B5 polytope
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![]() 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[edit]
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[edit]
- 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[edit]
- Klitzing, Richard. "5D uniform polytopes (polytera)".