Cantellated 5-cell

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4-simplex t0.svg
5-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4-simplex t02.svg
Cantellated 5-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4-simplex t012.svg
Cantitruncated 5-cell
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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.

There are 2 unique degrees of runcinations of the 5-cell including with permutations truncations.

Cantellated 5-cell[edit]

Cantellated 5-cell
Schlegel half-solid cantellated 5-cell.png
Schlegel diagram with
octahedral cells shown
Type Uniform 4-polytope
Schläfli symbol t0,2{3,3,3}
rr{3,3,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells 20 5 Cuboctahedron.png(3.4.3.4)
5 Octahedron.png(3.3.3.3)
10 Triangular prism.png(3.4.4)
Faces 80 50{3}
30{4}
Edges 90
Vertices 30
Vertex figure Cantellated 5-cell verf.png
Irreg. triangular prism
Symmetry group A4, [3,3,3], order 120
Properties convex, isogonal
Uniform index 3 4 5

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[edit]

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

Images[edit]

orthographic projections
Ak
Coxeter plane
A4 A3 A2
Graph 4-simplex t02.svg 4-simplex t02 A3.svg 4-simplex t02 A2.svg
Dihedral symmetry [5] [4] [3]
Cantel pentachoron1.png
Wireframe
Cantel pentachoron2.png
Ten triangular prisms colored green
Cantel pentachoron3.png
Five octahedra colored blue

Coordinates[edit]

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.

Cantitruncated 5-cell[edit]

Cantitruncated 5-cell
Schlegel half-solid cantitruncated 5-cell.png
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 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells 20 5 Truncated octahedron.png(4.6.6)
10 Triangular prism.png(3.4.4)
 5 Truncated tetrahedron.png(3.6.6)
Faces 80 20{3}
30{4}
30{6}
Edges 120
Vertices 60
Vertex figure Cantitruncated 5-cell verf.png
sphenoid
Symmetry group A4, [3,3,3], order 120
Properties convex, isogonal
Uniform index 6 7 8

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[edit]

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

Images[edit]

orthographic projections
Ak
Coxeter plane
A4 A3 A2
Graph 4-simplex t012.svg 4-simplex t012 A3.svg 4-simplex t012 A2.svg
Dihedral symmetry [5] [4] [3]
Cantitruncated 5 cell.png
Stereographic projection with its 10 triangular prisms.

Cartesian coordinates[edit]

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.

Related 4-polytopes[edit]

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
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel
diagram
Schlegel wireframe 5-cell.png Schlegel half-solid truncated pentachoron.png Schlegel half-solid rectified 5-cell.png Schlegel half-solid cantellated 5-cell.png Schlegel half-solid bitruncated 5-cell.png Schlegel half-solid cantitruncated 5-cell.png Schlegel half-solid runcinated 5-cell.png Schlegel half-solid runcitruncated 5-cell.png Schlegel half-solid omnitruncated 5-cell.png
A4
Coxeter plane
Graph
4-simplex t0.svg 4-simplex t01.svg 4-simplex t1.svg 4-simplex t02.svg 4-simplex t12.svg 4-simplex t012.svg 4-simplex t03.svg 4-simplex t013.svg 4-simplex t0123.svg
A3 Coxeter plane
Graph
4-simplex t0 A3.svg 4-simplex t01 A3.svg 4-simplex t1 A3.svg 4-simplex t02 A3.svg 4-simplex t12 A3.svg 4-simplex t012 A3.svg 4-simplex t03 A3.svg 4-simplex t013 A3.svg 4-simplex t0123 A3.svg
A2 Coxeter plane
Graph
4-simplex t0 A2.svg 4-simplex t01 A2.svg 4-simplex t1 A2.svg 4-simplex t02 A2.svg 4-simplex t12 A2.svg 4-simplex t012 A2.svg 4-simplex t03 A2.svg 4-simplex t013 A2.svg 4-simplex t0123 A2.svg

References[edit]

  • 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
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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