Orthographic projection

Orthographic projection

Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...

s in the D₆ Coxeter plane

6-demicube

6-orthoplex

In 6-dimensional geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, there are 64 uniform polytopes with D₆ symmetry, 16 are unique, and 48 are shared with the B₆ symmetry. There are two regular forms, the 6-orthoplex, and 6-demicube with 12 and 32 vertices respectively.

They can be visualized as symmetric orthographic projection

Orthographic projection

Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...

s in Coxeter planes of the D₆ Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projection

Orthographic projection

Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...

s of these 16 polytopes can be made in the D₆, D₅, D₄, D₃, A₅, A₃, Coxeter planes. A_k has [k+1] symmetry, D_k has [2(k-1)] symmetry. B₆ is also included although only half of its [12] symmetry exists in these polytopes.

These 16 polytopes are each shown in these 7 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

#	Coxeter plane graphs							Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... Schläfli symbol Names
	B6 [12/2]	D6 [10]	D5 [8]	D4 [6]	D3 [4]	A5 [6]	A3 [4]
1								6-demicube Hemihexeract (hax)
2								Truncated 6-demicube Truncated hemihexeract (thax)
3								Cantellated 6-demicube Cantellated 6-demicube In six-dimensional geometry, a cantellated 6-demicube is a convex uniform 6-polytope, being a cantellation of the uniform 6-demicube. There are 2 unique cantellation for the 6-demicube including a truncation.- Cantellated 6-demicube:... Small rhombated hemihexeract (sirhax)
4								Runcinated 6-demicube Runcinated 6-demicube In six-dimensional geometry, a runcinated 6-demicube is a convex uniform 6-polytope with 3rd order truncations of the uniform 6-demicube.There are unique 4 runcinations of the 6-demicube, including permutations of truncations, and cantellations.... Small prismated hemihexeract (sophax)
5								tericated demihexeract Small cellated demihexeract (sochax)
6								Cantitruncated 6-demicube Great rhombated hemihexeract (girhax)
7								Runcitruncated 6-demicube Prismatotruncated hemihexeract (pithax)
8								Runcicantellated 6-demicube Prismatorhombated hemihexeract (prohax)
9								Runcitruncated 6-demicube Cellitruncated hemihexeract (cathix)
10								Stericantellated 6-demicube Cellirhombated hemihexeract (crohax)
11								Steriruncinated 6-demicube Celliprismated hemihexeract (cophix)
12								Runcicantitruncated 6-demicube Great prismated hemihexeract (gophax)
13								Stericantitruncated 6-demicube Celligreatorhombated hemihexeract (cagrohax)
14								Steriruncitruncated 6-demicube Celliprismatotruncated hemihexeract (capthix)
15								Steriruncicantellated 6-demicube Celliprismatorhombated hemihexeract (caprohax)
16								Steriruncicantitruncated 6-demicube Great cellated hemihexeract (gochax)