- Source: Hexicated 7-simplexes
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 a omnitruncated 7-simplex with all of the nodes ringed.
Hexicated 7-simplex
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
= 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
= 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
= Alternate names
=Petiprismated 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
= 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
= 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
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
= 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
= 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
= 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
= 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
= 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
= 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
= 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
= 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
= 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
= 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
= 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
= 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
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
=Related polytopes
These polytope are a part of 71 uniform 7-polytopes with A7 symmetry.
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, ISBN 978-0-471-01003-6, 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)
Klitzing, Richard. "7D". 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
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
