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# Cantellated 5-cell

In four dimensional geometry, 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

* Srip (Jonathan Bowers: for small rhombated pentachoron)

Images

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)

Related uniform polychora

The cantellated pentachoron is one 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 |
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SchlĂ¤fli symbol |
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Coxeter-Dynkin diagram |
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Schlegel diagram |
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Coxeter plane projection |

Images: *Robert Webb's Great Stella software*

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

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