Rectified 7-simplexes

File:7-simplex t0.svg 7-simplex Template:CDD	File:7-simplex t1.svg Rectified 7-simplex Template:CDD
File:7-simplex t2.svg Birectified 7-simplex Template:CDD	File:7-simplex t3.svg Trirectified 7-simplex Template:CDD
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
Type	uniform 7-polytope
Coxeter symbol	051
Schläfli symbol	r{36} = {35,1} or $\left\{\begin{array}{c}3,3,3,3,3\\ 3\end{array}\right\}$
Coxeter diagrams	Template:CDD or Template:CDD
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 Template:CDD.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S¹
₇.

Alternate names

• Rectified octaexon (Acronym: roc) (Jonathan Bowers)

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	File:7-simplex t1.svg	File:7-simplex t1 A6.svg	File:7-simplex t1 A5.svg
Dihedral symmetry	[8]	[7]	[6]
Ak 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} = {34,2} or $\left\{\begin{array}{c}3,3,3,3\\ 3,3\end{array}\right\}$
Coxeter diagrams	Template:CDD or Template:CDD
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 Template:CDD.

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	File:7-simplex t2.svg	File:7-simplex t2 A6.svg	File:7-simplex t2 A5.svg
Dihedral symmetry	[8]	[7]	[6]
Ak 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} = {33,3} or $\left\{\begin{array}{c}3,3,3\\ 3,3,3\end{array}\right\}$
Coxeter diagrams	Template:CDD or Template:CDD
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 Template:CDD.

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	File:7-simplex t3.svg	File:7-simplex t3 A6.svg	File:7-simplex t3 A5.svg
Dihedral symmetry	[8]	[[7]]	[6]
Ak 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 Template:CDD = Template:CDD t{3} = {6}	Octahedron Template:CDD = Template:CDD r{3,3} = {31,1} = {3,4} $\left\{\begin{array}{c}3\\ 3\end{array}\right\}$	Decachoron Template:CDD 2t{33}	Dodecateron Template:CDD 2r{34} = {32,2} $\left\{\begin{array}{c}3,3\\ 3,3\end{array}\right\}$	Tetradecapeton Template:CDD 3t{35}	Hexadecaexon Template:CDD 3r{36} = {33,3} $\left\{\begin{array}{c}3,3,3\\ 3,3,3\end{array}\right\}$	Octadecazetton Template:CDD 4t{37}
Images	File:Truncated triangle.svg	File:3-cube t2.svgFile:Uniform polyhedron-33-t1.svg	File:4-simplex t12.svg File:Schlegel half-solid bitruncated 5-cell.png	File:5-simplex t2.svg File:5-simplex t2 A4.svg	File:6-simplex t23.svg File:6-simplex t23 A5.svg	File:7-simplex t3.svg File:7-simplex t3 A5.svg	File:8-simplex t34.svg File: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.svg	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 Template:CDD ∩ Template:CDD	File:Stellated octahedron A4 A5 skew.png Template:CDD ∩ Template:CDD	File:Compound dual 5-cells and bitruncated 5-cell intersection A4 coxeter plane.png Template:CDD ∩ Template:CDD	File:Dual 5-simplex intersection graph a5.pngFile:Dual 5-simplex intersection graph a4.png Template:CDD ∩ Template:CDD	Template:CDD ∩ Template:CDD	Template:CDD ∩ Template:CDD	Template:CDD ∩ Template:CDD

Related polytopes

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

See also

• List of A7 polytopes

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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, Template:Isbn
    • (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) with acronyms". o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - broc, o3o3o3x3o3o3o - he Template:Sfn whitelist

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary