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 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)