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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 5-demicube t0 B5.svg 5-demicube t0 D5.svg 5-demicube t0 D4.svg 5-demicube t0 D3.svg 5-demicube t0 A3.svg CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
h{4,3,3,3}
5-demicube
Hemipenteract (hin)
2 5-cube t0.svg 4-cube t0.svg 5-cube t0 B3.svg 5-cube t0 B2.svg 5-cube t0 A3.svg CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{4,3,3,3}
5-cube
Penteract (pent)
3 5-cube t1.svg 5-cube t1 B4.svg 5-cube t1 B3.svg 5-cube t1 B2.svg 5-cube t1 A3.svg CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t1{4,3,3,3} = r{4,3,3,3}
Rectified 5-cube
Rectified penteract (rin)
4 5-cube t2.svg 5-cube t2 B4.svg 5-cube t2 B3.svg 5-cube t2 B2.svg 5-cube t2 A3.svg CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t2{4,3,3,3} = 2r{4,3,3,3}
Birectified 5-cube
Penteractitriacontiditeron (nit)
5 5-cube t3.svg 5-cube t3 B4.svg 5-cube t3 B3.svg 5-cube t3 B2.svg 5-cube t3 A3.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t1{3,3,3,4} = r{3,3,3,4}
Rectified 5-orthoplex
Rectified triacontiditeron (rat)
6 5-cube t4.svg 5-cube t4 B4.svg 5-cube t4 B3.svg 5-cube t4 B2.svg 5-cube t4 A3.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{3,3,3,4}
5-orthoplex
Triacontiditeron (tac)
7 5-cube t01.svg 5-cube t01 B4.svg 5-cube t01 B3.svg 5-cube t01 B2.svg 5-cube t01 A3.svg CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t0,1{4,3,3,3} = t{3,3,3,4}
Truncated 5-cube
Truncated penteract (tan)
8 5-cube t12.svg 5-cube t12 B4.svg 5-cube t12 B3.svg 5-cube t12 B2.svg 5-cube t12 A3.svg CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t1,2{4,3,3,3} = 2t{4,3,3,3}
Bitruncated 5-cube
Bitruncated penteract (bittin)
9 5-cube t02.svg 5-cube t02 B4.svg 5-cube t02 B3.svg 5-cube t02 B2.svg 5-cube t02 A3.svg CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t0,2{4,3,3,3} = rr{4,3,3,3}
Cantellated 5-cube
Rhombated penteract (sirn)
10 5-cube t13.svg 5-cube t13 B4.svg 5-cube t13 B3.svg 5-cube t13 B2.svg 5-cube t13 A3.svg CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,3{4,3,3,3} = 2rr{4,3,3,3}
Bicantellated 5-cube
Small birhombi-penteractitriacontiditeron (sibrant)
11 5-cube t03.svg 5-cube t03 B4.svg 5-cube t03 B3.svg 5-cube t03 B2.svg 5-cube t03 A3.svg CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,3{4,3,3,3}
Runcinated 5-cube
Prismated penteract (span)
12 5-cube t04.svg 5-cube t04 B4.svg 5-cube t04 B3.svg 5-cube t04 B2.svg 5-cube t04 A3.svg CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,4{4,3,3,3} = 2r2r{4,3,3,3}
Stericated 5-cube
Small celli-penteractitriacontiditeron (scant)
13 5-cube t34.svg 5-cube t34 B4.svg 5-cube t34 B3.svg 5-cube t34 B2.svg 5-cube t34 A3.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t0,1{3,3,3,4} = t{3,3,3,4}
Truncated 5-orthoplex
Truncated triacontiditeron (tot)
14 5-cube t23.svg 5-cube t23 B4.svg 5-cube t23 B3.svg 5-cube t23 B2.svg 5-cube t23 A3.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t1,2{3,3,3,4} = 2t{3,3,3,4}
Bitruncated 5-orthoplex
Bitruncated triacontiditeron (bittit)
15 5-cube t24.svg 5-cube t24 B4.svg 5-cube t24 B3.svg 5-cube t24 B2.svg 5-cube t24 A3.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t0,2{3,3,3,4} = rr{3,3,3,4}
Cantellated 5-orthoplex
Small rhombated triacontiditeron (sart)
16 5-cube t14.svg 5-cube t14 B4.svg 5-cube t14 B3.svg 5-cube t14 B2.svg 5-cube t14 A3.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,3{3,3,3,4}
Runcinated 5-orthoplex
Small prismated triacontiditeron (spat)
17 5-cube t012.svg 5-cube t012 B4.svg 5-cube t012 B3.svg 5-cube t012 B2.svg 5-cube t012 A3.svg CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t0,1,2{4,3,3,3} = tr{4,3,3,3}
Cantitruncated 5-cube
Great rhombated penteract (girn)
18 5-cube t123.svg 5-cube t123 B4.svg 5-cube t123 B3.svg 5-cube t123 B2.svg 5-cube t123 A3.svg CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,2,3{4,3,3,3} = tr{4,3,3,3}
Bicantitruncated 5-cube
Great birhombi-penteractitriacontiditeron (gibrant)
19 5-cube t013.svg 5-cube t013 B4.svg 5-cube t013 B3.svg 5-cube t013 B2.svg 5-cube t013 A3.svg CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,3{4,3,3,3}
Runcitruncated 5-cube
Prismatotruncated penteract (pattin)
20 5-cube t023.svg 5-cube t023 B4.svg 5-cube t023 B3.svg 5-cube t023 B2.svg 5-cube t023 A3.svg CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,2,3{4,3,3,3}
Runcicantellated 5-cube
Prismatorhomated penteract (prin)
21 5-cube t014.svg 5-cube t014 B4.svg 5-cube t014 B3.svg 5-cube t014 B2.svg 5-cube t014 A3.svg CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,4{4,3,3,3}
Steritruncated 5-cube
Cellitruncated penteract (capt)
22 5-cube t024.svg 5-cube t024 B4.svg 5-cube t024 B3.svg 5-cube t024 B2.svg 5-cube t024 A3.svg CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2,4{4,3,3,3}
Stericantellated 5-cube
Cellirhombi-penteractitriacontiditeron (carnit)
23 5-cube t0123.svg 5-cube t0123 B4.svg 5-cube t0123 B3.svg 5-cube t0123 B2.svg 5-cube t0123 A3.svg CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,2,3{4,3,3,3}
Runcicantitruncated 5-cube
Great primated penteract (gippin)
24 5-cube t0124.svg 5-cube t0124 B4.svg 5-cube t0124 B3.svg 5-cube t0124 B2.svg 5-cube t0124 A3.svg CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,2,4{4,3,3,3}
Stericantitruncated 5-cube
Celligreatorhombated penteract (cogrin)
25 5-cube t0134.svg 5-cube t0134 B4.svg 5-cube t0134 B3.svg 5-cube t0134 B2.svg 5-cube t0134 A3.svg CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,3,4{4,3,3,3}
Steriruncitruncated 5-cube
Celliprismatotrunki-penteractitriacontiditeron (captint)
26 5-cube t01234.svg 5-cube t01234 B4.svg 5-cube t01234 B3.svg 5-cube t01234 B2.svg 5-cube t01234 A3.svg CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3,4{4,3,3,3}
Omnitruncated 5-cube
Great celli-penteractitriacontiditeron (gacnet)
27 5-cube t234.svg 5-cube t234 B4.svg 5-cube t234 B3.svg 5-cube t234 B2.svg 5-cube t234 A3.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t0,1,2{3,3,3,4} = tr{3,3,3,4}
Cantitruncated 5-orthoplex
Great rhombated triacontiditeron (gart)
28 5-cube t134.svg 5-cube t134 B4.svg 5-cube t134 B3.svg 5-cube t134 B2.svg 5-cube t134 A3.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,1,3{3,3,3,4}
Runcitruncated 5-orthoplex
Prismatotruncated triacontiditeron (pattit)
29 5-cube t124.svg 5-cube t124 B4.svg 5-cube t124 B3.svg 5-cube t124 B2.svg 5-cube t124 A3.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,2,3{3,3,3,4}
Runcicantellated 5-orthoplex
Prismatorhombated triacontiditeron (pirt)
30 5-cube t034.svg 5-cube t034 B4.svg 5-cube t034 B3.svg 5-cube t034 B2.svg 5-cube t034 A3.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,1,4{3,3,3,4}
Steritruncated 5-orthoplex
Cellitruncated triacontiditeron (cappin)
31 5-cube t1234.svg 5-cube t1234 B4.svg 5-cube t1234 B3.svg 5-cube t1234 B2.svg 5-cube t1234 A3.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,1,2,3{3,3,3,4}
Runcicantitruncated 5-orthoplex
Great prismatorhombated triacontiditeron (gippit)
32 5-cube t0234.svg 5-cube t0234 B4.svg 5-cube t0234 B3.svg 5-cube t0234 B2.svg 5-cube t0234 A3.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
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

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 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

Mathematics Encyclopedia

Hellenica World - Scientific Library

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