# 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.

There are 2 unique degrees of runcinations of the 5-cell including with permutations truncations.

## 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
Irreg. triangular prism
Symmetry group A4, [3,3,3], order 120
Properties convex, isogonal
Uniform index 3 4 5

The cantellated 5-cell or small rhombated pentachroron 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)

### 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:

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.

## 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

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.

### 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:

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 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}
2t{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

• 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)". x3o3x3o - srip, x3x3x3o - grip
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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds