Rectified 7-simplexes

Orthogonal projections in A₇ Coxeter plane
File:7-simplex t0.svg 7-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.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:7-simplex t1.svg Rectified 7-simplex File:CDel node.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
File:7-simplex t2.svg Birectified 7-simplex File:CDel node.pngFile:CDel 3.pngFile:CDel node.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:7-simplex t3.svg Trirectified 7-simplex 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.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png

In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex. There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex. Vertices of the birectified 7-simplex are located in the triangular face centers of the 7-simplex. Vertices of the trirectified 7-simplex are located in the tetrahedral cell centers of the 7-simplex.

Rectified 7-simplex

Rectified 7-simplex
Type	uniform 7-polytope
Coxeter symbol	0₅₁
Schläfli symbol	r{3⁶} = {3^5,1} or ${\begin{cases} 3, 3, 3, 3, 3 \\ 3 \end{cases}}$
Coxeter diagrams	File:CDel node.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 Or File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png
6-faces	16
5-faces	84
4-faces	224
Cells	350
Faces	336
Edges	168
Vertices	28
Vertex figure	6-simplex prism
Petrie polygon	Octagon
Coxeter group	A₇, [3⁶], order 40320
Properties	convex

The rectified 7-simplex is the edge figure of the 2₅₁ honeycomb. It is called 0_5,1 for its branching Coxeter-Dynkin diagram, shown as File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S¹
₇.

Alternate names

Rectified octaexon (Acronym: roc) (Jonathan Bowers)

Coordinates

The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 8-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₇	A₆	A₅
Graph	File:7-simplex t1.svg	File:7-simplex t1 A6.svg	File:7-simplex t1 A5.svg
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:7-simplex t1 A4.svg	File:7-simplex t1 A3.svg	File:7-simplex t1 A2.svg
Dihedral symmetry	[5]	[4]	[3]

Birectified 7-simplex

Birectified 7-simplex
Type	uniform 7-polytope
Coxeter symbol	0₄₂
Schläfli symbol	2r{3,3,3,3,3,3} = {3^4,2} or ${\begin{cases} 3, 3, 3, 3 \\ 3, 3 \end{cases}}$
Coxeter diagrams	File:CDel node.pngFile:CDel 3.pngFile:CDel node.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 Or File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png
6-faces	16: 8 r{3⁵} File:6-simplex t1.svg 8 2r{3⁵} File:6-simplex t2.svg
5-faces	112: 28 {3⁴} File:5-simplex t0.svg 56 r{3⁴} File:Rectified 5-simplex.png 28 2r{3⁴} File:5-simplex t2.svg
4-faces	392: 168 {3³} File:4-simplex t0.svg (56+168) r{3³} File:5-simplex t1.svg
Cells	770: (420+70) {3,3} File:3-simplex t0.svg 280 {3,4} File:3-simplex t1.svg
Faces	840: (280+560) {3}
Edges	420
Vertices	56
Vertex figure	{3}x{3,3,3}
Coxeter group	A₇, [3⁶], order 40320
Properties	convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S²
₇. It is also called 0_4,2 for its branching Coxeter-Dynkin diagram, shown as File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png.

Alternate names

Birectified octaexon (Acronym: broc) (Jonathan Bowers)

Coordinates

The vertices of the birectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 8-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₇	A₆	A₅
Graph	File:7-simplex t2.svg	File:7-simplex t2 A6.svg	File:7-simplex t2 A5.svg
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:7-simplex t2 A4.svg	File:7-simplex t2 A3.svg	File:7-simplex t2 A2.svg
Dihedral symmetry	[5]	[4]	[3]

Trirectified 7-simplex

Trirectified 7-simplex
Type	uniform 7-polytope
Coxeter symbol	0₃₃
Schläfli symbol	3r{3⁶} = {3^3,3} or ${\begin{cases} 3, 3, 3 \\ 3, 3, 3 \end{cases}}$
Coxeter diagrams	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.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png Or File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.png
6-faces	16 2r{3⁵}
5-faces	112
4-faces	448
Cells	980
Faces	1120
Edges	560
Vertices	70
Vertex figure	{3,3}x{3,3}
Coxeter group	A₇×2, [[3⁶]], order 80640
Properties	convex, isotopic

The trirectified 7-simplex is the intersection of two regular 7-simplexes in dual configuration. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S³
₇. This polytope is the vertex figure of the 1₃₃ honeycomb. It is called 0_3,3 for its branching Coxeter-Dynkin diagram, shown as File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.png.

Alternate names

Hexadecaexon (Acronym: he) (Jonathan Bowers)

Coordinates

The vertices of the trirectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 8-orthoplex. The trirectified 7-simplex is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space: the 70 distinct permutations of (1,1,1,1,−1,−1,−1,-1).

Images

orthographic projections
A_k Coxeter plane	A₇	A₆	A₅
Graph	File:7-simplex t3.svg	File:7-simplex t3 A6.svg	File:7-simplex t3 A5.svg
Dihedral symmetry	[8]	[[7]]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:7-simplex t3 A4.svg	File:7-simplex t3 A3.svg	File:7-simplex t3 A2.svg
Dihedral symmetry	[[5]]	[4]	[[3]]

Related polytopes

Isotopic uniform truncated simplices
Dim.	2	3	4	5	6	7	8
Name Coxeter	Hexagon File:CDel branch 11.png = File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.png t{3} = {6}	Octahedron File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png r{3,3} = {3^1,1} = {3,4} ${\begin{cases} 3 \\ 3 \end{cases}}$	Decachoron File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.png 2t{3³}	Dodecateron File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.png 2r{3⁴} = {3^2,2} ${\begin{cases} 3, 3 \\ 3, 3 \end{cases}}$	Tetradecapeton File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.png 3t{3⁵}	Hexadecaexon File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.png 3r{3⁶} = {3^3,3} ${\begin{cases} 3, 3, 3 \\ 3, 3, 3 \end{cases}}$	Octadecazetton File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.png 4t{3⁷}
Images	File:Truncated triangle.svg	File:3-cube t2.svgFile:Uniform polyhedron-33-t1.svg	File:4-simplex t12.svgFile:Schlegel half-solid bitruncated 5-cell.png	File:5-simplex t2.svgFile:5-simplex t2 A4.svg	File:6-simplex t23.svgFile:6-simplex t23 A5.svg	File:7-simplex t3.svgFile:7-simplex t3 A5.svg	File:8-simplex t34.svgFile:8-simplex t34 A7.svg
Vertex figure	( )∨( )	File:Octahedron vertfig.svg { }×{ }	File:Bitruncated 5-cell verf.png { }∨{ }	File:Birectified hexateron verf.png {3}×{3}	File:Tritruncated 6-simplex verf.png {3}∨{3}	{3,3}×{3,3}	File:Quadritruncated 8-simplex verf.png {3,3}∨{3,3}
Facets		{3} File:Regular polygon 3 annotated.svg	t{3,3} File:Uniform polyhedron-33-t01.png	r{3,3,3} File:Schlegel half-solid rectified 5-cell.png	2t{3,3,3,3} File:5-simplex t12.svg	2r{3,3,3,3,3} File:6-simplex t2.svg	3t{3,3,3,3,3,3} File:7-simplex t23.svg
As intersecting dual simplexes	File:Regular hexagon as intersection of two triangles.png File:CDel branch 10.png ∩ File:CDel branch 01.png	File:Stellated octahedron A4 A5 skew.png File:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png ∩ File:CDel node.pngFile:CDel split1.pngFile:CDel nodes 01ld.png	File:Compound dual 5-cells and bitruncated 5-cell intersection A4 coxeter plane.png File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png ∩ File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes 01l.png	File:Dual 5-simplex intersection graph a5.pngFile:Dual 5-simplex intersection graph a4.png File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png ∩ File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 01l.png	File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png ∩ File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 01l.png	File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png ∩ File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 01l.png	File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png ∩ File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 01l.png

Related polytopes

These polytopes are three of 71 uniform 7-polytopes with A₇ symmetry.

A7 polytopes
File:7-simplex t0.svg t₀	File:7-simplex t1.svg t₁	File:7-simplex t2.svg t₂	File:7-simplex t3.svg t₃	File:7-simplex t01.svg t_0,1	File:7-simplex t02.svg t_0,2	File:7-simplex t12.svg t_1,2	File:7-simplex t03.svg t_0,3
File:7-simplex t13.svg t_1,3	File:7-simplex t23.svg t_2,3	File:7-simplex t04.svg t_0,4	File:7-simplex t14.svg t_1,4	File:7-simplex t24.svg t_2,4	File:7-simplex t05.svg t_0,5	File:7-simplex t15.svg t_1,5	File:7-simplex t06.svg t_0,6
File:7-simplex t012.svg t_0,1,2	File:7-simplex t013.svg t_0,1,3	File:7-simplex t023.svg t_0,2,3	File:7-simplex t123.svg t_1,2,3	File:7-simplex t014.svg t_0,1,4	File:7-simplex t024.svg t_0,2,4	File:7-simplex t124.svg t_1,2,4	File:7-simplex t034.svg t_0,3,4
File:7-simplex t134.svg t_1,3,4	File:7-simplex t234.svg t_2,3,4	File:7-simplex t015.svg t_0,1,5	File:7-simplex t025.svg t_0,2,5	File:7-simplex t125.svg t_1,2,5	File:7-simplex t035.svg t_0,3,5	File:7-simplex t135.svg t_1,3,5	File:7-simplex t045.svg t_0,4,5
File:7-simplex t016.svg t_0,1,6	File:7-simplex t026.svg t_0,2,6	File:7-simplex t036.svg t_0,3,6	File:7-simplex t0123.svg t_0,1,2,3	File:7-simplex t0124.svg t_0,1,2,4	File:7-simplex t0134.svg t_0,1,3,4	File:7-simplex t0234.svg t_0,2,3,4	File:7-simplex t1234.svg t_1,2,3,4
File:7-simplex t0125.svg t_0,1,2,5	File:7-simplex t0135.svg t_0,1,3,5	File:7-simplex t0235.svg t_0,2,3,5	File:7-simplex t1235.svg t_1,2,3,5	File:7-simplex t0145.svg t_0,1,4,5	File:7-simplex t0245.svg t_0,2,4,5	File:7-simplex t1245.svg t_1,2,4,5	File:7-simplex t0345.svg t_0,3,4,5
File:7-simplex t0126.svg t_0,1,2,6	File:7-simplex t0136.svg t_0,1,3,6	File:7-simplex t0236.svg t_0,2,3,6	File:7-simplex t0146.svg t_0,1,4,6	File:7-simplex t0246.svg t_0,2,4,6	File:7-simplex t0156.svg t_0,1,5,6	File:7-simplex t01234.svg t_0,1,2,3,4	File:7-simplex t01235.svg t_0,1,2,3,5
File:7-simplex t01245.svg t_0,1,2,4,5	File:7-simplex t01345.svg t_0,1,3,4,5	File:7-simplex t02345.svg t_0,2,3,4,5	File:7-simplex t12345.svg t_1,2,3,4,5	File:7-simplex t01236.svg t_0,1,2,3,6	File:7-simplex t01246.svg t_0,1,2,4,6	File:7-simplex t01346.svg t_0,1,3,4,6	File:7-simplex t02346.svg t_0,2,3,4,6
File:7-simplex t01256.svg t_0,1,2,5,6	File:7-simplex t01356.svg t_0,1,3,5,6	File:7-simplex t012345.svg t_0,1,2,3,4,5	File:7-simplex t012346.svg t_0,1,2,3,4,6	File:7-simplex t012356.svg t_0,1,2,3,5,6	File:7-simplex t012456.svg t_0,1,2,4,5,6	File:7-simplex t0123456.svg t_{0,1,2,3,4,5,6}

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. "7D uniform polytopes (polyexa)". o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he

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

Rectified 7-simplexes

Contents