In 5dimensional geometry, there are 31 uniform polytopes with B5 symmetry. There are two regular forms, the 5orthoplex, and 5cube with 10 and 32 vertices respectively. The 5demicube is added as an alternation of the 5cube.
They can be visualized as symmetric orthographic projections in Coxeter planes of the B5 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 32 polytopes can be made in the B5, B4, B3, B2, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.
These 32 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
Graph B_{5} / A_{4} [10] 
Graph B_{4} / D_{5} [8] 
Graph B_{3} / A_{2} [6] 
Graph B_{2} [4] 
Graph A_{3} [4] 
CoxeterDynkin diagram and Schläfli symbol Johnson and Bowers names 


1  h{4,3,3,3} 5demicube Hemipenteract (hin) 

2  {4,3,3,3} 5cube Penteract (pent) 

3  t_{1}{4,3,3,3} = r{4,3,3,3} Rectified 5cube Rectified penteract (rin) 

4  t_{2}{4,3,3,3} = 2r{4,3,3,3} Birectified 5cube Penteractitriacontiditeron (nit) 

5  t_{1}{3,3,3,4} = r{3,3,3,4} Rectified 5orthoplex Rectified triacontiditeron (rat) 

6  {3,3,3,4} 5orthoplex Triacontiditeron (tac) 

7  t_{0,1}{4,3,3,3} = t{3,3,3,4} Truncated 5cube Truncated penteract (tan) 

8  t_{1,2}{4,3,3,3} = 2t{4,3,3,3} Bitruncated 5cube Bitruncated penteract (bittin) 

9  t_{0,2}{4,3,3,3} = rr{4,3,3,3} Cantellated 5cube Rhombated penteract (sirn) 

10  t_{1,3}{4,3,3,3} = 2rr{4,3,3,3} Bicantellated 5cube Small birhombipenteractitriacontiditeron (sibrant) 

11  t_{0,3}{4,3,3,3} Runcinated 5cube Prismated penteract (span) 

12  t_{0,4}{4,3,3,3} = 2r2r{4,3,3,3} Stericated 5cube Small cellipenteractitriacontiditeron (scant) 

13  t_{0,1}{3,3,3,4} = t{3,3,3,4} Truncated 5orthoplex Truncated triacontiditeron (tot) 

14  t_{1,2}{3,3,3,4} = 2t{3,3,3,4} Bitruncated 5orthoplex Bitruncated triacontiditeron (bittit) 

15  t_{0,2}{3,3,3,4} = rr{3,3,3,4} Cantellated 5orthoplex Small rhombated triacontiditeron (sart) 

16  t_{0,3}{3,3,3,4} Runcinated 5orthoplex Small prismated triacontiditeron (spat) 

17  t_{0,1,2}{4,3,3,3} = tr{4,3,3,3} Cantitruncated 5cube Great rhombated penteract (girn) 

18  t_{1,2,3}{4,3,3,3} = tr{4,3,3,3} Bicantitruncated 5cube Great birhombipenteractitriacontiditeron (gibrant) 

19  t_{0,1,3}{4,3,3,3} Runcitruncated 5cube Prismatotruncated penteract (pattin) 

20  t_{0,2,3}{4,3,3,3} Runcicantellated 5cube Prismatorhomated penteract (prin) 

21  t_{0,1,4}{4,3,3,3} Steritruncated 5cube Cellitruncated penteract (capt) 

22  t_{0,2,4}{4,3,3,3} Stericantellated 5cube Cellirhombipenteractitriacontiditeron (carnit) 

23  t_{0,1,2,3}{4,3,3,3} Runcicantitruncated 5cube Great primated penteract (gippin) 

24  t_{0,1,2,4}{4,3,3,3} Stericantitruncated 5cube Celligreatorhombated penteract (cogrin) 

25  t_{0,1,3,4}{4,3,3,3} Steriruncitruncated 5cube Celliprismatotrunkipenteractitriacontiditeron (captint) 

26  t_{0,1,2,3,4}{4,3,3,3} Omnitruncated 5cube Great cellipenteractitriacontiditeron (gacnet) 

27  t_{0,1,2}{3,3,3,4} = tr{3,3,3,4} Cantitruncated 5orthoplex Great rhombated triacontiditeron (gart) 

28  t_{0,1,3}{3,3,3,4} Runcitruncated 5orthoplex Prismatotruncated triacontiditeron (pattit) 

29  t_{0,2,3}{3,3,3,4} Runcicantellated 5orthoplex Prismatorhombated triacontiditeron (pirt) 

30  t_{0,1,4}{3,3,3,4} Steritruncated 5orthoplex Cellitruncated triacontiditeron (cappin) 

31  t_{0,1,2,3}{3,3,3,4} Runcicantitruncated 5orthoplex Great prismatorhombated triacontiditeron (gippit) 

32  t_{0,1,2,4}{3,3,3,4} Stericantitruncated 5orthoplex Celligreatorhombated triacontiditeron (cogart) 
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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