Cantellated 5-cell

Orthogonal projections in A₄ Coxeter plane
5-cell	Cantellated 5-cell	Cantitruncated 5-cell

In four-dimensional geometry, a cantellated 5-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation, up to edge-planing) of the regular 5-cell.

Cantellated 5-cell
Schlegel diagram with octahedral cells shown
Type	Uniform 4-polytope
Schläfli symbol	t_0,2{3,3,3} rr{3,3,3}
Coxeter diagram
Cells	20	5 (3.4.3.4) 5 (3.3.3.3) 10 (3.4.4)
Faces	80	50{3} 30{4}
Edges	90
Vertices	30
Vertex figure	Square wedge
Symmetry group	A₄, [3,3,3], order 120
Properties	convex, isogonal
Uniform index	3 4 5

The cantellated 5-cell or small rhombated pentachoron is a uniform 4-polytope. It has 30 vertices, 90 edges, 80 faces, and 20 cells. The cells are 5 cuboctahedra, 5 octahedra, and 10 triangular prisms. Each vertex is surrounded by 2 cuboctahedra, 2 triangular prisms, and 1 octahedron; the vertex figure is a nonuniform triangular prism.

Alternate names

Cantellated pentachoron
Cantellated 4-simplex
(small) prismatodispentachoron
Rectified dispentachoron
Small rhombated pentachoron (Acronym: Srip) (Jonathan Bowers)

Configuration

Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.^[1]

f_k	f₀	f₁		f₂				f₃
f₀	30	2	4	1	4	2	2	2	2	1
f₁	2	30	*	1	2	0	0	2	1	0
f₁	2	*	60	0	1	1	1	1	1	1
f₂	3	3	0	10	*	*	*	2	0	0
	4	2	2	*	30	*	*	1	1	0
	3	0	3	*	*	20	*	1	0	1
	3	0	3	*	*	*	20	0	1	1
f₃	12	12	12	4	6	4	0	5	*	*
	6	3	6	0	3	0	2	*	10	*
	6	0	12	0	0	4	4	*	*	5

Images

orthographic projections
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Wireframe	Ten triangular prisms colored green	Five octahedra colored blue

Coordinates

The Cartesian coordinates of the vertices of the origin-centered cantellated 5-cell having edge length 2 are:

Coordinates
$\left(2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)$ $\left(2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)$ $\left(2{\sqrt {\frac {2}{5}}},\ 0,\ \pm {\sqrt {3}},\ \pm 1\right)$ $\left(2{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ \pm 2\right)$ $\left(2{\sqrt {\frac {2}{5}}},\ -2{\sqrt {\frac {2}{3}}},\ {\frac {2}{\sqrt {3}}},\ 0\right)$ $\left(2{\sqrt {\frac {2}{5}}},\ -2{\sqrt {\frac {2}{3}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 1\right)$ $\left({\frac {-1}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ \pm {\sqrt {3}},\ \pm 1\right)$	$\left({\frac {-1}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ 0,\ \pm 2\right)$ $\left({\frac {-1}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)$ $\left({\frac {-1}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ 0\right)$ $\left({\frac {-1}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)$ $\left({\frac {-1}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)$ $\left(-3{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)\pm \left(0,\ {\sqrt {\frac {2}{3}}},\ {\frac {2}{\sqrt {3}}},\ 0\right)$ $\left(-3{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)\pm \left(0,\ {\sqrt {\frac {2}{3}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 1\right)$

The vertices of the cantellated 5-cell can be most simply positioned in 5-space as permutations of:

(0,0,1,1,2)

This construction is from the positive orthant facet of the cantellated 5-orthoplex.

Related polytopes

The convex hull of two cantellated 5-cells in opposite positions is a nonuniform polychoron composed of 100 cells: three kinds of 70 octahedra (10 rectified tetrahedra, 20 triangular antiprisms, 40 triangular antipodiums), 30 tetrahedra (as tetragonal disphenoids), and 60 vertices. Its vertex figure is a shape topologically equivalent to a cube with a triangular prism attached to one of its square faces.

Vertex figure

Cantitruncated 5-cell

Cantitruncated 5-cell
Schlegel diagram with Truncated tetrahedral cells shown
Type	Uniform 4-polytope
Schläfli symbol	t_0,1,2{3,3,3} tr{3,3,3}
Coxeter diagram
Cells	20	5 (4.6.6) 10 (3.4.4) 5 (3.6.6)
Faces	80	20{3} 30{4} 30{6}
Edges	120
Vertices	60
Vertex figure	sphenoid
Symmetry group	A₄, [3,3,3], order 120
Properties	convex, isogonal
Uniform index	6 7 8

The cantitruncated 5-cell or great rhombated pentachoron is a uniform 4-polytope. It is composed of 60 vertices, 120 edges, 80 faces, and 20 cells. The cells are: 5 truncated octahedra, 10 triangular prisms, and 5 truncated tetrahedra. Each vertex is surrounded by 2 truncated octahedra, one triangular prism, and one truncated tetrahedron.