Truncated 5-simplexes

Orthogonal projections in A₅ Coxeter plane
File:5-simplex t0.svg 5-simplex 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.png	File:5-simplex t01.svg Truncated 5-simplex 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.png	File:5-simplex t12.svg Bitruncated 5-simplex 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.png

In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.

There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex.

Truncated 5-simplex

Truncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t{3,3,3,3}
Coxeter-Dynkin diagram	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.png File:CDel branch 11.pngFile:CDel 3b.pngFile:CDel nodeb.pngFile:CDel 3b.pngFile:CDel nodeb.pngFile:CDel 3b.pngFile:CDel nodeb.png
4-faces	12	6 {3,3,3}File:Schlegel wireframe 5-cell.png 6 t{3,3,3}File:Schlegel half-solid rectified 5-cell.png
Cells	45	30 {3,3}File:Tetrahedron.png 15 t{3,3}File:Truncated tetrahedron.png
Faces	80	60 {3} 20 {6}
Edges	75
Vertices	30
Vertex figure	File:Truncated 5-simplex verf.png ( )v{3,3}
Coxeter group	A₅ [3,3,3,3], order 720
Properties	convex

The truncated 5-simplex has 30 vertices, 75 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 truncated tetrahedron), and 12 4-faces (6 5-cell and 6 truncated 5-cells).

Alternate names

Truncated hexateron (Acronym: tix) (Jonathan Bowers)^[1]

Coordinates

The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) or of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively.

Images

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph	File:5-simplex t01.svg	File:5-simplex t01 A4.svg
Dihedral symmetry	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph	File:5-simplex t01 A3.svg	File:5-simplex t01 A2.svg
Dihedral symmetry	[4]	[3]

Bitruncated 5-simplex

bitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	2t{3,3,3,3}
Coxeter-Dynkin diagram	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.png File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3b.pngFile:CDel nodeb.png
4-faces	12	6 2t{3,3,3}File:4-simplex t12.svg 6 t{3,3,3}File:4-simplex t01.svg
Cells	60	45 {3,3}File:3-simplex t0.svg 15 t{3,3}File:3-simplex t01.svg
Faces	140	80 {3}File:2-simplex t0.svg 60 {6}File:2-simplex t01.svg
Edges	150
Vertices	60
Vertex figure	File:Bitruncated 5-simplex verf.svg { }v{3}
Coxeter group	A₅ [3,3,3,3], order 720
Properties	convex

Alternate names

Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers)^[2]

Coordinates

The vertices of the bitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,2) or of (0,0,1,2,2,2). These represent positive orthant facets of the bitruncated 6-orthoplex, and the tritruncated 6-cube respectively.

Images

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph	File:5-simplex t12.svg	File:5-simplex t12 A4.svg
Dihedral symmetry	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph	File:5-simplex t12 A3.svg	File:5-simplex t12 A2.svg
Dihedral symmetry	[4]	[3]

Related uniform 5-polytopes

The truncated 5-simplex is one of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A₅ Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

A5 polytopes
File:5-simplex t0.svg t₀	File:5-simplex t1.svg t₁	File:5-simplex t2.svg t₂	File:5-simplex t01.svg t_0,1	File:5-simplex t02.svg t_0,2	File:5-simplex t12.svg t_1,2	File:5-simplex t03.svg t_0,3
File:5-simplex t13.svg t_1,3	File:5-simplex t04.svg t_0,4	File:5-simplex t012.svg t_0,1,2	File:5-simplex t013.svg t_0,1,3	File:5-simplex t023.svg t_0,2,3	File:5-simplex t123.svg t_1,2,3	File:5-simplex t014.svg t_0,1,4
File:5-simplex t024.svg t_0,2,4	File:5-simplex t0123.svg t_0,1,2,3	File:5-simplex t0124.svg t_0,1,2,4	File:5-simplex t0134.svg t_0,1,3,4	File:5-simplex t01234.svg t_0,1,2,3,4

Notes

^ Klitizing, (x3x3o3o3o - tix)
^ Klitizing, (o3x3x3o3o - bittix)

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, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). [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.
Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). x3x3o3o3o - tix, o3x3x3o3o - bittix

External links

Glossary for hyperspace, George Olshevsky.
Polytopes of Various Dimensions, Jonathan Bowers
- Truncated uniform polytera (tix), Jonathan Bowers
Multi-dimensional Glossary

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
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 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 • Polytope operations

[1] Klitizing, (x3x3o3o3o - tix)

[2] Klitizing, (o3x3x3o3o - bittix)

[1]

[2]

Truncated 5-simplexes

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