Alternate names

• Cantellated pentachoron
• Cantellated 4-simplex
• (small) prismatodispentachoron
• Rectified dispentachoron
• Small rhombated pentachoron (Acronym: Srip) (Jonathan Bowers)

Configuration

Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.^[1]

More information Element, fk ...

Element	fk	f0	f1		f2				f3
	f0	30	2	4	1	4	2	2	2	2	1
	f1	2	30	*	1	2	0	0	2	1	0
		2	*	60	0	1	1	1	1	1	1
	f2	3	3	0	10	*	*	*	2	0	0
		4	2	2	*	30	*	*	1	1	0
		3	0	3	*	*	20	*	1	0	1
		3	0	3	*	*	*	20	0	1	1
	f3	12	12	12	4	6	4	0	5	*	*
		6	3	6	0	3	0	2	*	10	*
		6	0	12	0	0	4	4	*	*	5

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Images

More information Ak Coxeter plane, A4 ...

orthographic projections

Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

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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:

More information , ...

Coordinates

${\displaystyle \left(2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)}$ ${\displaystyle \left(2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)}$ ${\displaystyle \left(2{\sqrt {\frac {2}{5}}},\ 0,\ \pm {\sqrt {3}},\ \pm 1\right)}$ ${\displaystyle \left(2{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ \pm 2\right)}$ ${\displaystyle \left(2{\sqrt {\frac {2}{5}}},\ -2{\sqrt {\frac {2}{3}}},\ {\frac {2}{\sqrt {3}}},\ 0\right)}$ ${\displaystyle \left(2{\sqrt {\frac {2}{5}}},\ -2{\sqrt {\frac {2}{3}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 1\right)}$ ${\displaystyle \left({\frac {-1}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ \pm {\sqrt {3}},\ \pm 1\right)}$

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

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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.

Related polytopes

The convex hull of two cantellated 5-cells in opposite positions is a nonuniform polychoron composed of 100 cells: three kinds of 70 octahedra (10 rectified tetrahedra, 20 triangular antiprisms, 40 triangular antipodiums), 30 tetrahedra (as tetragonal disphenoids), and 60 vertices. Its vertex figure is a shape topologically equivalent to a cube with a triangular prism attached to one of its square faces.

Vertex figure

Configuration

Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.^[2]

More information Element, fk ...

Element	fk	f0	f1			f2				f3
	f0	60	1	1	2	1	2	2	1	2	1	1
	f1	2	30	*	*	1	2	0	0	2	1	0
		2	*	30	*	1	0	2	0	2	0	1
		2	*	*	60	0	1	1	1	1	1	1
	f2	6	3	3	0	10	*	*	*	2	0	0
		4	2	0	2	*	30	*	*	1	1	0
		6	0	3	3	*	*	20	*	1	0	1
		3	0	0	3	*	*	*	20	0	1	1
	f3	24	12	12	12	4	6	4	0	5	*	*
		6	3	0	6	0	3	0	2	*	10	*
		12	0	6	12	0	0	4	4	*	*	5

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Alternative names

• Cantitruncated pentachoron
• Cantitruncated 4-simplex
• Great prismatodispentachoron
• Truncated dispentachoron
• Great rhombated pentachoron (Acronym: grip) (Jonathan Bowers)

Images

Cartesian coordinates

The Cartesian coordinates of an origin-centered cantitruncated 5-cell having edge length 2 are:

More information , ...

Coordinates

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

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

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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 polytopes

A double symmetry construction can be made by placing truncated tetrahedra on the truncated octahedra, resulting in a nonuniform polychoron with 10 truncated tetrahedra, 20 hexagonal prisms (as ditrigonal trapezoprisms), two kinds of 80 triangular prisms (20 with D_3h symmetry and 60 C_2v-symmetric wedges), and 30 tetrahedra (as tetragonal disphenoids). Its vertex figure is topologically equivalent to the octahedron.

Vertex figure