Orthographic projections in the B5 Coxeter plane

5-cube

5-orthoplex

5-demicube

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

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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

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  • 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
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Notes

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Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds