Rectified 7-simplexes


7-simplex

Rectified 7-simplex

Birectified 7-simplex

Trirectified 7-simplex
Orthogonal projections in A7 Coxeter plane

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
Typeuniform 7-polytope
Coxeter symbol 051
Schläfli symbol r{36} = {35,1}
or
Coxeter diagrams
Or
6-faces16
5-faces84
4-faces224
Cells350
Faces336
Edges168
Vertices28
Vertex figure6-simplex prism
Petrie polygonOctagon
Coxeter groupA7, [36], order 40320
Propertiesconvex

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

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
Typeuniform 7-polytope
Coxeter symbol 042
Schläfli symbol 2r{3,3,3,3,3,3} = {34,2}
or
Coxeter diagrams
Or
6-faces16:
8 r{35}
8 2r{35}
5-faces112:
28 {34}
56 r{34}
28 2r{34}
4-faces392:
168 {33}
(56+168) r{33}
Cells770:
(420+70) {3,3}
280 {3,4}
Faces840:
(280+560) {3}
Edges420
Vertices56
Vertex figure{3}x{3,3,3}
Coxeter groupA7, [36], order 40320
Propertiesconvex

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

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
Typeuniform 7-polytope
Coxeter symbol 033
Schläfli symbol 3r{36} = {33,3}
or
Coxeter diagrams
Or
6-faces16 2r{35}
5-faces112
4-faces448
Cells980
Faces1120
Edges560
Vertices70
Vertex figure{3,3}x{3,3}
Coxeter groupA7×2, [[36]], order 80640
Propertiesconvex, 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

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}
Decachoron

2t{33}
Dodecateron

2r{34} = {32,2}
Tetradecapeton

3t{35}
Hexadecaexon

3r{36} = {33,3}
Octadecazetton

4t{37}
Images
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.

See also

References

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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