Truncated 5-orthoplexes

In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.


5-orthoplex
Truncated 5-orthoplex
Bitruncated 5-orthoplex

5-cube
Truncated 5-cube
Bitruncated 5-cube
Orthogonal projections in B5 Coxeter plane

There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube.

Truncated 5-orthoplex
Truncated 5-orthoplex
Typeuniform 5-polytope
Schläfli symbolt{3,3,3,4}
t{3,31,1}
Coxeter-Dynkin diagrams
4-faces4210
32
Cells240160
80
Faces400320
80
Edges280240
40
Vertices80
Vertex figure
( )v{3,4}
Coxeter groupsB5, [3,3,3,4], order 3840
D5, [32,1,1], order 1920
Propertiesconvex

Alternate names
  • Truncated pentacross
  • Truncated triacontaditeron (Acronym: tot) (Jonathan Bowers)[1]

Coordinates

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

(±2,±1,0,0,0)

Images

The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.

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]

Bitruncated 5-orthoplex
Bitruncated 5-orthoplex
Typeuniform 5-polytope
Schläfli symbol2t{3,3,3,4}
2t{3,31,1}
Coxeter-Dynkin diagrams
4-faces4210
32
Cells28040
160
80
Faces720320
320
80
Edges720480
240
Vertices240
Vertex figure
{ }v{4}
Coxeter groupsB5, [3,3,3,4], order 3840
D5, [32,1,1], order 1920
Propertiesconvex

The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb.

Alternate names
  • Bitruncated pentacross
  • Bitruncated triacontiditeron (acronym: bittit) (Jonathan Bowers)[2]

Coordinates

Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign and coordinate permutations of

(±2,±2,±1,0,0)

Images

The bitrunacted 5-orthoplex is constructed by a bitruncation operation applied to the 5-orthoplex.

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]

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, (x3x3o3o4o - tot)
  2. Klitzing, (o3x3x3o4o - bittit)

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)". x3x3o3o4o - tot, o3x3x3o4o - bittit
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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