Truncated 8-cubes

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

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

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
7-faces
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)

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
7-faces
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)

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
7-faces
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)

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
7-faces
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, [3^5,1,1]
Properties	convex

Alternate names

Quadritruncated octeract (acronym: oke) (Jonathan Bowers)

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

↑ Klitzing, (o3o3o3o3o3o3x4x – tocto).
↑ Klitzing, (o3o3o3o3o3x3x4o – bato).
↑ Klitizing, (o3o3o3o3x3x3o4o – tato)
↑ 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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, 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) with acronyms". o3o3o3o3o3o3x4x – tocto, o3o3o3o3o3x3x4o – bato, o3o3o3o3x3x3o4o – tato, o3o3o3x3x3o3o4o – oke

External links

v t e Fundamental convex regular and uniform polytopes
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
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 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 • Polytope operations