Cantellated 5-simplexes

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

More information Orthogonal projections in A5 Coxeter plane ...

Orthogonal projections in A₅ Coxeter plane
5-simplex	Cantellated 5-simplex	Bicantellated 5-simplex
Birectified 5-simplex	Cantitruncated 5-simplex	Bicantitruncated 5-simplex

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There are unique 4 degrees of cantellation for the 5-simplex, including truncations.

Cantellated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	rr{3,3,3,3} = $r\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}$
Coxeter-Dynkin diagram	or
4-faces	27	6 r{3,3,3} 6 rr{3,3,3} 15 {}x{3,3}
Cells	135	30 {3,3} 30 r{3,3} 15 rr{3,3} 60 {}x{3}
Faces	290	200 {3} 90 {4}
Edges	240
Vertices	60
Vertex figure	Tetrahedral prism
Coxeter group	A₅ [3,3,3,3], order 720
Properties	convex

Bicantellated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	2rr{3,3,3,3} = $r\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$
Coxeter-Dynkin diagram	or
4-faces	32	12 t02{3,3,3} 20 {3}x{3}
Cells	180	30 t1{3,3} 120 {}x{3} 30 t02{3,3}
Faces	420	240 {3} 180 {4}
Edges	360
Vertices	90
Vertex figure
Coxeter group	A₅×2, [[3,3,3,3]], order 1440
Properties	convex, isogonal

Cantitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	tr{3,3,3,3} = $t\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}$
Coxeter-Dynkin diagram	or
4-faces	27	6 t012{3,3,3} 6 t{3,3,3} 15 {}x{3,3}
Cells	135	15 t012{3,3} 30 t{3,3} 60 {}x{3} 30 {3,3}
Faces	290	120 {3} 80 {6} 90 {}x{}
Edges	300
Vertices	120
Vertex figure	Irr. 5-cell
Coxeter group	A₅ [3,3,3,3], order 720
Properties	convex

Cantellated 5-simplex

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