Truncated 6-cubes

Orthogonal projections in B₆ Coxeter plane
File:6-cube t0.svg 6-cube File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	File:6-cube t01.svg Truncated 6-cube File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	File:6-cube t12.svg Bitruncated 6-cube File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	File:6-cube t23.svg Tritruncated 6-cube File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png
File:6-cube t5.svg 6-orthoplex File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png	File:6-cube t45.svg Truncated 6-orthoplex File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png	File:6-cube t34.svg Bitruncated 6-orthoplex File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png

In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube. There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube.

Truncated 6-cube

Truncated 6-cube
Type	uniform 6-polytope
Class	B6 polytope
Schläfli symbol	t{4,3,3,3,3}
Coxeter-Dynkin diagrams	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
5-faces	76
4-faces	464
Cells	1120
Faces	1520
Edges	1152
Vertices	384
Vertex figure	File:Truncated 6-cube verf.png ( )v{3,3,3}
Coxeter groups	B₆, [3,3,3,3,4]
Properties	convex

Alternate names

Truncated hexeract (Acronym: tox) (Jonathan Bowers)^[1]

Construction and coordinates

The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at $1 / (\sqrt{2} + 2)$ of the edge length. A regular 5-simplex replaces each original vertex. The Cartesian coordinates of the vertices of a truncated 6-cube having edge length 2 are the permutations of:

(\pm 1, \pm (1 + \sqrt{2}), \pm (1 + \sqrt{2}), \pm (1 + \sqrt{2}), \pm (1 + \sqrt{2}), \pm (1 + \sqrt{2}))

Images

orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph	File:6-cube t01.svg	File:6-cube t01 B5.svg	File:6-cube t01 B4.svg
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph	File:6-cube t01 B3.svg	File:6-cube t01 B2.svg
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph	File:6-cube t01 A5.svg	File:6-cube t01 A3.svg
Dihedral symmetry	[6]	[4]

Related polytopes

The truncated 6-cube, is fifth in a sequence of truncated hypercubes:

Truncated hypercubes
Image	File:Regular polygon 8 annotated.svg	File:3-cube t01.svgFile:Truncated hexahedron.png	File:4-cube t01.svgFile:Schlegel half-solid truncated tesseract.png	File:5-cube t01.svgFile:5-cube t01 A3.svg	File:6-cube t01.svgFile:6-cube t01 A5.svg	File:7-cube t01.svgFile:7-cube t01 A5.svg	File:8-cube t01.svgFile:8-cube t01 A7.svg	...
Name	Octagon	Truncated cube	Truncated tesseract	Truncated 5-cube	Truncated 6-cube	Truncated 7-cube	Truncated 8-cube
Coxeter diagram	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Vertex figure	( )v( )	File:Truncated cube vertfig.svg ( )v{ }	File:Truncated 8-cell verf.png ( )v{3}	File:Truncated 5-cube verf.png ( )v{3,3}	( )v{3,3,3}	( )v{3,3,3,3}	( )v{3,3,3,3,3}

Bitruncated 6-cube

Bitruncated 6-cube
Type	uniform 6-polytope
Class	B6 polytope
Schläfli symbol	2t{4,3,3,3,3}
Coxeter-Dynkin diagrams	File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	File:Bitruncated 6-cube verf.png { }v{3,3}
Coxeter groups	B₆, [3,3,3,3,4]
Properties	convex

Alternate names

Bitruncated hexeract (Acronym: botox) (Jonathan Bowers)^[2]

Construction and coordinates

The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge length 2 are the permutations of:

(0, \pm 1, \pm 2, \pm 2, \pm 2, \pm 2)

Images

orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph	File:6-cube t12.svg	File:6-cube t12 B5.svg	File:6-cube t12 B4.svg
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph	File:6-cube t12 B3.svg	File:6-cube t12 B2.svg
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph	File:6-cube t12 A5.svg	File:6-cube t12 A3.svg
Dihedral symmetry	[6]	[4]

Related polytopes

The bitruncated 6-cube is fourth in a sequence of bitruncated hypercubes:

Bitruncated hypercubes
Image	File:3-cube t12.svgFile:Truncated octahedron.png	File:4-cube t12.svgFile:Schlegel half-solid bitruncated 8-cell.png	File:5-cube t12.svgFile:5-cube t12 A3.svg	File:6-cube t12.svgFile:6-cube t12 A5.svg	File:7-cube t12.svgFile:7-cube t12 A5.svg	File:8-cube t12.svgFile:8-cube t12 A7.svg	...
Name	Bitruncated cube	Bitruncated tesseract	Bitruncated 5-cube	Bitruncated 6-cube	Bitruncated 7-cube	Bitruncated 8-cube
Coxeter	File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Vertex figure	File:Truncated octahedron vertfig.png ( )v{ }	File:Bitruncated 8-cell verf.png { }v{ }	File:Bitruncated penteract verf.png { }v{3}	File:Bitruncated 6-cube verf.png { }v{3,3}	{ }v{3,3,3}	{ }v{3,3,3,3}

Tritruncated 6-cube

Tritruncated 6-cube
Type	uniform 6-polytope
Class	B6 polytope
Schläfli symbol	3t{4,3,3,3,3}
Coxeter-Dynkin diagrams	File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	File:Tritruncated 6-cube verf.png {3}v{4}^[3]
Coxeter groups	B₆, [3,3,3,3,4]
Properties	convex

Alternate names

Tritruncated hexeract (Acronym: xog) (Jonathan Bowers)^[4]

Construction and coordinates

The Cartesian coordinates of the vertices of a tritruncated 6-cube having edge length 2 are the permutations of:

(0, 0, \pm 1, \pm 2, \pm 2, \pm 2)

Images

orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph	File:6-cube t23.svg	File:6-cube t23 B5.svg	File:6-cube t23 B4.svg
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph	File:6-cube t23 B3.svg	File:6-cube t23 B2.svg
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph	File:6-cube t23 A5.svg	File:6-cube t23 A3.svg
Dihedral symmetry	[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	File:CDel label4.pngFile:CDel branch 11.png	File:CDel node 1.pngFile:CDel split1-43.pngFile:CDel nodes.png	File:CDel branch 11.pngFile:CDel 4a3b.pngFile:CDel nodes.png	File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png	File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png	File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png	File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png
Images	File:Truncated square.png	File:3-cube t1.svgFile:Cuboctahedron.png	File:4-cube t12.svgFile:Schlegel half-solid bitruncated 8-cell.png	File:5-cube t2.svgFile:5-cube t2 A3.svg	File:6-cube t23.svgFile:6-cube t23 A5.svg	File:7-cube t3.svgFile:7-cube t3 A5.svg	File:8-cube t34.svgFile:8-cube t34 A7.svg
Facets		{3} File:Regular polygon 3 annotated.svg {4} File:Regular polygon 4 annotated.svg	t{3,3} File:Uniform polyhedron-33-t01.png t{3,4} File:Uniform polyhedron-43-t12.png	r{3,3,3} File:Schlegel half-solid rectified 5-cell.png r{3,3,4} File:Schlegel wireframe 24-cell.png	2t{3,3,3,3} File:5-simplex t12.svg 2t{3,3,3,4} File:5-cube t23.svg	2r{3,3,3,3,3} File:6-simplex t2.svg 2r{3,3,3,3,4} File:6-cube t4.svg	3t{3,3,3,3,3,3} File:7-simplex t23.svg 3t{3,3,3,3,3,4} File:7-cube t45.svg
Vertex figure	( )v( )	File:Cuboctahedron vertfig.png { }×{ }	File:Bitruncated 8-cell verf.png { }v{ }	File:Birectified penteract verf.png {3}×{4}	File:Tritruncated 6-cube verf.png {3}v{4}	{3,3}×{3,4}	{3,3}v{3,4}

Related polytopes

These polytopes are from a set of 63 Uniform 6-polytopes generated from the B₆ Coxeter plane, including the regular 6-cube or 6-orthoplex.

B6 polytopes
File:6-cube t5.svg β₆	File:6-cube t4.svg t₁β₆	File:6-cube t3.svg t₂β₆	File:6-cube t2.svg t₂γ₆	File:6-cube t1.svg t₁γ₆	File:6-cube t0.svg γ₆	File:6-cube t45.svg t_0,1β₆	File:6-cube t35.svg t_0,2β₆
File:6-cube t34.svg t_1,2β₆	File:6-cube t25.svg t_0,3β₆	File:6-cube t24.svg t_1,3β₆	File:6-cube t23.svg t_2,3γ₆	File:6-cube t15.svg t_0,4β₆	File:6-cube t14.svg t_1,4γ₆	File:6-cube t13.svg t_1,3γ₆	File:6-cube t12.svg t_1,2γ₆
File:6-cube t05.svg t_0,5γ₆	File:6-cube t04.svg t_0,4γ₆	File:6-cube t03.svg t_0,3γ₆	File:6-cube t02.svg t_0,2γ₆	File:6-cube t01.svg t_0,1γ₆	File:6-cube t345.svg t_0,1,2β₆	File:6-cube t245.svg t_0,1,3β₆	File:6-cube t235.svg t_0,2,3β₆
File:6-cube t234.svg t_1,2,3β₆	File:6-cube t145.svg t_0,1,4β₆	File:6-cube t135.svg t_0,2,4β₆	File:6-cube t134.svg t_1,2,4β₆	File:6-cube t125.svg t_0,3,4β₆	File:6-cube t124.svg t_1,2,4γ₆	File:6-cube t123.svg t_1,2,3γ₆	File:6-cube t045.svg t_0,1,5β₆
File:6-cube t035.svg t_0,2,5β₆	File:6-cube t034.svg t_0,3,4γ₆	File:6-cube t025.svg t_0,2,5γ₆	File:6-cube t024.svg t_0,2,4γ₆	File:6-cube t023.svg t_0,2,3γ₆	File:6-cube t015.svg t_0,1,5γ₆	File:6-cube t014.svg t_0,1,4γ₆	File:6-cube t013.svg t_0,1,3γ₆
File:6-cube t012.svg t_0,1,2γ₆	File:6-cube t2345.svg t_0,1,2,3β₆	File:6-cube t1345.svg t_0,1,2,4β₆	File:6-cube t1245.svg t_0,1,3,4β₆	File:6-cube t1235.svg t_0,2,3,4β₆	File:6-cube t1234.svg t_1,2,3,4γ₆	File:6-cube t0345.svg t_0,1,2,5β₆	File:6-cube t0245.svg t_0,1,3,5β₆
File:6-cube t0235.svg t_0,2,3,5γ₆	File:6-cube t0234.svg t_0,2,3,4γ₆	File:6-cube t0145.svg t_0,1,4,5γ₆	File:6-cube t0135.svg t_0,1,3,5γ₆	File:6-cube t0134.svg t_0,1,3,4γ₆	File:6-cube t0125.svg t_0,1,2,5γ₆	File:6-cube t0124.svg t_0,1,2,4γ₆	File:6-cube t0123.svg t_0,1,2,3γ₆
File:6-cube t12345.svg t_0,1,2,3,4β₆	File:6-cube t02345.svg t_0,1,2,3,5β₆	File:6-cube t01345.svg t_0,1,2,4,5β₆	File:6-cube t01245.svg t_0,1,2,4,5γ₆	File:6-cube t01235.svg t_0,1,2,3,5γ₆	File:6-cube t01234.svg t_0,1,2,3,4γ₆	File:6-cube t012345.svg t_0,1,2,3,4,5γ₆

Notes

↑ Klitzing, (o3o3o3o3x4x - tox)
↑ Klitzing, (o3o3o3x3x4o - botox)
↑ "Squete".
↑ Klitzing, (o3o3x3x3o4o - xog)

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. "6D uniform polytopes (polypeta)". o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog

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

[1] Klitzing, (o3o3o3o3x4x - tox)

[2] Klitzing, (o3o3o3x3x4o - botox)

[3] "Squete".

[4] Klitzing, (o3o3x3x3o4o - xog)

[1]

[2]

[3]

[4]

Truncated 6-cubes

Contents

Truncated 6-cube

Alternate names

Construction and coordinates

Images

Related polytopes

Bitruncated 6-cube

Alternate names

Construction and coordinates

Images

Related polytopes

Tritruncated 6-cube

Alternate names

Construction and coordinates

Images

Related polytopes

Related polytopes

Notes

References

External links

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