Rectified 7-simplexes

Orthogonal projections in A7 Coxeter plane
7-simplex	Rectified 7-simplex
Birectified 7-simplex	Trirectified 7-simplex

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} = {35,1} or { 3 , 3 , 3 , 3 , 3 3 } {\displaystyle \left\{{\begin{array}{l}3,3,3,3,3\\3\end{array}}\right\}}
Coxeter diagrams	or
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 05,1 for its branching Coxeter-Dynkin diagram, shown as .

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
Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
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} = {34,2} or { 3 , 3 , 3 , 3 3 , 3 } {\displaystyle \left\{{\begin{array}{l}3,3,3,3\\3,3\end{array}}\right\}}
Coxeter diagrams	or
6-faces	16: 8 r{35} 8 2r{35}
5-faces	112: 28 {34} 56 r{34} 28 2r{34}
4-faces	392: 168 {33} (56+168) r{33}
Cells	770: (420+70) {3,3} 280 {3,4}
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 04,2 for its branching Coxeter-Dynkin diagram, shown as .

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
Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Trirectified 7-simplex

Trirectified 7-simplex
Type	uniform 7-polytope
Coxeter symbol	033
Schläfli symbol	3r{36} = {33,3} or { 3 , 3 , 3 3 , 3 , 3 } {\displaystyle \left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}}
Coxeter diagrams	or
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 03,3 for its branching Coxeter-Dynkin diagram, shown as .

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
Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[[7]]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Related polytopes

Isotopic uniform truncated simplices
Dim.	2	3	4	5	6	7	8
Name Coxeter	Hexagon = t{3} = {6}	Octahedron = r{3,3} = {31,1} = {3,4} { 3 3 } {\displaystyle \left\{{\begin{array}{l}3\\3\end{array}}\right\}}	Decachoron 2t{33}	Dodecateron 2r{34} = {32,2} { 3 , 3 3 , 3 } {\displaystyle \left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}}	Tetradecapeton 3t{35}	Hexadecaexon 3r{36} = {33,3} { 3 , 3 , 3 3 , 3 , 3 } {\displaystyle \left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}}	Octadecazetton 4t{37}
Images
Vertex figure	( )∨( )	{ }×{ }	{ }∨{ }	{3}×{3}	{3}∨{3}	{3,3}×{3,3}	{3,3}∨{3,3}
Facets		{3}	t{3,3}	r{3,3,3}	2t{3,3,3,3}	2r{3,3,3,3,3}	3t{3,3,3,3,3,3}
As intersecting dual simplexes	∩	∩	∩	∩	∩	∩	∩

Related polytopes

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

A7 polytopes
t0	t1	t2	t3	t0,1	t0,2	t1,2	t0,3
t1,3	t2,3	t0,4	t1,4	t2,4	t0,5	t1,5	t0,6
t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4	t0,2,4	t1,2,4	t0,3,4
t1,3,4	t2,3,4	t0,1,5	t0,2,5	t1,2,5	t0,3,5	t1,3,5	t0,4,5
t0,1,6	t0,2,6	t0,3,6	t0,1,2,3	t0,1,2,4	t0,1,3,4	t0,2,3,4	t1,2,3,4
t0,1,2,5	t0,1,3,5	t0,2,3,5	t1,2,3,5	t0,1,4,5	t0,2,4,5	t1,2,4,5	t0,3,4,5
t0,1,2,6	t0,1,3,6	t0,2,3,6	t0,1,4,6	t0,2,4,6	t0,1,5,6	t0,1,2,3,4	t0,1,2,3,5
t0,1,2,4,5	t0,1,3,4,5	t0,2,3,4,5	t1,2,3,4,5	t0,1,2,3,6	t0,1,2,4,6	t0,1,3,4,6	t0,2,3,4,6
t0,1,2,5,6	t0,1,3,5,6	t0,1,2,3,4,5	t0,1,2,3,4,6	t0,1,2,3,5,6	t0,1,2,4,5,6	t0,1,2,3,4,5,6

Notes

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, , 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. . o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - broc, o3o3o3x3o3o3o - he

External links

vteFundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

Rectified 7-simplexes