Cantellated 5-cell

Orthogonal projections in A₄ Coxeter plane
File:4-simplex t0.svg 5-cell Template:CDD	File:4-simplex t02.svg Cantellated 5-cell Template:CDD	File:4-simplex t012.svg Cantitruncated 5-cell Template:CDD

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.

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Cantellated 5-cell

Cantellated 5-cell
File:Schlegel half-solid cantellated 5-cell.png Schlegel diagram with octahedral cells shown
Type	Uniform 4-polytope
Schläfli symbol	t_0,2{3,3,3} rr{3,3,3}
Coxeter diagram	Template:CDD
Cells	20	5 File:Cuboctahedron.png(3.4.3.4) 5 (3.3.3.3) 10 File:Triangular prism.png(3.4.4)
Faces	80	50{3} 30{4}
Edges	90
Vertices	30
Vertex figure	Error creating thumbnail: Square wedge
Symmetry group	A₄, [3,3,3], order 120
Properties	convex, isogonal
Uniform index	3 4 5

Error creating thumbnail:

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

Images

Template:4-simplex Coxeter plane graphs

File:Cantel pentachoron1.png Wireframe	Ten triangular prisms colored green	Error creating thumbnail: Five octahedra colored blue

Coordinates

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

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

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.

File:Birhombatodecachoron vertex figure.png
Vertex figure

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Cantitruncated 5-cell

Cantitruncated 5-cell
Error creating thumbnail: 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	Template:CDD
Cells	20	5 File:Truncated octahedron.png(4.6.6) 10 File:Triangular prism.png(3.4.4) 5 Error creating thumbnail: (3.6.6)
Faces	80	20{3} 30{4} 30{6}
Edges	120
Vertices	60
Vertex figure	File:Cantitruncated 5-cell verf.png sphenoid
Symmetry group	A₄, [3,3,3], order 120
Properties	convex, isogonal
Uniform index	6 7 8

File:Great rhombated pentachoron net.png

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	f_k	f₀	f₁			f₂				f₃
Template:CDD	f₀	60	1	1	2	1	2	2	1	2	1	1
Template:CDD	f₁	2	30	*	*	1	2	0	0	2	1	0
Template:CDD		2	*	30	*	1	0	2	0	2	0	1
Template:CDD		2	*	*	60	0	1	1	1	1	1	1
Template:CDD	f₂	6	3	3	0	10	*	*	*	2	0	0
Template:CDD		4	2	0	2	*	30	*	*	1	1	0
Template:CDD		6	0	3	3	*	*	20	*	1	0	1
Template:CDD		3	0	0	3	*	*	*	20	0	1	1
Template:CDD	f₃	24	12	12	12	4	6	4	0	5	*	*
Template:CDD		6	3	0	6	0	3	0	2	*	10	*
Template:CDD		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

Template:4-simplex Coxeter plane graphs

File:Cantitruncated 5 cell.png 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
$(3 \sqrt{\frac{2}{5}}, \pm \sqrt{6}, \pm \sqrt{3}, \pm 1)$ $(3 \sqrt{\frac{2}{5}}, \pm \sqrt{6}, 0, \pm 2)$ $(3 \sqrt{\frac{2}{5}}, 0, 0, 0) \pm (0, \sqrt{\frac{2}{3}}, \frac{5}{\sqrt{3}}, \pm 1)$ $(3 \sqrt{\frac{2}{5}}, 0, 0, 0) \pm (0, \sqrt{\frac{2}{3}}, \frac{- 1}{\sqrt{3}}, \pm 3)$ $(3 \sqrt{\frac{2}{5}}, 0, 0, 0) \pm (0, \sqrt{\frac{2}{3}}, \frac{- 4}{\sqrt{3}}, \pm 2)$ $(\frac{1}{\sqrt{10}}, \frac{5}{\sqrt{6}}, \frac{5}{\sqrt{3}}, \pm 1)$ $(\frac{1}{\sqrt{10}}, \frac{5}{\sqrt{6}}, \frac{- 1}{\sqrt{3}}, \pm 3)$ $(\frac{1}{\sqrt{10}}, \frac{5}{\sqrt{6}}, \frac{- 4}{\sqrt{3}}, \pm 2)$ $(\frac{1}{\sqrt{10}}, - \sqrt{\frac{3}{2}}, \sqrt{3}, \pm 3)$ $(\frac{1}{\sqrt{10}}, - \sqrt{\frac{3}{2}}, - 2 \sqrt{3}, 0)$ $(\frac{1}{\sqrt{10}}, \frac{- 7}{\sqrt{6}}, \frac{2}{\sqrt{3}}, \pm 2)$ $(\frac{1}{\sqrt{10}}, \frac{- 7}{\sqrt{6}}, \frac{- 4}{\sqrt{3}}, 0)$ $(- 2 \sqrt{\frac{2}{5}}, 2 \sqrt{\frac{2}{3}}, \frac{4}{\sqrt{3}}, \pm 2)$	$(- 2 \sqrt{\frac{2}{5}}, 2 \sqrt{\frac{2}{3}}, \frac{1}{\sqrt{3}}, \pm 3)$ $(- 2 \sqrt{\frac{2}{5}}, 2 \sqrt{\frac{2}{3}}, \frac{- 5}{\sqrt{3}}, \pm 1)$ $(- 2 \sqrt{\frac{2}{5}}, 0, \sqrt{3}, \pm 3)$ $(- 2 \sqrt{\frac{2}{5}}, 0, - 2 \sqrt{3}, 0)$ $(- 2 \sqrt{\frac{2}{5}}, - 4 \sqrt{\frac{2}{3}}, \frac{1}{\sqrt{3}}, \pm 1)$ $(- 2 \sqrt{\frac{2}{5}}, - 4 \sqrt{\frac{2}{3}}, \frac{- 2}{\sqrt{3}}, 0)$ $(\frac{- 9}{\sqrt{10}}, \sqrt{\frac{3}{2}}, \pm \sqrt{3}, \pm 1)$ $(\frac{- 9}{\sqrt{10}}, \sqrt{\frac{3}{2}}, 0, \pm 2)$ $(\frac{- 9}{\sqrt{10}}, \frac{- 1}{\sqrt{6}}, \frac{2}{\sqrt{3}}, \pm 2)$ $(\frac{- 9}{\sqrt{10}}, \frac{- 1}{\sqrt{6}}, \frac{- 4}{\sqrt{3}}, 0)$ $(\frac{- 9}{\sqrt{10}}, \frac{- 5}{\sqrt{6}}, \frac{1}{\sqrt{3}}, \pm 1)$ $(\frac{- 9}{\sqrt{10}}, \frac{- 5}{\sqrt{6}}, \frac{- 2}{\sqrt{3}}, 0)$

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.

File:Bicantitruncatodecachoron vertex figure.png
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. Template:Pentachoron family

References

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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, Template:ISBN [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)
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Cantellated 5-cell

Contents