Cantellated 5-cell

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

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

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.

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.

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

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