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B5 polytope

[[File:5-cube t0.svg120px]][5-cube](5-cube)[[File:5-cube t4.svg120px]][5-orthoplex](5-orthoplex)[[File:5-demicube t0 B5.svg120px]][5-demicube](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

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.

#GraphB5 / A4[10]GraphB4 / D5[8]GraphB3 / A2[6]GraphB2[4]GraphA3[4]Coxeter-Dynkin diagram and Schläfli symbolJohnson and Bowers names
1[[File:5-demicube t0 B5.svg80px]][[File:5-demicube t0 D5.svg80px]][[File:5-demicube t0 D4.svg80px]][[File:5-demicube t0 D3.svg80px]][[File:5-demicube t0 A3.svg80px]]h{4,3,3,3}[5-demicube](5-demicube)Hemipenteract (hin)
2[[File:5-cube t0.svg80px]][[File:4-cube t0.svg80px]][[File:5-cube t0 B3.svg80px]][[File:5-cube t0 B2.svg80px]][[File:5-cube t0 A3.svg80px]]{4,3,3,3}[5-cube](5-cube)Penteract (pent)
3[[File:5-cube t1.svg80px]][[File:5-cube t1 B4.svg80px]][[File:5-cube t1 B3.svg80px]][[File:5-cube t1 B2.svg80px]][[File:5-cube t1 A3.svg80px]]t1{4,3,3,3} = r{4,3,3,3}Rectified 5-cubeRectified penteract (rin)
4[[File:5-cube t2.svg80px]][[File:5-cube t2 B4.svg80px]][[File:5-cube t2 B3.svg80px]][[File:5-cube t2 B2.svg80px]][[File:5-cube t2 A3.svg80px]]t2{4,3,3,3} = 2r{4,3,3,3}Birectified 5-cubePenteractitriacontiditeron (nit)
5[[File:5-cube t3.svg80px]][[File:5-cube t3 B4.svg80px]][[File:5-cube t3 B3.svg80px]][[File:5-cube t3 B2.svg80px]][[File:5-cube t3 A3.svg80px]]t1{3,3,3,4} = r{3,3,3,4}Rectified 5-orthoplexRectified triacontiditeron (rat)
6[[File:5-cube t4.svg80px]][[File:5-cube t4 B4.svg80px]][[File:5-cube t4 B3.svg80px]][[File:5-cube t4 B2.svg80px]][[File:5-cube t4 A3.svg80px]]{3,3,3,4}[5-orthoplex](5-orthoplex)Triacontiditeron (tac)
7[[File:5-cube t01.svg80px]][[File:5-cube t01 B4.svg80px]][[File:5-cube t01 B3.svg80px]][[File:5-cube t01 B2.svg80px]][[File:5-cube t01 A3.svg80px]]t0,1{4,3,3,3} = t{3,3,3,4}Truncated 5-cubeTruncated penteract (tan)
8[[File:5-cube t12.svg80px]][[File:5-cube t12 B4.svg80px]][[File:5-cube t12 B3.svg80px]][[File:5-cube t12 B2.svg80px]][[File:5-cube t12 A3.svg80px]]t1,2{4,3,3,3} = 2t{4,3,3,3}Bitruncated 5-cubeBitruncated penteract (bittin)
9[[File:5-cube t02.svg80px]][[File:5-cube t02 B4.svg80px]][[File:5-cube t02 B3.svg80px]][[File:5-cube t02 B2.svg80px]][[File:5-cube t02 A3.svg80px]]t0,2{4,3,3,3} = rr{4,3,3,3}Cantellated 5-cubeRhombated penteract (sirn)
10[[File:5-cube t13.svg80px]][[File:5-cube t13 B4.svg80px]][[File:5-cube t13 B3.svg80px]][[File:5-cube t13 B2.svg80px]][[File:5-cube t13 A3.svg80px]]t1,3{4,3,3,3} = 2rr{4,3,3,3}Bicantellated 5-cubeSmall birhombi-penteractitriacontiditeron (sibrant)
11[[File:5-cube t03.svg80px]][[File:5-cube t03 B4.svg80px]][[File:5-cube t03 B3.svg80px]][[File:5-cube t03 B2.svg80px]][[File:5-cube t03 A3.svg80px]]t0,3{4,3,3,3}Runcinated 5-cubePrismated penteract (span)
12[[File:5-cube t04.svg80px]][[File:5-cube t04 B4.svg80px]][[File:5-cube t04 B3.svg80px]][[File:5-cube t04 B2.svg80px]][[File:5-cube t04 A3.svg80px]]t0,4{4,3,3,3} = 2r2r{4,3,3,3}Stericated 5-cubeSmall celli-penteractitriacontiditeron (scant)
13[[File:5-cube t34.svg80px]][[File:5-cube t34 B4.svg80px]][[File:5-cube t34 B3.svg80px]][[File:5-cube t34 B2.svg80px]][[File:5-cube t34 A3.svg80px]]t0,1{3,3,3,4} = t{3,3,3,4}Truncated 5-orthoplexTruncated triacontiditeron (tot)
14[[File:5-cube t23.svg80px]][[File:5-cube t23 B4.svg80px]][[File:5-cube t23 B3.svg80px]][[File:5-cube t23 B2.svg80px]][[File:5-cube t23 A3.svg80px]]t1,2{3,3,3,4} = 2t{3,3,3,4}Bitruncated 5-orthoplexBitruncated triacontiditeron (bittit)
15[[File:5-cube t24.svg80px]][[File:5-cube t24 B4.svg80px]][[File:5-cube t24 B3.svg80px]][[File:5-cube t24 B2.svg80px]][[File:5-cube t24 A3.svg80px]]t0,2{3,3,3,4} = rr{3,3,3,4}Cantellated 5-orthoplexSmall rhombated triacontiditeron (sart)
16[[File:5-cube t14.svg80px]][[File:5-cube t14 B4.svg80px]][[File:5-cube t14 B3.svg80px]][[File:5-cube t14 B2.svg80px]][[File:5-cube t14 A3.svg80px]]t0,3{3,3,3,4}Runcinated 5-orthoplexSmall prismated triacontiditeron (spat)
17[[File:5-cube t012.svg80px]][[File:5-cube t012 B4.svg80px]][[File:5-cube t012 B3.svg80px]][[File:5-cube t012 B2.svg80px]][[File:5-cube t012 A3.svg80px]]t0,1,2{4,3,3,3} = tr{4,3,3,3}Cantitruncated 5-cubeGreat rhombated penteract (girn)
18[[File:5-cube t123.svg80px]][[File:5-cube t123 B4.svg80px]][[File:5-cube t123 B3.svg80px]][[File:5-cube t123 B2.svg80px]][[File:5-cube t123 A3.svg80px]]t1,2,3{4,3,3,3} = tr{4,3,3,3}Bicantitruncated 5-cubeGreat birhombi-penteractitriacontiditeron (gibrant)
19[[File:5-cube t013.svg80px]][[File:5-cube t013 B4.svg80px]][[File:5-cube t013 B3.svg80px]][[File:5-cube t013 B2.svg80px]][[File:5-cube t013 A3.svg80px]]t0,1,3{4,3,3,3}Runcitruncated 5-cubePrismatotruncated penteract (pattin)
20[[File:5-cube t023.svg80px]][[File:5-cube t023 B4.svg80px]][[File:5-cube t023 B3.svg80px]][[File:5-cube t023 B2.svg80px]][[File:5-cube t023 A3.svg80px]]t0,2,3{4,3,3,3}Runcicantellated 5-cubePrismatorhomated penteract (prin)
21[[File:5-cube t014.svg80px]][[File:5-cube t014 B4.svg80px]][[File:5-cube t014 B3.svg80px]][[File:5-cube t014 B2.svg80px]][[File:5-cube t014 A3.svg80px]]t0,1,4{4,3,3,3}Steritruncated 5-cubeCellitruncated penteract (capt)
22[[File:5-cube t024.svg80px]][[File:5-cube t024 B4.svg80px]][[File:5-cube t024 B3.svg80px]][[File:5-cube t024 B2.svg80px]][[File:5-cube t024 A3.svg80px]]t0,2,4{4,3,3,3}Stericantellated 5-cubeCellirhombi-penteractitriacontiditeron (carnit)
23[[File:5-cube t0123.svg80px]][[File:5-cube t0123 B4.svg80px]][[File:5-cube t0123 B3.svg80px]][[File:5-cube t0123 B2.svg80px]][[File:5-cube t0123 A3.svg80px]]t0,1,2,3{4,3,3,3}Runcicantitruncated 5-cubeGreat primated penteract (gippin)
24[[File:5-cube t0124.svg80px]][[File:5-cube t0124 B4.svg80px]][[File:5-cube t0124 B3.svg80px]][[File:5-cube t0124 B2.svg80px]][[File:5-cube t0124 A3.svg80px]]t0,1,2,4{4,3,3,3}Stericantitruncated 5-cubeCelligreatorhombated penteract (cogrin)
25[[File:5-cube t0134.svg80px]][[File:5-cube t0134 B4.svg80px]][[File:5-cube t0134 B3.svg80px]][[File:5-cube t0134 B2.svg80px]][[File:5-cube t0134 A3.svg80px]]t0,1,3,4{4,3,3,3}Steriruncitruncated 5-cubeCelliprismatotrunki-penteractitriacontiditeron (captint)
26[[File:5-cube t01234.svg80px]][[File:5-cube t01234 B4.svg80px]][[File:5-cube t01234 B3.svg80px]][[File:5-cube t01234 B2.svg80px]][[File:5-cube t01234 A3.svg80px]]t0,1,2,3,4{4,3,3,3}Omnitruncated 5-cubeGreat celli-penteractitriacontiditeron (gacnet)
27[[File:5-cube t234.svg80px]][[File:5-cube t234 B4.svg80px]][[File:5-cube t234 B3.svg80px]][[File:5-cube t234 B2.svg80px]][[File:5-cube t234 A3.svg80px]]t0,1,2{3,3,3,4} = tr{3,3,3,4}Cantitruncated 5-orthoplexGreat rhombated triacontiditeron (gart)
28[[File:5-cube t134.svg80px]][[File:5-cube t134 B4.svg80px]][[File:5-cube t134 B3.svg80px]][[File:5-cube t134 B2.svg80px]][[File:5-cube t134 A3.svg80px]]t0,1,3{3,3,3,4}Runcitruncated 5-orthoplexPrismatotruncated triacontiditeron (pattit)
29[[File:5-cube t124.svg80px]][[File:5-cube t124 B4.svg80px]][[File:5-cube t124 B3.svg80px]][[File:5-cube t124 B2.svg80px]][[File:5-cube t124 A3.svg80px]]t0,2,3{3,3,3,4}Runcicantellated 5-orthoplexPrismatorhombated triacontiditeron (pirt)
30[[File:5-cube t034.svg80px]][[File:5-cube t034 B4.svg80px]][[File:5-cube t034 B3.svg80px]][[File:5-cube t034 B2.svg80px]][[File:5-cube t034 A3.svg80px]]t0,1,4{3,3,3,4}Steritruncated 5-orthoplexCellitruncated triacontiditeron (cappin)
31[[File:5-cube t1234.svg80px]][[File:5-cube t1234 B4.svg80px]][[File:5-cube t1234 B3.svg80px]][[File:5-cube t1234 B2.svg80px]][[File:5-cube t1234 A3.svg80px]]t0,1,2,3{3,3,3,4}Runcicantitruncated 5-orthoplexGreat prismatorhombated triacontiditeron (gippit)
32[[File:5-cube t0234.svg80px]][[File:5-cube t0234 B4.svg80px]][[File:5-cube t0234 B3.svg80px]][[File:5-cube t0234 B2.svg80px]][[File:5-cube t0234 A3.svg80px]]t0,1,2,4{3,3,3,4}Stericantitruncated 5-orthoplexCelligreatorhombated 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,
    • (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

References

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