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, up to edge-planing) of the regular 5-cell.

Cantellated 5-cell

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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  
Square wedge
Symmetry group A4, [3,3,3], order 120
Properties convex, isogonal
Uniform index 3 4 5
 
Net

The cantellated 5-cell or 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

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

Configuration

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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

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orthographic projections
Ak
Coxeter plane
A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]
 
Wireframe
 
Ten triangular prisms colored green
 
Five octahedra colored blue

Coordinates

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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)

This construction is from the positive orthant facet of the cantellated 5-orthoplex.

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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

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

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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

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

Images

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orthographic projections
Ak
Coxeter plane
A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]
 
Stereographic projection with its 10 triangular prisms.

Cartesian coordinates

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

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.

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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

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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
                 
A2 Coxeter plane
Graph
                 

References

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  1. ^ Klitzing, Richard. "o3x4x3o - deca".
  2. ^ Klitzing, Richard. "x3x4x3o - grip".
  • 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)". x3o3x3o - srip, x3x3x3o - grip
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds