# B5 polytope

Orthographic projections in the B$_5$ Coxeter plane :
| 5-cube | 5-orthoplex | 5-demicube |
In 5-dimensional geometry, there are 31 uniform polytopes with B$_5$ 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 B$_5$ Coxeter group, and other subgroups.

## Graphs

Symmetric orthographic projections of these 32 polytopes can be made in the B$_5$, B$_4$, B$_3$, B$_2$, A$_3$, Coxeter planes. A$_k$ has [k+1] symmetry, and B$_k$ 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.
| # | 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 | t$_1${4,3,3,3} = r{4,3,3,3} Rectified 5-cube Rectified penteract (rin) |
| 4 | t$_2${4,3,3,3} = 2r{4,3,3,3} Birectified 5-cube Penteractitriacontiditeron (nit) |
| 5 | t$_1${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 | t$_{0,1}${4,3,3,3} = t{3,3,3,4} Truncated 5-cube Truncated penteract (tan) |
| 8 | t$_{1,2}${4,3,3,3} = 2t{4,3,3,3} Bitruncated 5-cube Bitruncated penteract (bittin) |
| 9 | t$_{0,2}${4,3,3,3} = rr{4,3,3,3} Cantellated 5-cube Rhombated penteract (sirn) |
| 10 | t$_{1,3}${4,3,3,3} = 2rr{4,3,3,3} Bicantellated 5-cube Small birhombi-penteractitriacontiditeron (sibrant) |
| 11 | t$_{0,3}${4,3,3,3} Runcinated 5-cube Prismated penteract (span) |
| 12 | t$_{0,4}${4,3,3,3} = 2r2r{4,3,3,3} Stericated 5-cube Small celli-penteractitriacontiditeron (scant) |
| 13 | t$_{0,1}${3,3,3,4} = t{3,3,3,4} Truncated 5-orthoplex Truncated triacontiditeron (tot) |
| 14 | t$_{1,2}${3,3,3,4} = 2t{3,3,3,4} Bitruncated 5-orthoplex Bitruncated triacontiditeron (bittit) |
| 15 | t$_{0,2}${3,3,3,4} = rr{3,3,3,4} Cantellated 5-orthoplex Small rhombated triacontiditeron (sart) |
| 16 | t$_{0,3}${3,3,3,4} Runcinated 5-orthoplex Small prismated triacontiditeron (spat) |
| 17 | t$_{0,1,2}${4,3,3,3} = tr{4,3,3,3} Cantitruncated 5-cube Great rhombated penteract (girn) |
| 18 | t$_{1,2,3}${4,3,3,3} = tr{4,3,3,3} Bicantitruncated 5-cube Great birhombi-penteractitriacontiditeron (gibrant) |
| 19 | t$_{0,1,3}${4,3,3,3} Runcitruncated 5-cube Prismatotruncated penteract (pattin) |
| 20 | t$_{0,2,3}${4,3,3,3} Runcicantellated 5-cube Prismatorhomated penteract (prin) |
| 21 | t$_{0,1,4}${4,3,3,3} Steritruncated 5-cube Cellitruncated penteract (capt) |
| 22 | t$_{0,2,4}${4,3,3,3} Stericantellated 5-cube Cellirhombi-penteractitriacontiditeron (carnit) |
| 23 | t$_{0,1,2,3}${4,3,3,3} Runcicantitruncated 5-cube Great primated penteract (gippin) |
| 24 | t$_{0,1,2,4}${4,3,3,3} Stericantitruncated 5-cube Celligreatorhombated penteract (cogrin) |
| 25 | t$_{0,1,3,4}${4,3,3,3} Steriruncitruncated 5-cube Celliprismatotrunki-penteractitriacontiditeron (captint) |
| 26 | t$_{0,1,2,3,4}${4,3,3,3} Omnitruncated 5-cube Great celli-penteractitriacontiditeron (gacnet) |
| 27 | t$_{0,1,2}${3,3,3,4} = tr{3,3,3,4} Cantitruncated 5-orthoplex Great rhombated triacontiditeron (gart) |
| 28 | t$_{0,1,3}${3,3,3,4} Runcitruncated 5-orthoplex Prismatotruncated triacontiditeron (pattit) |
| 29 | t$_{0,2,3}${3,3,3,4} Runcicantellated 5-orthoplex Prismatorhombated triacontiditeron (pirt) |
| 30 | t$_{0,1,4}${3,3,3,4} Steritruncated 5-orthoplex Cellitruncated triacontiditeron (cappin) |
| 31 | t$_{0,1,2,3}${3,3,3,4} Runcicantitruncated 5-orthoplex Great prismatorhombated triacontiditeron (gippit) |
| 32 | t$_{0,1,2,4}${3,3,3,4} Stericantitruncated 5-orthoplex Celligreatorhombated triacontiditeron (cogart) |