A5 polytope

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Orthographic projection in
A5 Coxeter plane
5-simplex

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.

Each can be visualized as symmetric orthographic projections in the 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

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|

[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  | 80px  | 80px  | 80px  | 80px  | Template:Dark mode invert
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  | 80px  | 80px  | 80px  | 80px  | Template:Dark mode invert
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  | 80px  | 80px  | 80px  | 80px  | Template:Dark mode invert
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  | 80px  | 80px  | 80px  | 80px  | Template:Dark mode invert
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  | 80px  | 80px  | 80px  | 80px  | Template:Dark mode invert
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  | 80px  | 80px  | 80px  | 80px  | Template:Dark mode invert
t0,1,3,4{3,3,3,3}
Steriruncitruncated 5-simplex (captid)
19  | 80px  | 80px  | 80px  | 80px  | Template:Dark mode invert
t0,1,2,3,4{3,3,3,3}
Omnitruncated 5-simplex (gocad)

Template:Hexateron family

## 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, and Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com, 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]
* N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

## External links

* Klitzing, Richard. "5D uniform polytopes (polytera)". https://bendwavy.org/klitzing/dimensions/polytera.htm.

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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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