A5 polytope

In 5-dimensional geometry, there are 19 uniform polytopes with A5 symmetry. There is one self-dual regular form, the 5-simplex with 6 vertices.

Orthographic projections
A5 Coxeter plane

5-simplex

Each can be visualized as symmetric orthographic projections in Coxeter planes of the A5 Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 19 polytopes can be made in the A5, A4, A3, A2 Coxeter planes. Ak graphs have [k+1] symmetry. For even k and symmetrically nodea_1ed-diagrams, symmetry doubles to [2(k+1)].

These 19 polytopes are each shown in these 4 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

# Coxeter plane graphs Coxeter-Dynkin diagram
Schläfli symbol
Name
[6] [5] [4] [3]
A5 A4 A3 A2
1
{3,3,3,3}
5-simplex (hix)
2
t1{3,3,3,3} or r{3,3,3,3}
Rectified 5-simplex (rix)
3
t2{3,3,3,3} or 2r{3,3,3,3}
Birectified 5-simplex (dot)
4
t0,1{3,3,3,3} or t{3,3,3,3}
Truncated 5-simplex (tix)
5
t1,2{3,3,3,3} or 2t{3,3,3,3}
Bitruncated 5-simplex (bittix)
6
t0,2{3,3,3,3} or rr{3,3,3,3}
Cantellated 5-simplex (sarx)
7
t1,3{3,3,3,3} or 2rr{3,3,3,3}
Bicantellated 5-simplex (sibrid)
8
t0,3{3,3,3,3}
Runcinated 5-simplex (spix)
9
t0,4{3,3,3,3} or 2r2r{3,3,3,3}
Stericated 5-simplex (scad)
10
t0,1,2{3,3,3,3} or tr{3,3,3,3}
Cantitruncated 5-simplex (garx)
11
t1,2,3{3,3,3,3} or 2tr{3,3,3,3}
Bicantitruncated 5-simplex (gibrid)
12
t0,1,3{3,3,3,3}
Runcitruncated 5-simplex (pattix)
13
t0,2,3{3,3,3,3}
Runcicantellated 5-simplex (pirx)
14
t0,1,4{3,3,3,3}
Steritruncated 5-simplex (cappix)
15
t0,2,4{3,3,3,3}
Stericantellated 5-simplex (card)
16
t0,1,2,3{3,3,3,3}
Runcicantitruncated 5-simplex (gippix)
17
t0,1,2,4{3,3,3,3}
Stericantitruncated 5-simplex (cograx)
18
t0,1,3,4{3,3,3,3}
Steriruncitruncated 5-simplex (captid)
19
t0,1,2,3,4{3,3,3,3}
Omnitruncated 5-simplex (gocad)


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]
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

Notes
  1. Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
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 4-polytope 5-cell 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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