User:Tomruen/List of D5 polytopes

Orthographic projections in the D₅ Coxeter plane
5-demicube	5-orthoplex

In 5-dimensional geometry, there are 23 uniform polytopes with D₅ symmetry, 8 are unique, and 15 are shared with the B₅ symmetry. There are two special forms, the 5-orthoplex, and 5-demicube with 10 and 16 vertices respectively.

They can be visualized as symmetric orthographic projections in Coxeter planes of the D₆ Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 8 polytopes can be made in the D₅, D₄, D₃, A₃, Coxeter planes. A_k has [k+1] symmetry, D_k has [2(k-1)] symmetry. The B₅ plane is included, with only half the [10] symmetry displayed.

These 8 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.

#	Coxeter plane projections					Coxeter-Dynkin diagram Schläfli symbol symbols Johnson and Bowers names
	[10/2]	[8]	[6]	[4]	[4]
	B₅	D₅	D₄	D₃	A₃
1						(1₂₁) 5-demicube Hemipenteract (hin)
2						t_0,1(1₂₁) Truncated 5-demicube Truncated hemipenteract (thin)
3						t_0,2(1₂₁) Cantellated 5-demicube Small rhombated hemipenteract (sirhin)
4						t_0,3(1₂₁) Runcinated 5-demicube Small prismated hemipenteract (siphin)
5						t_0,1,2(1₂₁) Cantitruncated 5-demicube Great rhombated hemipenteract (girhin)
6						t_0,1,3(1₂₁) Runcitruncated 5-demicube Prismatotruncated hemipenteract (pithin)
7						t_0,2,3(1₂₁) Runcicantellated 5-demicube Prismatorhombated hemipenteract (pirhin)
8						t_0,1,2,3(1₂₁) Runcicantitruncated 5-demicube Great prismated hemipenteract (giphin)

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
Klitzing, Richard. "5D uniform polytopes (polytera)".

Notes

^ Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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

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

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