# Cantellated 5-cell

 Orthogonal projections in A4Coxeter plane 5-cell Cantellated 5-cell Cantitruncated 5-cell

In four-dimensional geometry, a cantellated 5-cell is a convex uniform polychoron, being a cantellation (a 2nd order truncation) of the regular 5-cell.

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

## Cantellation 5-cell

Cantellated 5-cell

Schlegel diagram with
octahedral cells shown
Type Uniform polychoron
Schläfli symbol t0,2{3,3,3}
Coxeter-Dynkin 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 is a uniform polychoron. 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:

 $\left(2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\ \pm1\right)$ $\left(2\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ 0\right)$ $\left(2\sqrt{\frac{2}{5}},\ 0,\ \pm\sqrt{3},\ \pm1\right)$ $\left(2\sqrt{\frac{2}{5}},\ 0,\ 0,\ \pm2\right)$ $\left(2\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{2}{\sqrt{3}},\ 0\right)$ $\left(2\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm1\right)$ $\left(\frac{-1}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ \pm\sqrt{3},\ \pm1\right)$ $\left(\frac{-1}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ 0,\ \pm2\right)$ $\left(\frac{-1}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ \pm2\right)$ $\left(\frac{-1}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ 0\right)$ $\left(\frac{-1}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)$ $\left(\frac{-1}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right)$ $\left(-3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right) \pm \left(0,\ \sqrt{\frac{2}{3}},\ \frac{2}{\sqrt{3}},\ 0\right)$ $\left(-3\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0\right) \pm \left(0,\ \sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm1\right)$

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.

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## Cantitruncated 5-cell

Cantitruncated 5-cell

Schlegel diagram with Truncated tetrahedral cells shown
Type Uniform polychoron
Schläfli symbol t0,1,2{3,3,3}
Coxeter-Dynkin 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 is a uniform polychoron. 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:

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

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.

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## Related polychora

These polytopes are art of a set of 9 uniform polychora 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} t0,1{3,3,3} t1{3,3,3} t0,2{3,3,3} t1,2{3,3,3} t0,1,2{3,3,3} t0,3{3,3,3} t0,1,3{3,3,3} t0,1,2,3{3,3,3}
Coxeter-Dynkin
diagram
Schlegel
diagram
A4
Coxeter plane
Graph
A3 Coxeter plane
Graph
A2 Coxeter plane
Graph
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## 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, editied 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.
• Richard Klitzing, 4D, uniform polytopes (polychora) x3o3x3o - srip, x3x3x3o - grip
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