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.

Orthographic projections in the D₅ Coxeter plane

5-demicube

5-orthoplex

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

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.

More information #, Coxeter plane projections ...

#	Coxeter plane projections					Coxeter diagram = Schläfli symbol Johnson and Bowers names
	[10/2]	[8]	[6]	[4]	[4]
	B5	D5	D4	D3	A3
1						= h{4,3,3,3} 5-demicube Hemipenteract (hin)
2						= h2{4,3,3,3} Cantic 5-cube Truncated hemipenteract (thin)
3						= h3{4,3,3,3} Runcic 5-cube Small rhombated hemipenteract (sirhin)
4						= h4{4,3,3,3} Steric 5-cube Small prismated hemipenteract (siphin)
5						= h2,3{4,3,3,3} Runcicantic 5-cube Great rhombated hemipenteract (girhin)
6						= h2,4{4,3,3,3} Stericantic 5-cube Prismatotruncated hemipenteract (pithin)
7						= h3,4{4,3,3,3} Steriruncic 5-cube Prismatorhombated hemipenteract (pirhin)
8						= h2,3,4{4,3,3,3} Steriruncicantic 5-cube Great prismated hemipenteract (giphin)

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• 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)".