Cantellated 5-cell

5-cell	Cantellated 5-cell	Cantitruncated 5-cell
Orthogonal projections in A4 Coxeter plane

In four-dimensional geometry, a cantellated 5-cell is a convex uniform 4-polytope, 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.

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	80px Irreg. triangular prism
Symmetry group	A4, [3,3,3], order 120
Properties	convex, isogonal
Uniform index	3 4 5

File:Small rhombated pentachoron net.png

Net

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		100px	100px
Dihedral symmetry	[5]	[4]	[3]

160px Wireframe

160px Ten triangular prisms colored green

160px 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 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	80px sphenoid
Symmetry group	A4, [3,3,3], order 120
Properties	convex, isogonal
Uniform index	6 7 8

File:Great rhombated pentachoron net.png

Net

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		100px	100px
Dihedral symmetry	[5]	[4]	[3]

240px 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 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		80px	80px	80px	80px	80px	80px	80px	80px
A2 Coxeter plane Graph		80px	80px	80px	80px	80px	80px	80px	80px

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.
• Richard Klitzing, 4D, uniform polytopes (polychora) x3o3x3o - srip, x3x3x3o - grip