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Hexicated 7-simplexes
Type of 7-polytope
Type of 7-polytope
In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.
There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations.
The simple hexicated 7-simplex is also called an expanded 7-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-simplex. The highest form, the hexipentisteriruncicantitruncated 7-simplex is more simply called an omnitruncated 7-simplex with all of the nodes ringed.
| Orthogonal projections in A7 Coxeter plane |
|---|
Hexicated 7-simplex
| Hexicated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-simplex, or alternately can be seen as an expansion operation.
Root vectors
Its 56 vertices represent the root vectors of the simple Lie group A7.
Alternate names
- Expanded 7-simplex
- Small petated hexadecaexon (Acronym: suph) (Jonathan Bowers)
Coordinates
The vertices of the hexicated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,1,2). This construction is based on facets of the hexicated 8-orthoplex, .
A second construction in 8-space, from the center of a rectified 8-orthoplex is given by coordinate permutations of: : (1,-1,0,0,0,0,0,0)
Images
Hexitruncated 7-simplex
| hexitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Petitruncated octaexon (Acronym: puto) (Jonathan Bowers)
Coordinates
The vertices of the hexitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,2,3). This construction is based on facets of the hexitruncated 8-orthoplex, .
Images
Hexicantellated 7-simplex
| Hexicantellated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Petirhombated octaexon (Acronym: puro) (Jonathan Bowers)
Coordinates
The vertices of the hexicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,2,3). This construction is based on facets of the hexicantellated 8-orthoplex, .
Images
Hexiruncinated 7-simplex
| Hexiruncinated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Petaprismated hexadecaexon (Acronym: puph) (Jonathan Bowers)
Coordinates
The vertices of the hexiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, .
Images
Hexicantitruncated 7-simplex
| Hexicantitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Petigreatorhombated octaexon (Acronym: pugro) (Jonathan Bowers)
Coordinates
The vertices of the hexicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,3,4). This construction is based on facets of the hexicantitruncated 8-orthoplex, .
Images
Hexiruncitruncated 7-simplex
| Hexiruncitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Petiprismatotruncated octaexon (Acronym: pupato) (Jonathan Bowers)
Coordinates
The vertices of the hexiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,3,4). This construction is based on facets of the hexiruncitruncated 8-orthoplex, .
Images
Hexiruncicantellated 7-simplex
| Hexiruncicantellated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
In seven-dimensional geometry, a hexiruncicantellated 7-simplex is a uniform 7-polytope.
Alternate names
- Petiprismatorhombated octaexon (Acronym: pupro) (Jonathan Bowers)
Coordinates
The vertices of the hexiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,3,3,4). This construction is based on facets of the hexiruncicantellated 8-orthoplex, .
Images
Hexisteritruncated 7-simplex
| hexisteritruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Peticellitruncated octaexon (Acronym: pucto) (Jonathan Bowers)
Coordinates
The vertices of the hexisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,2,3,4). This construction is based on facets of the hexisteritruncated 8-orthoplex, .
Images
Hexistericantellated 7-simplex
| hexistericantellated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Peticellirhombihexadecaexon (Acronym: pucroh) (Jonathan Bowers)
Coordinates
The vertices of the hexistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,3,4). This construction is based on facets of the hexistericantellated 8-orthoplex, .
Images
Hexipentitruncated 7-simplex
| Hexipentitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Petiteritruncated hexadecaexon (Acronym: putath) (Jonathan Bowers)
Coordinates
The vertices of the hexipentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,2,3,4). This construction is based on facets of the hexipentitruncated 8-orthoplex, .
Images
Hexiruncicantitruncated 7-simplex
| Hexiruncicantitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Petigreatoprismated octaexon (Acronym: pugopo) (Jonathan Bowers)
Coordinates
The vertices of the hexiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexiruncicantitruncated 8-orthoplex, .
Images
Hexistericantitruncated 7-simplex
| Hexistericantitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Peticelligreatorhombated octaexon (Acronym: pucagro) (Jonathan Bowers)
Coordinates
The vertices of the hexistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexistericantitruncated 8-orthoplex, .
Images
Hexisteriruncitruncated 7-simplex
| Hexisteriruncitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Peticelliprismatotruncated octaexon (Acronym: pucpato) (Jonathan Bowers)
Coordinates
The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,3,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .
Images
Hexisteriruncicantellated 7-simplex
| Hexisteriruncicantellated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Peticelliprismatorhombihexadecaexon (Acronym: pucproh) (Jonathan Bowers)
Coordinates
The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .
Images
Hexipenticantitruncated 7-simplex
| hexipenticantitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Petiterigreatorhombated octaexon (Acronym: putagro) (Jonathan Bowers)
Coordinates
The vertices of the hexipenticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,3,4,5). This construction is based on facets of the hexipenticantitruncated 8-orthoplex, .
Images
Hexipentiruncitruncated 7-simplex
| Hexipentiruncitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Petiteriprismatotruncated hexadecaexon (Acronym: putpath) (Jonathan Bowers)
Coordinates
The vertices of the hexipentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,4,5). This construction is based on facets of the hexipentiruncitruncated 8-orthoplex, .
Images
Hexisteriruncicantitruncated 7-simplex
| Hexisteriruncicantitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Petigreatocellated octaexon (Acronym: pugaco) (Jonathan Bowers)
Coordinates
The vertices of the hexisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,5,6). This construction is based on facets of the hexisteriruncicantitruncated 8-orthoplex, .
Images
Hexipentiruncicantitruncated 7-simplex
| Hexipentiruncicantitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Petiterigreatoprismated octaexon (Acronym: putgapo) (Jonathan Bowers)
Coordinates
The vertices of the hexipentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,5,6). This construction is based on facets of the hexipentiruncicantitruncated 8-orthoplex, .
Images
Hexipentistericantitruncated 7-simplex
| Hexipentistericantitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Petitericelligreatorhombihexadecaexon (Acronym: putcagroh) (Jonathan Bowers)
Coordinates
The vertices of the hexipentistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,3,4,5,6). This construction is based on facets of the hexipentistericantitruncated 8-orthoplex, .
Images
Omnitruncated 7-simplex
| Omnitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
The omnitruncated 7-simplex is composed of 40320 (8 factorial) vertices and is the largest uniform 7-polytope in the A7 symmetry of the regular 7-simplex. It can also be called the hexipentisteriruncicantitruncated 7-simplex which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active.
Permutohedron and related tessellation
The omnitruncated 7-simplex is the permutohedron of order 8. The omnitruncated 7-simplex is a zonotope, the Minkowski sum of eight line segments parallel to the eight lines through the origin and the eight vertices of the 7-simplex.
Like all uniform omnitruncated n-simplices, the omnitruncated 7-simplex can tessellate space by itself, in this case 7-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of .
Alternate names
- Great petated hexadecaexon (Acronym: guph) (Jonathan Bowers)
Coordinates
The vertices of the omnitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,4,5,6,7). This construction is based on facets of the hexipentisteriruncicantitruncated 8-orthoplex, t0,1,2,3,4,5,6{36,4}, .
Images
Notes
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, wiley.com,
- (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, PhD (1966)
- x3o3o3o3o3o3x - suph, x3x3o3o3o3o3x - puto, x3o3x3o3o3o3x - puro, x3o3o3x3o3o3x - puph, x3o3o3o3x3o3x - pugro, x3x3x3o3o3o3x - pupato, x3o3x3x3o3o3x - pupro, x3x3o3o3x3o3x - pucto, x3o3x3o3x3o3x - pucroh, x3x3o3o3o3x3x - putath, x3x3x3x3o3o3x - pugopo, x3x3x3o3x3o3x - pucagro, x3x3o3x3x3o3x - pucpato, x3o3x3x3x3o3x - pucproh, x3x3x3o3o3x3x - putagro, x3x3x3x3o3x3x - putpath, x3x3x3x3x3o3x - pugaco, x3x3x3x3o3x3x - putgapo, x3x3x3o3x3x3x - putcagroh, x3x3x3x3x3x3x - guph
References
- Klitzing, (x3x3o3o3o3o3x- puto)
- Klitzing, (x3o3x3o3o3o3x - puro)
- Klitzing, (x3o3o3o3x3o3x - pugro)
- Klitzing, (x3x3x3o3o3o3x - pupato)
- Klitzing, (x3o3x3x3o3o3x - pupro)
- Klitzing, (x3x3o3o3x3o3x - pucto)
- Klitzing, (x3o3x3o3x3o3x - pucroh)
- Klitzing, (x3x3o3o3o3x3x - putath)
- Klitzing, (x3x3x3x3o3o3x - pugopo)
- Klitzing, (x3x3x3o3x3o3x - pucagro)
- Klitzing, (x3x3o3x3x3o3x - pucpato)
- Klitzing, (x3o3x3x3x3o3x - pucproh)
- Klitzing, (x3x3x3o3o3x3x - putagro)
- Klitzing, (x3x3o3x3o3x3x - putpath)
- Klitzing, (x3x3x3x3x3o3x - pugaco)
- Klitzing, (x3x3x3x3o3x3x - putgapo)
- Klitzing, (x3x3x3o3x3x3x - putcagroh)
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