Rectified 8-cubes

In eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-cube.

8-cube	Rectified 8-cube	Birectified 8-cube	Trirectified 8-cube
Trirectified 8-orthoplex	Birectified 8-orthoplex	Rectified 8-orthoplex	8-orthoplex
Orthogonal projections in B8 Coxeter plane

There are unique 8 degrees of rectifications, the zeroth being the 8-cube, and the 7th and last being the 8-orthoplex. Vertices of the rectified 8-cube are located at the edge-centers of the 8-cube. Vertices of the birectified 8-cube are located in the square face centers of the 8-cube. Vertices of the trirectified 8-cube are located in the 7-cube cell centers of the 8-cube.

Rectified 8-cube

Rectified 8-cube
Type	uniform 8-polytope
Schläfli symbol	t1{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces	256 + 16
6-faces	2048 + 112
5-faces	7168 + 448
4-faces	14336 + 1120
Cells	17920 +* 1792
Faces	4336 + 1792
Edges	7168
Vertices	1024
Vertex figure	6-simplex prism {3,3,3,3,3}×{}
Coxeter groups	B8, [36,4] D8, [35,1,1]
Properties	convex

Alternate names

• rectified octeract

Images

orthographic projections

B8			B7
[16]			[14]
B6			B5
[12]			[10]
B4		B3		B2
[8]		[6]		[4]
A7		A5		A3
[8]		[6]		[4]

Birectified 8-cube

Birectified 8-cube
Type	uniform 8-polytope
Coxeter symbol	0511
Schläfli symbol	t2{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces	256 + 16
6-faces	1024 + 2048 + 112
5-faces	7168 + 7168 + 448
4-faces	21504 + 14336 + 1120
Cells	35840 + 17920 + 1792
Faces	35840 + 14336
Edges	21504
Vertices	1792
Vertex figure	{3,3,3,3}x{4}
Coxeter groups	B8, [36,4] D8, [35,1,1]
Properties	convex

Alternate names

• Birectified octeract
• Rectified 8-demicube

Images

orthographic projections

B8			B7
[16]			[14]
B6			B5
[12]			[10]
B4		B3		B2
[8]		[6]		[4]
A7		A5		A3
[8]		[6]		[4]

Trirectified 8-cube

Triectified 8-cube
Type	uniform 8-polytope
Schläfli symbol	t3{4,3,3,3,3,3,3}
Coxeter diagrams
7-faces	16+256
6-faces	1024 + 2048 + 112
5-faces	1792 + 7168 + 7168 + 448
4-faces	1792 + 10752 + 21504 +14336
Cells	8960 + 26880 + 35840
Faces	17920+35840
Edges	17920
Vertices	1152
Vertex figure	{3,3,3}x{3,4}
Coxeter groups	B8, [36,4] D8, [35,1,1]
Properties	convex

Alternate names

• trirectified octeract

Images

orthographic projections

B8			B7
[16]			[14]
B6			B5
[12]			[10]
B4		B3		B2
[8]		[6]		[4]
A7		A5		A3
[8]		[6]		[4]