B5 polytope

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

Symmetric orthographic projections of these 32 polytopes can be made in the B₅, B₄, B₃, B₂, A₃, 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.

#	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						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)

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	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / E₉ / E₁₀ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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