In 5-dimensional geometry, there are 31 uniform polytopes with B5 symmetry. There are two regular forms, the 5-orthoplex, and 5-cube with 10 and 32 vertices respectively. The 5-demicube is added as an alternation of the 5-cube.

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
B5 / A4
[10]
Graph
B4 / D5
[8]
Graph
B3 / A2
[6]
Graph
B2
[4]
Graph
A3
[4]
Coxeter-Dynkin diagram
and Schläfli symbol
Johnson and Bowers names
1
h{4,3,3,3}
5-demicube
Hemipenteract (hin)
2
{4,3,3,3}
5-cube
Penteract (pent)
3
t1{4,3,3,3} = r{4,3,3,3}
Rectified 5-cube
Rectified penteract (rin)
4
t2{4,3,3,3} = 2r{4,3,3,3}
Birectified 5-cube
Penteractitriacontiditeron (nit)
5
t1{3,3,3,4} = r{3,3,3,4}
Rectified 5-orthoplex
Rectified triacontiditeron (rat)
6
{3,3,3,4}
5-orthoplex
Triacontiditeron (tac)
7
t0,1{4,3,3,3} = t{3,3,3,4}
Truncated 5-cube
Truncated penteract (tan)
8
t1,2{4,3,3,3} = 2t{4,3,3,3}
Bitruncated 5-cube
Bitruncated penteract (bittin)
9
t0,2{4,3,3,3} = rr{4,3,3,3}
Cantellated 5-cube
Rhombated penteract (sirn)
10
t1,3{4,3,3,3} = 2rr{4,3,3,3}
Bicantellated 5-cube
Small birhombi-penteractitriacontiditeron (sibrant)
11
t0,3{4,3,3,3}
Runcinated 5-cube
Prismated penteract (span)
12
t0,4{4,3,3,3} = 2r2r{4,3,3,3}
Stericated 5-cube
Small celli-penteractitriacontiditeron (scant)
13
t0,1{3,3,3,4} = t{3,3,3,4}
Truncated 5-orthoplex
Truncated triacontiditeron (tot)
14
t1,2{3,3,3,4} = 2t{3,3,3,4}
Bitruncated 5-orthoplex
Bitruncated triacontiditeron (bittit)
15
t0,2{3,3,3,4} = rr{3,3,3,4}
Cantellated 5-orthoplex
Small rhombated triacontiditeron (sart)
16
t0,3{3,3,3,4}
Runcinated 5-orthoplex
Small prismated triacontiditeron (spat)
17
t0,1,2{4,3,3,3} = tr{4,3,3,3}
Cantitruncated 5-cube
Great rhombated penteract (girn)
18
t1,2,3{4,3,3,3} = tr{4,3,3,3}
Bicantitruncated 5-cube
Great birhombi-penteractitriacontiditeron (gibrant)
19
t0,1,3{4,3,3,3}
Runcitruncated 5-cube
Prismatotruncated penteract (pattin)
20
t0,2,3{4,3,3,3}
Runcicantellated 5-cube
Prismatorhomated penteract (prin)
21
t0,1,4{4,3,3,3}
Steritruncated 5-cube
Cellitruncated penteract (capt)
22
t0,2,4{4,3,3,3}
Stericantellated 5-cube
Cellirhombi-penteractitriacontiditeron (carnit)
23
t0,1,2,3{4,3,3,3}
Runcicantitruncated 5-cube
Great primated penteract (gippin)
24
t0,1,2,4{4,3,3,3}
Stericantitruncated 5-cube
Celligreatorhombated penteract (cogrin)
25
t0,1,3,4{4,3,3,3}
Steriruncitruncated 5-cube
Celliprismatotrunki-penteractitriacontiditeron (captint)
26
t0,1,2,3,4{4,3,3,3}
Omnitruncated 5-cube
Great celli-penteractitriacontiditeron (gacnet)
27
t0,1,2{3,3,3,4} = tr{3,3,3,4}
Cantitruncated 5-orthoplex
Great rhombated triacontiditeron (gart)
28
t0,1,3{3,3,3,4}
Runcitruncated 5-orthoplex
Prismatotruncated triacontiditeron (pattit)
29
t0,2,3{3,3,3,4}
Runcicantellated 5-orthoplex
Prismatorhombated triacontiditeron (pirt)
30
t0,1,4{3,3,3,4}
Steritruncated 5-orthoplex
Cellitruncated triacontiditeron (cappin)
31
t0,1,2,3{3,3,3,4}
Runcicantitruncated 5-orthoplex
Great prismatorhombated triacontiditeron (gippit)
32
t0,1,2,4{3,3,3,4}
Stericantitruncated 5-orthoplex
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, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6[1]
(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

Notes

Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter

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