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

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 20px(3.3.3.3) 10 20px(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]

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

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
[math]\displaystyle{ \left(2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\ \pm1\right) }[/math] [math]\displaystyle{ \left(2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ 0\right) }[/math] [math]\displaystyle{ \left(2\sqrt{\frac{2}{5}},\ 0,\ \pm\sqrt{3},\ \pm1\right) }[/math] [math]\displaystyle{ \left(2\sqrt{\frac{2}{5}},\ 0,\ 0,\ \pm2\right) }[/math] [math]\displaystyle{ \left(2\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{2}{\sqrt{3}},\ 0\right) }[/math] [math]\displaystyle{ \left(2\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm1\right) }[/math] [math]\displaystyle{ \left(\frac{-1}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ \pm\sqrt{3},\ \pm1\right) }[/math]	[math]\displaystyle{ \left(\frac{-1}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ 0,\ \pm2\right) }[/math] [math]\displaystyle{ \left(\frac{-1}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ \pm2\right) }[/math] [math]\displaystyle{ \left(\frac{-1}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ 0\right) }[/math] [math]\displaystyle{ \left(\frac{-1}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right) }[/math] [math]\displaystyle{ \left(\frac{-1}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right) }[/math] [math]\displaystyle{ \left(-3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right) \pm \left(0,\ \sqrt{\frac{2}{3}},\ \frac{2}{\sqrt{3}},\ 0\right) }[/math] [math]\displaystyle{ \left(-3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right) \pm \left(0,\ \sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm1\right) }[/math]

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 20px(3.4.4) 5 20px(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]

f_k	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

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
[math]\displaystyle{ \left(3\sqrt{\frac{2}{5}},\ \pm\sqrt{6},\ \pm\sqrt{3},\ \pm1\right) }[/math] [math]\displaystyle{ \left(3\sqrt{\frac{2}{5}},\ \pm\sqrt{6},\ 0,\ \pm2\right) }[/math] [math]\displaystyle{ \left(3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right) \pm \left(0,\ \sqrt{\frac{2}{3}},\ \frac{5}{\sqrt{3}},\ \pm1\right) }[/math] [math]\displaystyle{ \left(3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right) \pm \left(0,\ \sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm3\right) }[/math] [math]\displaystyle{ \left(3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right) \pm \left(0,\ \sqrt{\frac{2}{3}},\ \frac{-4}{\sqrt{3}},\ \pm2\right) }[/math] [math]\displaystyle{ \left(\frac{1}{\sqrt{10}},\ \frac{5}{\sqrt{6}},\ \frac{5}{\sqrt{3}},\ \pm1\right) }[/math] [math]\displaystyle{ \left(\frac{1}{\sqrt{10}},\ \frac{5}{\sqrt{6}},\ \frac{-1}{\sqrt{3}},\ \pm3\right) }[/math] [math]\displaystyle{ \left(\frac{1}{\sqrt{10}},\ \frac{5}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ \pm2\right) }[/math] [math]\displaystyle{ \left(\frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ \sqrt{3},\ \pm3\right) }[/math] [math]\displaystyle{ \left(\frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ -2\sqrt{3},\ 0\right) }[/math] [math]\displaystyle{ \left(\frac{1}{\sqrt{10}},\ \frac{-7}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ \pm2\right) }[/math] [math]\displaystyle{ \left(\frac{1}{\sqrt{10}},\ \frac{-7}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ 0\right) }[/math] [math]\displaystyle{ \left(-2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{4}{\sqrt{3}},\ \pm2\right) }[/math]	[math]\displaystyle{ \left(-2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\ \pm3\right) }[/math] [math]\displaystyle{ \left(-2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{-5}{\sqrt{3}},\ \pm1\right) }[/math] [math]\displaystyle{ \left(-2\sqrt{\frac{2}{5}},\ 0,\ \sqrt{3},\ \pm3\right) }[/math] [math]\displaystyle{ \left(-2\sqrt{\frac{2}{5}},\ 0,\ -2\sqrt{3},\ 0\right) }[/math] [math]\displaystyle{ \left(-2\sqrt{\frac{2}{5}},\ -4\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\ \pm1\right) }[/math] [math]\displaystyle{ \left(-2\sqrt{\frac{2}{5}},\ -4\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ 0\right) }[/math] [math]\displaystyle{ \left(\frac{-9}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ \pm\sqrt{3},\ \pm1\right) }[/math] [math]\displaystyle{ \left(\frac{-9}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ 0,\ \pm2\right) }[/math] [math]\displaystyle{ \left(\frac{-9}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ \pm2\right) }[/math] [math]\displaystyle{ \left(\frac{-9}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ 0\right) }[/math] [math]\displaystyle{ \left(\frac{-9}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right) }[/math] [math]\displaystyle{ \left(\frac{-9}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right) }[/math]

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.

References

↑ Klitzing, Richard. "o3x4x3o - deca". https://bendwavy.org/klitzing/dimensions/../incmats/deca.htm.
↑ Klitzing, Richard. "x3x4x3o - grip". https://bendwavy.org/klitzing/dimensions/../incmats/grip.htm.

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 [1]
  - (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)". https://bendwavy.org/klitzing/dimensions/polychora.htm. x3o3x3o - srip, x3x3x3o - grip

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform 4-polytope	5-cell	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

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Original source: https://en.wikipedia.org/wiki/Cantellated 5-cell. Read more

[1] Klitzing, Richard. "o3x4x3o - deca". https://bendwavy.org/klitzing/dimensions/../incmats/deca.htm.

[2] Klitzing, Richard. "x3x4x3o - grip". https://bendwavy.org/klitzing/dimensions/../incmats/grip.htm.

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