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]

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
    • (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

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds