Cantellated 5-cell
In four-dimensional geometry , a cantellated 5-cell is a convex uniform 4-polytope , being a cantellation (a 2nd order truncation, up to edge -planing) of the regular 5-cell .
Cantellated 5-cell
Cantellated 5-cell
Schlegel diagram with
Type
Uniform 4-polytope
Schläfli symbol
t0,2 {3,3,3}
Coxeter diagram
Cells
20
5 (3.4.3.4) (3.3.3.3) (3.4.4)
Faces
80
50{3} {4}
Edges
90
Vertices
30
Vertex figure
Symmetry group
A4 , [3,3,3], order 120
Properties
convex , isogonal
Uniform index
3 5
Net The cantellated 5-cell small rhombated pentachoron 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)
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]
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
Images
Coordinates
The Cartesian coordinates of the vertices of the origin-centered cantellated 5-cell having edge length 2 are:
Coordinates
(
2
2
5
,
2
2
3
,
1
3
,
± ± -->
1
)
{\displaystyle \left(2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)}
(
2
2
5
,
2
2
3
,
− − -->
2
3
,
0
)
{\displaystyle \left(2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)}
(
2
2
5
,
0
,
± ± -->
3
,
± ± -->
1
)
{\displaystyle \left(2{\sqrt {\frac {2}{5}}},\ 0,\ \pm {\sqrt {3}},\ \pm 1\right)}
(
2
2
5
,
0
,
0
,
± ± -->
2
)
{\displaystyle \left(2{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ \pm 2\right)}
(
2
2
5
,
− − -->
2
2
3
,
2
3
,
0
)
{\displaystyle \left(2{\sqrt {\frac {2}{5}}},\ -2{\sqrt {\frac {2}{3}}},\ {\frac {2}{\sqrt {3}}},\ 0\right)}
(
2
2
5
,
− − -->
2
2
3
,
− − -->
1
3
,
± ± -->
1
)
{\displaystyle \left(2{\sqrt {\frac {2}{5}}},\ -2{\sqrt {\frac {2}{3}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 1\right)}
(
− − -->
1
10
,
3
2
,
± ± -->
3
,
± ± -->
1
)
{\displaystyle \left({\frac {-1}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ \pm {\sqrt {3}},\ \pm 1\right)}
(
− − -->
1
10
,
3
2
,
0
,
± ± -->
2
)
{\displaystyle \left({\frac {-1}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ 0,\ \pm 2\right)}
(
− − -->
1
10
,
− − -->
1
6
,
2
3
,
± ± -->
2
)
{\displaystyle \left({\frac {-1}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)}
(
− − -->
1
10
,
− − -->
1
6
,
− − -->
4
3
,
0
)
{\displaystyle \left({\frac {-1}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ 0\right)}
(
− − -->
1
10
,
− − -->
5
6
,
1
3
,
± ± -->
1
)
{\displaystyle \left({\frac {-1}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)}
(
− − -->
1
10
,
− − -->
5
6
,
− − -->
2
3
,
0
)
{\displaystyle \left({\frac {-1}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)}
(
− − -->
3
2
5
,
0
,
0
,
0
)
± ± -->
(
0
,
2
3
,
2
3
,
0
)
{\displaystyle \left(-3{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)\pm \left(0,\ {\sqrt {\frac {2}{3}}},\ {\frac {2}{\sqrt {3}}},\ 0\right)}
(
− − -->
3
2
5
,
0
,
0
,
0
)
± ± -->
(
0
,
2
3
,
− − -->
1
3
,
± ± -->
1
)
{\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)}
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 .
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
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}
Coxeter diagram
Cells
20
5 (4.6.6) (3.4.4) (3.6.6)
Faces
80
20{3}
Edges
120
Vertices
60
Vertex figure
sphenoid
Symmetry group
A4 , [3,3,3], order 120
Properties
convex , isogonal
Uniform index
6 8
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.
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]
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
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:
Coordinates
(
3
2
5
,
± ± -->
6
,
± ± -->
3
,
± ± -->
1
)
{\displaystyle \left(3{\sqrt {\frac {2}{5}}},\ \pm {\sqrt {6}},\ \pm {\sqrt {3}},\ \pm 1\right)}
(
3
2
5
,
± ± -->
6
,
0
,
± ± -->
2
)
{\displaystyle \left(3{\sqrt {\frac {2}{5}}},\ \pm {\sqrt {6}},\ 0,\ \pm 2\right)}
(
3
2
5
,
0
,
0
,
0
)
± ± -->
(
0
,
2
3
,
5
3
,
± ± -->
1
)
{\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)}
(
3
2
5
,
0
,
0
,
0
)
± ± -->
(
0
,
2
3
,
− − -->
1
3
,
± ± -->
3
)
{\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)}
(
3
2
5
,
0
,
0
,
0
)
± ± -->
(
0
,
2
3
,
− − -->
4
3
,
± ± -->
2
)
{\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)}
(
1
10
,
5
6
,
5
3
,
± ± -->
1
)
{\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {5}{\sqrt {3}}},\ \pm 1\right)}
(
1
10
,
5
6
,
− − -->
1
3
,
± ± -->
3
)
{\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 3\right)}
(
1
10
,
5
6
,
− − -->
4
3
,
± ± -->
2
)
{\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ \pm 2\right)}
(
1
10
,
− − -->
3
2
,
3
,
± ± -->
3
)
{\displaystyle \left({\frac {1}{\sqrt {10}}},\ -{\sqrt {\frac {3}{2}}},\ {\sqrt {3}},\ \pm 3\right)}
(
1
10
,
− − -->
3
2
,
− − -->
2
3
,
0
)
{\displaystyle \left({\frac {1}{\sqrt {10}}},\ -{\sqrt {\frac {3}{2}}},\ -2{\sqrt {3}},\ 0\right)}
(
1
10
,
− − -->
7
6
,
2
3
,
± ± -->
2
)
{\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {-7}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)}
(
1
10
,
− − -->
7
6
,
− − -->
4
3
,
0
)
{\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {-7}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ 0\right)}
(
− − -->
2
2
5
,
2
2
3
,
4
3
,
± ± -->
2
)
{\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {4}{\sqrt {3}}},\ \pm 2\right)}
(
− − -->
2
2
5
,
2
2
3
,
1
3
,
± ± -->
3
)
{\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {1}{\sqrt {3}}},\ \pm 3\right)}
(
− − -->
2
2
5
,
2
2
3
,
− − -->
5
3
,
± ± -->
1
)
{\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {-5}{\sqrt {3}}},\ \pm 1\right)}
(
− − -->
2
2
5
,
0
,
3
,
± ± -->
3
)
{\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ 0,\ {\sqrt {3}},\ \pm 3\right)}
(
− − -->
2
2
5
,
0
,
− − -->
2
3
,
0
)
{\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ 0,\ -2{\sqrt {3}},\ 0\right)}
(
− − -->
2
2
5
,
− − -->
4
2
3
,
1
3
,
± ± -->
1
)
{\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ -4{\sqrt {\frac {2}{3}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)}
(
− − -->
2
2
5
,
− − -->
4
2
3
,
− − -->
2
3
,
0
)
{\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ -4{\sqrt {\frac {2}{3}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)}
(
− − -->
9
10
,
3
2
,
± ± -->
3
,
± ± -->
1
)
{\displaystyle \left({\frac {-9}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ \pm {\sqrt {3}},\ \pm 1\right)}
(
− − -->
9
10
,
3
2
,
0
,
± ± -->
2
)
{\displaystyle \left({\frac {-9}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ 0,\ \pm 2\right)}
(
− − -->
9
10
,
− − -->
1
6
,
2
3
,
± ± -->
2
)
{\displaystyle \left({\frac {-9}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)}
(
− − -->
9
10
,
− − -->
1
6
,
− − -->
4
3
,
0
)
{\displaystyle \left({\frac {-9}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ 0\right)}
(
− − -->
9
10
,
− − -->
5
6
,
1
3
,
± ± -->
1
)
{\displaystyle \left({\frac {-9}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)}
(
− − -->
9
10
,
− − -->
5
6
,
− − -->
2
3
,
0
)
{\displaystyle \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 .
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 D3h symmetry and 60 C2v -symmetric wedges), and 30 tetrahedra (as tetragonal disphenoids). Its vertex figure is topologically equivalent to the octahedron .
Vertex figure
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
{3,3,3}
t{3,3,3}
r{3,3,3}
rr{3,3,3}
2t{3,3,3}
tr{3,3,3}
t0,3 {3,3,3}
t0,1,3 {3,3,3}0,2,3 {3,3,3}
t0,1,2,3 {3,3,3}
Coxeter
Schlegel
A4 Coxeter plane
A3 Coxeter plane
A2 Coxeter plane
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.Klitzing, Richard. "4D uniform polytopes (polychora)" .
