1,2,3γ₅	t_0,1,2,3,4γ₅

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
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 • Polytope operations