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Hexicated 7-simplexes
Type of 7-polytope From Wikipedia, the free encyclopedia
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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.
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Hexicated 7-simplex
Summarize
Perspective
| Hexicated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,6{36} |
| Coxeter-Dynkin diagrams |    |
| 6-faces | 254: 8+8 {35}  28+28 {}x{34} 56+56 {3}x{3,3,3} 70 {3,3}x{3,3} |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 336 |
| Vertices | 56 |
| Vertex figure | 5-simplex antiprism |
| Coxeter group | A7×2, [[36]], order 80640 |
| Properties | convex |
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)[1]
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
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Hexitruncated 7-simplex
Alternate names
- Petitruncated octaexon (Acronym: puto) (Jonathan Bowers)[2]
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
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Hexicantellated 7-simplex
Alternate names
- Petirhombated octaexon (Acronym: puro) (Jonathan Bowers)[3]
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
Alternate names
- Petaprismated hexadecaexon (Acronym: puph) (Jonathan Bowers)[4]
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
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Hexicantitruncated 7-simplex
Alternate names
- Petigreatorhombated octaexon (Acronym: pugro) (Jonathan Bowers)[5]
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
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Hexiruncitruncated 7-simplex
Alternate names
- Petiprismatotruncated octaexon (Acronym: pupato) (Jonathan Bowers)[6]
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
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Hexiruncicantellated 7-simplex
In seven-dimensional geometry, a hexiruncicantellated 7-simplex is a uniform 7-polytope.
Alternate names
- Petiprismatorhombated octaexon (Acronym: pupro) (Jonathan Bowers)[7]
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
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Hexisteritruncated 7-simplex
Alternate names
- Peticellitruncated octaexon (Acronym: pucto) (Jonathan Bowers)[8]
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
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Hexistericantellated 7-simplex
Alternate names
- Peticellirhombihexadecaexon (Acronym: pucroh) (Jonathan Bowers)[9]
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
Alternate names
- Petiteritruncated hexadecaexon (Acronym: putath) (Jonathan Bowers)[10]
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
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Hexiruncicantitruncated 7-simplex
Alternate names
- Petigreatoprismated octaexon (Acronym: pugopo) (Jonathan Bowers)[11]
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
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Hexistericantitruncated 7-simplex
Alternate names
- Peticelligreatorhombated octaexon (Acronym: pucagro) (Jonathan Bowers)[12]
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
Alternate names
- Peticelliprismatotruncated octaexon (Acronym: pucpato) (Jonathan Bowers)[13]
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
Alternate names
- Peticelliprismatorhombihexadecaexon (Acronym: pucproh) (Jonathan Bowers)[14]
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
Alternate names
- Petiterigreatorhombated octaexon (Acronym: putagro) (Jonathan Bowers)[15]
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
Alternate names
- Petiteriprismatotruncated hexadecaexon (Acronym: putpath) (Jonathan Bowers)[16]
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
Alternate names
- Petigreatocellated octaexon (Acronym: pugaco) (Jonathan Bowers)[17]
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
Alternate names
- Petiterigreatoprismated octaexon (Acronym: putgapo) (Jonathan Bowers)[18]
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
Alternate names
- Petitericelligreatorhombihexadecaexon (Acronym: putcagroh) (Jonathan Bowers)[19]
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
Summarize
Perspective
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)[20]
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
Related polytopes
The 20 polytopes presented in this article are a part of 71 uniform 7-polytopes with A7 symmetry shown in the table below.
Notes
References
External links
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