In 5dimensional geometry, there are 19 uniform polytopes with A5 symmetry. There is one selfdual regular form, the 5simplex with 6 vertices.
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_1eddiagrams, 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  CoxeterDynkin diagram Schläfli symbol Name 


[6]  [5]  [4]  [3]  
A_{5}  A_{4}  A_{3}  A_{2}  
1  {3,3,3,3} 5simplex (hix) 

2  t_{1}{3,3,3,3} or r{3,3,3,3} Rectified 5simplex (rix) 

3  t_{2}{3,3,3,3} or 2r{3,3,3,3} Birectified 5simplex (dot) 

4  t_{0,1}{3,3,3,3} or t{3,3,3,3} Truncated 5simplex (tix) 

5  t_{1,2}{3,3,3,3} or 2t{3,3,3,3} Bitruncated 5simplex (bittix) 

6  t_{0,2}{3,3,3,3} or rr{3,3,3,3} Cantellated 5simplex (sarx) 

7  t_{1,3}{3,3,3,3} or 2rr{3,3,3,3} Bicantellated 5simplex (sibrid) 

8  t_{0,3}{3,3,3,3} Runcinated 5simplex (spix) 

9  t_{0,4}{3,3,3,3} or 2r2r{3,3,3,3} Stericated 5simplex (scad) 

10  t_{0,1,2}{3,3,3,3} or tr{3,3,3,3} Cantitruncated 5simplex (garx) 

11  t_{1,2,3}{3,3,3,3} or 2tr{3,3,3,3} Bicantitruncated 5simplex (gibrid) 

12  t_{0,1,3}{3,3,3,3} Runcitruncated 5simplex (pattix) 

13  t_{0,2,3}{3,3,3,3} Runcicantellated 5simplex (pirx) 

14  t_{0,1,4}{3,3,3,3} Steritruncated 5simplex (cappix) 

15  t_{0,2,4}{3,3,3,3} Stericantellated 5simplex (card) 

16  t_{0,1,2,3}{3,3,3,3} Runcicantitruncated 5simplex (gippix) 

17  t_{0,1,2,4}{3,3,3,3} Stericantitruncated 5simplex (cograx) 

18  t_{0,1,3,4}{3,3,3,3} Steriruncitruncated 5simplex (captid) 

19  t_{0,1,2,3,4}{3,3,3,3} Omnitruncated 5simplex (gocad) 
A5 polytopes  

t_{0} 
t_{1} 
t_{2} 
t_{0,1} 
t_{0,2} 
t_{1,2} 
t_{0,3} 

t_{1,3} 
t_{0,4} 
t_{0,1,2} 
t_{0,1,3} 
t_{0,2,3} 
t_{1,2,3} 
t_{0,1,4} 

t_{0,2,4} 
t_{0,1,2,3} 
t_{0,1,2,4} 
t_{0,1,3,4} 
t_{0,1,2,3,4} 
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, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
(Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
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)".
Notes
Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
Fundamental convex regular and uniform polytopes in dimensions 2–10



Family  A_{n}  B_{n}  I_{2}(p) / D_{n}  E_{6} / E_{7} / E_{8} / F_{4} / G_{2}  H_{n}  
Regular polygon  Triangle  Square  pgon  Hexagon  Pentagon  
Uniform polyhedron  Tetrahedron  Octahedron • Cube  Demicube  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  16cell • Tesseract  Demitesseract  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5orthoplex • 5cube  5demicube  
Uniform 6polytope  6simplex  6orthoplex • 6cube  6demicube  1_{22} • 2_{21}  
Uniform 7polytope  7simplex  7orthoplex • 7cube  7demicube  1_{32} • 2_{31} • 3_{21}  
Uniform 8polytope  8simplex  8orthoplex • 8cube  8demicube  1_{42} • 2_{41} • 4_{21}  
Uniform 9polytope  9simplex  9orthoplex • 9cube  9demicube  
Uniform 10polytope  10simplex  10orthoplex • 10cube  10demicube  
Uniform npolytope  nsimplex  northoplex • ncube  ndemicube  1_{k2} • 2_{k1} • k_{21}  npentagonal polytope  
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds 
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