Cantellated 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.

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

Cantellated 5-cell

Cantellated 5-cell
Schlegel diagram with octahedral cells shown
Type	Uniform 4-polytope
Schläfli symbol	t0,2{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

Net

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]

Element	fk	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
		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

Ak 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

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

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

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	t0,1,2{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

Net

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.

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.^[2]

Element	fk	f0	f1			f2				f3
	f0	60	1	1	2	1	2	2	1	2	1	1
	f1	2	30	*	*	1	2	0	0	2	1	0
		2	*	30	*	1	0	2	0	2	0	1
		2	*	*	60	0	1	1	1	1	1	1
	f2	6	3	3	0	10	*	*	*	2	0	0
		4	2	0	2	*	30	*	*	1	1	0
		6	0	3	3	*	*	20	*	1	0	1
		3	0	0	3	*	*	*	20	0	1	1
	f3	24	12	12	12	4	6	4	0	5	*	*
		6	3	0	6	0	3	0	2	*	10	*
		12	0	6	12	0	0	4	4	*	*	5

Alternative names

• Cantitruncated pentachoron
• Cantitruncated 4-simplex
• Great prismatodispentachoron
• Truncated dispentachoron
• Great rhombated pentachoron (Acronym: grip) (Jonathan Bowers)

Images

orthographic projections

Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Stereographic projection with its 10 triangular prisms.

Cartesian coordinates

The Cartesian coordinates of an origin-centered cantitruncated 5-cell having edge length 2 are:

Coordinates

${\displaystyle \left(3{\sqrt {\frac {2}{5}}},\ \pm {\sqrt {6}},\ \pm {\sqrt {3}},\ \pm 1\right)}$ ${\displaystyle \left(3{\sqrt {\frac {2}{5}}},\ \pm {\sqrt {6}},\ 0,\ \pm 2\right)}$ ${\displaystyle \left(3{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)\pm \left(0,\ {\sqrt {\frac {2}{3}}},\ {\frac {5}{\sqrt {3}}},\ \pm 1\right)}$ ${\displaystyle \left(3{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)\pm \left(0,\ {\sqrt {\frac {2}{3}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 3\right)}$ ${\displaystyle \left(3{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)\pm \left(0,\ {\sqrt {\frac {2}{3}}},\ {\frac {-4}{\sqrt {3}}},\ \pm 2\right)}$ ${\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {5}{\sqrt {3}}},\ \pm 1\right)}$ ${\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 3\right)}$ ${\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ \pm 2\right)}$ ${\displaystyle \left({\frac {1}{\sqrt {10}}},\ -{\sqrt {\frac {3}{2}}},\ {\sqrt {3}},\ \pm 3\right)}$ ${\displaystyle \left({\frac {1}{\sqrt {10}}},\ -{\sqrt {\frac {3}{2}}},\ -2{\sqrt {3}},\ 0\right)}$ ${\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {-7}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)}$ ${\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {-7}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ 0\right)}$ ${\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {4}{\sqrt {3}}},\ \pm 2\right)}$

${\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {1}{\sqrt {3}}},\ \pm 3\right)}$ ${\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {-5}{\sqrt {3}}},\ \pm 1\right)}$ ${\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ 0,\ {\sqrt {3}},\ \pm 3\right)}$ ${\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ 0,\ -2{\sqrt {3}},\ 0\right)}$ ${\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ -4{\sqrt {\frac {2}{3}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)}$ ${\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ -4{\sqrt {\frac {2}{3}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)}$ ${\displaystyle \left({\frac {-9}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ \pm {\sqrt {3}},\ \pm 1\right)}$ ${\displaystyle \left({\frac {-9}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ 0,\ \pm 2\right)}$ ${\displaystyle \left({\frac {-9}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)}$ ${\displaystyle \left({\frac {-9}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ 0\right)}$ ${\displaystyle \left({\frac {-9}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)}$ ${\displaystyle \left({\frac {-9}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)}$

These vertices can be more simply constructed on a hyperplane in 5-space, as the permutations of:

(0,0,1,2,3)

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

Related polytopes

A double symmetry construction can be made by placing truncated tetrahedra on the truncated octahedra, resulting in a nonuniform polychoron with 10 truncated tetrahedra, 20 hexagonal prisms (as ditrigonal trapezoprisms), two kinds of 80 triangular prisms (20 with D_3h symmetry and 60 C_2v-symmetric wedges), and 30 tetrahedra (as tetragonal disphenoids). Its vertex figure is topologically equivalent to the octahedron.

Vertex figure

Related 4-polytopes

These polytopes are art of a set of 9 Uniform 4-polytopes constructed from the [3,3,3] Coxeter group.

Name	5-cell	truncated 5-cell	rectified 5-cell	cantellated 5-cell	bitruncated 5-cell	cantitruncated 5-cell	runcinated 5-cell	runcitruncated 5-cell	omnitruncated 5-cell
Schläfli symbol	{3,3,3} 3r{3,3,3}	t{3,3,3} 3t{3,3,3}	r{3,3,3} 2r{3,3,3}	rr{3,3,3} r2r{3,3,3}	2t{3,3,3}	tr{3,3,3} t2r{3,3,3}	t0,3{3,3,3}	t0,1,3{3,3,3} t0,2,3{3,3,3}	t0,1,2,3{3,3,3}
Coxeter diagram
Schlegel diagram
A4 Coxeter plane Graph
A3 Coxeter plane Graph
A2 Coxeter plane Graph

References

1. Klitzing, Richard. "o3x4x3o - deca".
2. Klitzing, Richard. "x3x4x3o - grip".

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• 1. Convex uniform polychora based on the pentachoron - Model 4, 7, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora)". x3o3x3o - srip, x3x3x3o - grip

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds