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 D₈ Coxeter plane

8-demicube

8-orthoplex

In 8-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 256 uniform polytopes with D₈ symmetry, 64 are unique, and 192 are shared with the B₈ symmetry. There are two regular forms, the 8-orthoplex, and 8-demicube with 16 and 128 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 D₈ 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 64 polytopes can be made in the D₈, D₇, D₆, D₅, D₄, D₃, A₅, A₃, Coxeter planes. A_k has [k+1] symmetry, D_k has [2(k-1)] symmetry. B₈ is also included although only half of its [16] symmetry exists in these polytopes.

These 64 polytopes are each shown in these 10 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
	B8 [16/2]	D8 [14]	D7 [12]	D6 [10]	D5 [8]	D4 [6]	D3 [4]	A7 [8]	A5 [6]	A3 [4]
1											8-demicube (hocto)
2											Truncated 8-demicube Truncated 8-demicube In eight-dimensional geometry, a truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube.- Cartesian coordinates :The Cartesian coordinates for the vertices of a truncated 8-demicube centered at the origin and edge length 6√2 are coordinate permutations:with an odd number... (thocto)
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