Runcinated 5-orthoplexes

In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.


5-orthoplex
Runcinated 5-orthoplex
Runcinated 5-cube

Runcitruncated 5-orthoplex
Runcicantellated 5-orthoplex
Runcicantitruncated 5-orthoplex

Runcitruncated 5-cube
Runcicantellated 5-cube
Runcicantitruncated 5-cube
Orthogonal projections in B5 Coxeter plane

There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations. Four are more simply constructed relative to the 5-cube.

Runcinated 5-orthoplex
Runcinated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,3{3,3,3,4}
Coxeter-Dynkin diagram
4-faces 162
Cells 1200
Faces 2160
Edges 1440
Vertices 320
Vertex figure
Coxeter group B5 [4,3,3,3]
D5 [32,1,1]
Properties convex

Alternate names
  • Runcinated pentacross
  • Small prismated triacontiditeron (Acronym: spat) (Jonathan Bowers)[1]

Coordinates

The vertices of the can be made in 5-space, as permutations and sign combinations of:

(0,1,1,1,2)

Images
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Runcitruncated 5-orthoplex
Runcitruncated 5-orthoplex
Typeuniform 5-polytope
Schläfli symbolt0,1,3{3,3,3,4}
t0,1,3{3,31,1}
Coxeter-Dynkin diagrams
4-faces162
Cells1440
Faces3680
Edges3360
Vertices960
Vertex figure
Coxeter groupsB5, [3,3,3,4]
D5, [32,1,1]
Propertiesconvex

Alternate names
  • Runcitruncated pentacross
  • Prismatotruncated triacontiditeron (Acronym: pattit) (Jonathan Bowers)[2]

Coordinates

Cartesian coordinates for the vertices of a runcitruncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of

(±3,±2,±1,±1,0)

Images
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Runcicantellated 5-orthoplex
Runcicantellated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,2,3{3,3,3,4}
t0,2,3{3,3,31,1}
Coxeter-Dynkin diagram
4-faces162
Cells1200
Faces2960
Edges2880
Vertices960
Vertex figure
Coxeter group B5 [4,3,3,3]
D5 [32,1,1]
Properties convex

Alternate names
  • Runcicantellated pentacross
  • Prismatorhombated triacontiditeron (Acronym: pirt) (Jonathan Bowers)[3]

Coordinates

The vertices of the runcicantellated 5-orthoplex can be made in 5-space, as permutations and sign combinations of:

(0,1,2,2,3)

Images
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Runcicantitruncated 5-orthoplex
Runcicantitruncated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3{3,3,3,4}
Coxeter-Dynkin
diagram
4-faces162
Cells1440
Faces4160
Edges4800
Vertices1920
Vertex figure
Irregular 5-cell
Coxeter groups B5 [4,3,3,3]
D5 [32,1,1]
Properties convex, isogonal

Alternate names
  • Runcicantitruncated pentacross
  • Great prismated triacontiditeron (gippit) (Jonathan Bowers)[4]

Coordinates

The Cartesian coordinates of the vertices of a runcicantitruncated 5-orthoplex having an edge length of √2 are given by all permutations of coordinates and sign of:

Images
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Snub 5-demicube

The snub 5-demicube defined as an alternation of the omnitruncated 5-demicube is not uniform, but it can be given Coxeter diagram or and symmetry [32,1,1]+ or [4,(3,3,3)+], and constructed from 10 snub 24-cells, 32 snub 5-cells, 40 snub tetrahedral antiprisms, 80 2-3 duoantiprisms, and 960 irregular 5-cells filling the gaps at the deleted vertices.

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

B5 polytopes

β5
t1β5
t2γ5
t1γ5
γ5
t0,1β5
t0,2β5
t1,2β5

t0,3β5
t1,3γ5
t1,2γ5
t0,4γ5
t0,3γ5
t0,2γ5
t0,1γ5
t0,1,2β5

t0,1,3β5
t0,2,3β5
t1,2,3γ5
t0,1,4β5
t0,2,4γ5
t0,2,3γ5
t0,1,4γ5
t0,1,3γ5

t0,1,2γ5
t0,1,2,3β5
t0,1,2,4β5
t0,1,3,4γ5
t0,1,2,4γ5
t0,1,2,3γ5
t0,1,2,3,4γ5

Notes
  1. Klitzing, (x3o3o3x4o - spat)
  2. Klitzing, (x3x3o3x4o - pattit)
  3. Klitzing, (x3o3x3x4o - pirt)
  4. Klitzing, (x3x3x3x4o - gippit)

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
      • (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.
  • Klitzing, Richard. "5D uniform polytopes (polytera)". x3o3o3x4o - spat, x3x3o3x4o - pattit, x3o3x3x4o - pirt, x3x3x3x4o - gippit
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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