Rectified 8-cubes

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

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

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, [3^5,1,1]
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, [3^5,1,1]
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, [3^5,1,1]
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]

Notes

References

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]
Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
Klitzing, Richard. "8D uniform polytopes (polyzetta)". o3o3o3o3o3o3x4o, o3o3o3o3o3x3o4o, o3o3o3o3x3o3o4o

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds