In 4dimensional geometry, there are 9 uniform polytopes with A4 symmetry. There is one selfdual regular form, the 5cell with 5 vertices.
Symmetry
A4 symmetry, or [3,3,3] is order 120, with Conway quaternion notation +1/60[I×I].21. Its abstract structure is the symmetric group S5. Three forms with symmetric Coxeter diagrams have extended symmetry, [[3,3,3]] of order 240, and Conway notation ±1/60[I×I].2, and abstract structure S5×C2.
Visualizations
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A4 Coxeter group, and other subgroups. Three Coxeter plane 2D projections are given, for the A4, A3, A2 Coxeter groups, showing symmetry order 5,4,3, and doubled on even Ak orders to 10,4,6 for symmetric Coxeter diagrams.
The 3D picture are drawn as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid.
#  Name  Coxeter diagram and Schläfli symbols 
Coxeter plane graphs  Schlegel diagram  Net  

A_{4} [5] 
A_{3} [4] 
A_{2} [3] 
Tetrahedron centered 
Dual tetrahedron centered 

1  5cell pentachoron 
{3,3,3} 

2  rectified 5cell  r{3,3,3} 

3  truncated 5cell  t{3,3,3} 

4  cantellated 5cell  rr{3,3,3} 

7  cantitruncated 5cell  tr{3,3,3} 

8  runcitruncated 5cell  t_{0,1,3}{3,3,3} 
#  Name  Coxeter diagram and Schläfli symbols 
Coxeter plane graphs  Schlegel diagram  Net  

A_{4} [[5]] = [10] 
A_{3} [4] 
A_{2} [[3]] = [6] 
Tetrahedron centered 

5  *runcinated 5cell  t_{0,3}{3,3,3} 

6  *bitruncated 5cell decachoron 
2t{3,3,3} 

9  *omnitruncated 5cell  t_{0,1,2,3}{3,3,3} 
Coordinates
The coordinates of uniform 4polytopes with pentachoric symmetry can be generated as permutations of simple integers in 5space, all in hyperplanes with normal vector (1,1,1,1,1). The A4 Coxeter group is palindromic, so repeated polytopes exist in pairs of dual configurations. There are 3 symmetric positions, and 6 pairs making the total 15 permutations of one or more rings. All 15 are listed here in order of binary arithmetic for clarity of the coordinate generation from the rings in each corresponding Coxeter diagram.
The number of vertices can be deduced here from the permutations of the number of coordinates, peaking at 5 factorial for the omnitruncated form with 5 unique coordinate values.
#  Base point  Name (symmetric name) 
Coxeter diagram  Vertices  

1  (0, 0, 0, 0, 1) (1, 1, 1, 1, 0) 
5cell Trirectified 5cell 
5  5!/(4!)  
2  (0, 0, 0, 1, 1) (1, 1, 1, 0, 0) 
Rectified 5cell Birectified 5cell 
10  5!/(3!2!)  
3  (0, 0, 0, 1, 2) (2, 2, 2, 1, 0) 
Truncated 5cell Tritruncated 5cell 
20  5!/(3!)  
5  (0, 1, 1, 1, 2)  Runcinated 5cell  20  5!/(3!)  
4  (0, 0, 1, 1, 2) (2, 2, 1, 1, 0) 
Cantellated 5cell Bicantellated 5cell 
30  5!/(2!2!)  
6  (0, 0, 1, 2, 2)  Bitruncated 5cell  30  5!/(2!2!)  
7  (0, 0, 1, 2, 3) (3, 3, 2, 1, 0) 
Cantitruncated 5cell Bicantitruncated 5cell 
60  5!/2!  
8  (0, 1, 1, 2, 3) (3, 2, 2, 1, 0) 
Runcitruncated 5cell Runcicantellated 5cell 
60  5!/2!  
9  (0, 1, 2, 3, 4)  Omnitruncated 5cell  120  5! 
References
J.H. Conway and M.J.T. Guy: FourDimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, The Symmetries of Things 2008, ISBN 9781568812205 (Chapter 26)
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 Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
(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. "4D uniform 4polytopes".
Uniform, convex polytopes in four dimensions:, Marco Möller (in German)
Möller, Marco (2004). Vierdimensionale Archimedische Polytope (PDF) (Doctoral dissertation) (in German). University of Hamburg.
Uniform Polytopes in Four Dimensions, George Olshevsky.
Convex uniform polychora based on the pentachoron, George Olshevsky.
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 
Hellenica World  Scientific Library
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