Rectified 7-simplexes
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Short description: Convex uniform 7-polytope in seven-dimensional geometry
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 Rectified 7-simplex Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
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 Birectified 7-simplex Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
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 Trirectified 7-simplex Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
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| 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 [math]\displaystyle{ \left\{\begin{array}{l}3, 3, 3, 3, 3\\3\end{array}\right\} }[/math] |
| Coxeter diagrams | Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Or Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
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| 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
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E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S17.
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
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph |
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| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | 
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| 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 [math]\displaystyle{ \left\{\begin{array}{l}3, 3, 3, 3\\3, 3\end{array}\right\} }[/math] |
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| 6-faces | 16: 8 r{35}  8 2r{35} 
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| 5-faces | 112: 28 {34}  56 r{34} 25px 28 2r{34} 
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| 4-faces | 392: 168 {33}  (56+168) r{33} 
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| Cells | 770: (420+70) {3,3}  280 {3,4} 
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| 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 S27. It is also called 04,2 for its branching Coxeter-Dynkin diagram, shown as
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.
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
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph |
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| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph | 
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| 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 [math]\displaystyle{ \left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\} }[/math] |
| Coxeter diagrams | Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Or Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
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| 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 S37.
This polytope is the vertex figure of the 133 honeycomb. It is called 03,3 for its branching Coxeter-Dynkin diagram, shown as
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.
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
Related polytopes
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 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)". https://bendwavy.org/klitzing/dimensions/polyexa.htm. o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he
External links
Fundamental convex regular and uniform polytopes in dimensions 2–10
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|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 4-polytope | 5-cell | 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 | ||||||||||||
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