Truncated 6-orthoplexes

Orthogonal projections in B₆ Coxeter plane
File:6-cube t5.svg 6-orthoplex File: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 4.pngFile:CDel node.png	File:6-cube t45.svg Truncated 6-orthoplex File: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 4.pngFile:CDel node.png	File:6-cube t34.svg Bitruncated 6-orthoplex File: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.pngFile:CDel 4.pngFile:CDel node.png	File:6-cube t23.svg Tritruncated 6-cube File:CDel node.pngFile:CDel 3.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 t0.svg 6-cube File: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 4.pngFile:CDel node 1.png	File:6-cube t01.svg Truncated 6-cube File: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 1.pngFile:CDel 4.pngFile:CDel node 1.png	File:6-cube t12.svg Bitruncated 6-cube File:CDel node.pngFile:CDel 3.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-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex. There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the bitruncated 6-orthoplex are located on the triangular faces of the 6-orthoplex. Vertices of the tritruncated 6-orthoplex are located inside the tetrahedral cells of the 6-orthoplex.

Truncated 6-orthoplex

Truncated 6-orthoplex
Type	uniform 6-polytope
Schläfli symbol	t{3,3,3,3,4}
Coxeter-Dynkin diagrams	File: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 4.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png
5-faces	76
4-faces	576
Cells	1200
Faces	1120
Edges	540
Vertices	120
Vertex figure	File:Truncated 6-orthoplex verf.png ( )v{3,4}
Coxeter groups	B₆, [3,3,3,3,4] D₆, [3^3,1,1]
Properties	convex

Alternate names

Truncated hexacross
Truncated hexacontatetrapeton (Acronym: tag) (Jonathan Bowers)^[1]

Construction

There are two Coxeter groups associated with the truncated hexacross, one with the C₆ or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D₆ or [3^3,1,1] Coxeter group.

Coordinates

Cartesian coordinates for the vertices of a truncated 6-orthoplex, centered at the origin, are all 120 vertices are sign (4) and coordinate (30) permutations of

(±2,±1,0,0,0,0)

Images

orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph	File:6-cube t45.svg	File:6-cube t45 B5.svg	File:6-cube t45 B4.svg
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph	File:6-cube t45 B3.svg	File:6-cube t45 B2.svg
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph	File:6-cube t45 A5.svg	File:6-cube t45 A3.svg
Dihedral symmetry	[6]	[4]

Bitruncated 6-orthoplex

Bitruncated 6-orthoplex
Type	uniform 6-polytope
Schläfli symbol	2t{3,3,3,3,4}
Coxeter-Dynkin diagrams	File: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.pngFile:CDel 4.pngFile:CDel node.png File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	File:Bitruncated 6-orthoplex verf.png { }v{3,4}
Coxeter groups	B₆, [3,3,3,3,4] D₆, [3^3,1,1]
Properties	convex

Alternate names

Bitruncated hexacross
Bitruncated hexacontatetrapeton (Acronym: botag) (Jonathan Bowers)^[2]

Images

orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph	File:6-cube t34.svg	File:6-cube t34 B5.svg	File:6-cube t34 B4.svg
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph	File:6-cube t34 B3.svg	File:6-cube t34 B2.svg
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph	File:6-cube t34 A5.svg	File:6-cube t34 A3.svg
Dihedral symmetry	[6]	[4]

Related polytopes

These polytopes are a part of 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, (x3x3o3o3o4o - tag)
↑ Klitzing, (o3x3x3o3o4o - botag)

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)". x3x3o3o3o4o - tag, o3x3x3o3o4o - botag

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, (x3x3o3o3o4o - tag)

[2] Klitzing, (o3x3x3o3o4o - botag)

[1]

[2]

Truncated 6-orthoplexes

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