Cantellated 5-cubes

In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.

More information Orthogonal projections in B5 Coxeter plane ...

Orthogonal projections in B₅ Coxeter plane
5-cube	Cantellated 5-cube	Bicantellated 5-cube	Cantellated 5-orthoplex
5-orthoplex	Cantitruncated 5-cube	Bicantitruncated 5-cube	Cantitruncated 5-orthoplex

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There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex

Cantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	rr{4,3,3,3} = $r\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}$
Coxeter-Dynkin diagram	=
4-faces	122	10 80 32
Cells	680	40 320 160 160
Faces	1520	80 480 320 640
Edges	1280	320+960
Vertices	320
Vertex figure
Coxeter group	B₅ [4,3,3,3]
Properties	convex, uniform

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

Bicantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbols	2rr{4,3,3,3} = $r\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}$ r{3^2,1,1} = $r\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}$
Coxeter-Dynkin diagrams	=
4-faces	122	10 80 32
Cells	840	40 240 160 320 80
Faces	2160	240 320 960 320 320
Edges	1920	960+960
Vertices	480
Vertex figure
Coxeter groups	B₅, [3,3,3,4] D₅, [3^2,1,1]
Properties	convex, uniform

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

Cantitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	tr{4,3,3,3} = $t\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}$
Coxeter-Dynkin diagram	=
4-faces	122	10 80 32
Cells	680	40 320 160 160
Faces	1520	80 480 320 640
Edges	1600	320+320+960
Vertices	640
Vertex figure
Coxeter group	B₅ [4,3,3,3]
Properties	convex, uniform

Cantellated 5-cube

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