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In 4-dimensional geometry, there are 9 uniform polytopes with A4 symmetry. There is one self-dual regular form, the 5-cell 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.

Uniform polytopes with A4 symmetry
# Name Coxeter diagram
and Schläfli
symbols
Coxeter plane graphs Schlegel diagram Net
A4
[5]
A3
[4]
A2
[3]
Tetrahedron
centered
Dual tetrahedron
centered
1 5-cell
pentachoron

{3,3,3}
2 rectified 5-cell
r{3,3,3}
3 truncated 5-cell
t{3,3,3}
4 cantellated 5-cell
rr{3,3,3}
7 cantitruncated 5-cell
tr{3,3,3}
8 runcitruncated 5-cell
t0,1,3{3,3,3}
Uniform polytopes with extended A4 symmetry
# Name Coxeter diagram
and Schläfli
symbols
Coxeter plane graphs Schlegel diagram Net
A4
[[5]] = [10]
A3
[4]
A2
[[3]] = [6]
Tetrahedron
centered
5 *runcinated 5-cell
t0,3{3,3,3}
6 *bitruncated 5-cell
decachoron

2t{3,3,3}
9 *omnitruncated 5-cell
t0,1,2,3{3,3,3}

Coordinates

The coordinates of uniform 4-polytopes with pentachoric symmetry can be generated as permutations of simple integers in 5-space, 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.

5-cell truncations in 5-space:
# Base point Name
(symmetric name)
Coxeter diagram Vertices
1 (0, 0, 0, 0, 1)
(1, 1, 1, 1, 0)
5-cell
Trirectified 5-cell

5 5!/(4!)
2 (0, 0, 0, 1, 1)
(1, 1, 1, 0, 0)
Rectified 5-cell
Birectified 5-cell

10 5!/(3!2!)
3 (0, 0, 0, 1, 2)
(2, 2, 2, 1, 0)
Truncated 5-cell
Tritruncated 5-cell

20 5!/(3!)
5 (0, 1, 1, 1, 2) Runcinated 5-cell 20 5!/(3!)
4 (0, 0, 1, 1, 2)
(2, 2, 1, 1, 0)
Cantellated 5-cell
Bicantellated 5-cell

30 5!/(2!2!)
6 (0, 0, 1, 2, 2) Bitruncated 5-cell 30 5!/(2!2!)
7 (0, 0, 1, 2, 3)
(3, 3, 2, 1, 0)
Cantitruncated 5-cell
Bicantitruncated 5-cell

60 5!/2!
8 (0, 1, 1, 2, 3)
(3, 2, 2, 1, 0)
Runcitruncated 5-cell
Runcicantellated 5-cell

60 5!/2!
9 (0, 1, 2, 3, 4) Omnitruncated 5-cell 120 5!

References

J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (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, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 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) 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

Klitzing, Richard. "4D uniform 4-polytopes".
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 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