Truncated 8-cubes

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

8-cube	Truncated 8-cube	Bitruncated 8-cube
Quadritruncated 8-cube	Tritruncated 8-cube	Tritruncated 8-orthoplex
Bitruncated 8-orthoplex	Truncated 8-orthoplex	8-orthoplex
Orthogonal projections in B8 Coxeter plane

There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube are located on the square faces of the 8-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 8-cube. The final truncations are best expressed relative to the 8-orthoplex.

Truncated 8-cube

Truncated 8-cube
Type	uniform 8-polytope
Schläfli symbol	t{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	( )v{3,3,3,3,3}
Coxeter groups	B8, [3,3,3,3,3,3,4]
Properties	convex

Alternate names

• Truncated octeract (acronym tocto) (Jonathan Bowers)^[1]

Coordinates

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of

(±2,±2,±2,±2,±2,±2,±1,0)

Images

orthographic projections

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

Related polytopes

The truncated 8-cube, is seventh in a sequence of truncated hypercubes:

Truncated hypercubes

Image								...
Name	Octagon	Truncated cube	Truncated tesseract	Truncated 5-cube	Truncated 6-cube	Truncated 7-cube	Truncated 8-cube
Coxeter diagram
Vertex figure	( )v( )	( )v{ }	( )v{3}	( )v{3,3}	( )v{3,3,3}	( )v{3,3,3,3}	( )v{3,3,3,3,3}

Bitruncated 8-cube

Bitruncated 8-cube
Type	uniform 8-polytope
Schläfli symbol	2t{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	{ }v{3,3,3,3}
Coxeter groups	B8, [3,3,3,3,3,3,4]
Properties	convex

Alternate names

• Bitruncated octeract (acronym bato) (Jonathan Bowers)^[2]

Coordinates

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of

(±2,±2,±2,±2,±2,±1,0,0)

Images

orthographic projections

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

Related polytopes

The bitruncated 8-cube is sixth in a sequence of bitruncated hypercubes:

Bitruncated hypercubes

Image							...
Name	Bitruncated cube	Bitruncated tesseract	Bitruncated 5-cube	Bitruncated 6-cube	Bitruncated 7-cube	Bitruncated 8-cube
Coxeter
Vertex figure	( )v{ }	{ }v{ }	{ }v{3}	{ }v{3,3}	{ }v{3,3,3}	{ }v{3,3,3,3}

Tritruncated 8-cube

Tritruncated 8-cube
Type	uniform 8-polytope
Schläfli symbol	3t{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	{4}v{3,3,3}
Coxeter groups	B8, [3,3,3,3,3,3,4]
Properties	convex

Alternate names

• Tritruncated octeract (acronym tato) (Jonathan Bowers)^[3]

Coordinates

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of

(±2,±2,±2,±2,±1,0,0,0)

Images

orthographic projections

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

Quadritruncated 8-cube

Quadritruncated 8-cube
Type	uniform 8-polytope
Schläfli symbol	4t{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	{3,4}v{3,3}
Coxeter groups	B8, [3,3,3,3,3,3,4] D8, [35,1,1]
Properties	convex

Alternate names

• Quadritruncated octeract (acronym oke) (Jonathan Bowers)^[4]

Coordinates

Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±2,±2,±1,0,0,0)

Images

orthographic projections

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

Related polytopes

2-isotopic hypercubes

Dim.	2	3	4	5	6	7	8	n
Name	t{4}	r{4,3}	2t{4,3,3}	2r{4,3,3,3}	3t{4,3,3,3,3}	3r{4,3,3,3,3,3}	4t{4,3,3,3,3,3,3}	...
Coxeter diagram
Images
Facets		{3} {4}	t{3,3} t{3,4}	r{3,3,3} r{3,3,4}	2t{3,3,3,3} 2t{3,3,3,4}	2r{3,3,3,3,3} 2r{3,3,3,3,4}	3t{3,3,3,3,3,3} 3t{3,3,3,3,3,4}
Vertex figure	( )v( )	{ }×{ }	{ }v{ }	{3}×{4}	{3}v{4}	{3,3}×{3,4}	{3,3}v{3,4}

Notes

1. Klitizing, (o3o3o3o3o3o3x4x – tocto)
2. Klitizing, (o3o3o3o3o3x3x4o – bato)
3. Klitizing, (o3o3o3o3x3x3o4o – tato)
4. Klitizing, (o3o3o3x3x3o3o4o – oke)

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)". o3o3o3o3o3o3x4x – tocto, o3o3o3o3o3x3x4o – bato, o3o3o3o3x3x3o4o – tato, o3o3o3x3x3o3o4o – oke

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