Cantellated 5-cell explained

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

bgcolor=#e7dcc3 align=center colspan=3	Cantellated 5-cell
align=center colspan=3	Schlegel diagram with octahedral cells shown
Type	Uniform 4-polytope
Schläfli symbol	t_0,2 rr
Coxeter diagram
Cells	20	5 (3.4.3.4) 5 (3.3.3.3) 10 (3.4.4)
Faces	80	50 30
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.

Element	f_k	f₀	f₁		f₂				f₃
align=left bgcolor=#ffffe0		f₀	30	2	4	1	4	2	2	2	2	1
align=left bgcolor=#ffffe0		f₁	2	30		1	2	0	0	2	1	0
align=left bgcolor=#ffffe0		f₁	2		60	0	1	1	1	1	1	1
align=left bgcolor=#ffffe0		f₂	3	3	0	10				2	0	0
align=left bgcolor=#ffffe0			4	2	2		30			1	1	0
align=left bgcolor=#ffffe0			3	0	3			20		1	0	1
align=left bgcolor=#ffffe0			3	0	3				20	0	1	1
align=left bgcolor=#ffffe0		f₃	12	12	12	4	6	4	0	5
align=left bgcolor=#ffffe0			6	3	6	0	3	0	2		10
align=left bgcolor=#ffffe0			6	0	12	0	0	4	4			5

Coordinates

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

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

bgcolor=#e7dcc3 align=center colspan=3	Cantitruncated 5-cell
align=center colspan=3	Schlegel diagram with Truncated tetrahedral cells shown
Type	Uniform 4-polytope
Schläfli symbol	t_0,1,2 tr
Coxeter diagram
Cells	20	5 (4.6.6) 10 (3.4.4) 5 (3.6.6)
Faces	80	20 30 30
Edges	120
Vertices	60
bgcolor=#e7dcc3 valign=center	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.

Configuration

Element	f_k	f₀	f₁			f₂				f₃
align=left bgcolor=#ffffe0		f₀	60	1	1	2	1	2	2	1	2	1	1
align=left bgcolor=#ffffe0		f₁	2	30			1	2	0	0	2	1	0
align=left bgcolor=#ffffe0			2		30		1	0	2	0	2	0	1
align=left bgcolor=#ffffe0			2			60	0	1	1	1	1	1	1
align=left bgcolor=#ffffe0		f₂	6	3	3	0	10				2	0	0
align=left bgcolor=#ffffe0			4	2	0	2		30			1	1	0
align=left bgcolor=#ffffe0			6	0	3	3			20		1	0	1
align=left bgcolor=#ffffe0			3	0	0	3				20	0	1	1
align=left bgcolor=#ffffe0		f₃	24	12	12	12	4	6	4	0	5
align=left bgcolor=#ffffe0			6	3	0	6	0	3	0	2		10
align=left bgcolor=#ffffe0			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)

Cartesian coordinates

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

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

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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  - (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)
- x3o3x3o - srip, x3x3x3o - grip