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.
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.
File:Ammann-Beenker tiling example.png The vertices of the A7 2D orthogonal projection are seen in the Ammann–Beenker tiling . Its 56 vertices represent the root vectors of the simple Lie group A7 .
Expanded 7-simplex
Small petated hexadecaexon (Acronym: suph) (Jonathan Bowers) 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
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) Petitruncated octaexon (Acronym: puto) (Jonathan Bowers)[ 2] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
Petirhombated octaexon (Acronym: puro) (Jonathan Bowers)[ 3] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
Petaprismated hexadecaexon (Acronym: puph) (Jonathan Bowers) 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
Petigreatorhombated octaexon (Acronym: pugro) (Jonathan Bowers)[ 5] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
Petiprismatotruncated octaexon (Acronym: pupato) (Jonathan Bowers)[ 6] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
In seven-dimensional geometry , a hexiruncicantellated 7-simplex is a uniform 7-polytope .
Petiprismatorhombated octaexon (Acronym: pupro) (Jonathan Bowers)[ 7] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
Peticellitruncated octaexon (Acronym: pucto) (Jonathan Bowers)[ 8] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
hexistericantellated 7-simplex
Type
uniform 7-polytope
Schläfli symbol t0,2,4,6 {36 }
Coxeter-Dynkin diagrams File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
6-faces
t0,2,4 {3,3,3,3,3} {}xt0,2,4 {3,3,3,3}0,2 {3,3,3}0,2 {3,3}xt0,2 {3,3}
5-faces
4-faces
Cells
Faces
Edges
30240
Vertices
5040
Vertex figure
Coxeter group A7 ×2, [[36 ]], order 80640
Properties
convex
Peticellirhombihexadecaexon (Acronym: pucroh) (Jonathan Bowers)[ 9] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
Petiteritruncated hexadecaexon (Acronym: putath) (Jonathan Bowers)[ 10] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png
Petigreatoprismated octaexon (Acronym: pugopo) (Jonathan Bowers)[ 11] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
Peticelligreatorhombated octaexon (Acronym: pucagro) (Jonathan Bowers)[ 12] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
Peticelliprismatotruncated octaexon (Acronym: pucpato) (Jonathan Bowers)[ 13] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
Peticelliprismatorhombihexadecaexon (Acronym: pucproh) (Jonathan Bowers)[ 14] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
Petiterigreatorhombated octaexon (Acronym: putagro) (Jonathan Bowers)[ 15] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png
Petiteriprismatotruncated hexadecaexon (Acronym: putpath) (Jonathan Bowers)[ 16] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png
Petigreatocellated octaexon (Acronym: pugaco) (Jonathan Bowers)[ 17] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
Petiterigreatoprismated octaexon (Acronym: putgapo) (Jonathan Bowers)[ 18] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png
Petitericelligreatorhombihexadecaexon (Acronym: putcagroh) (Jonathan Bowers)[ 19] 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 , File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png
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.
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 File:CDel node 1.png File:CDel split1.png File:CDel nodes 11.png File:CDel 3ab.png File:CDel nodes 11.png File:CDel 3ab.png File:CDel nodes 11.png File:CDel split2.png File:CDel node 1.png
Great petated hexadecaexon (Acronym: guph) (Jonathan Bowers) 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}, File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png
The 20 polytopes presented in this article are a part of 71 uniform 7-polytopes with A7 symmetry shown in the table below.
A7 polytopes
Error creating thumbnail: t0
File:7-simplex t1.svg t1
File:7-simplex t2.svg t2
File:7-simplex t3.svg t3
File:7-simplex t01.svg t0,1
File:7-simplex t02.svg t0,2
File:7-simplex t12.svg t1,2
File:7-simplex t03.svg t0,3
File:7-simplex t13.svg t1,3
File:7-simplex t23.svg t2,3
File:7-simplex t04.svg t0,4
File:7-simplex t14.svg t1,4
File:7-simplex t24.svg t2,4
File:7-simplex t05.svg t0,5
File:7-simplex t15.svg t1,5
File:7-simplex t06.svg t0,6
File:7-simplex t012.svg t0,1,2
File:7-simplex t013.svg t0,1,3
File:7-simplex t023.svg t0,2,3
File:7-simplex t123.svg t1,2,3
File:7-simplex t014.svg t0,1,4
File:7-simplex t024.svg t0,2,4
File:7-simplex t124.svg t1,2,4
File:7-simplex t034.svg t0,3,4
File:7-simplex t134.svg t1,3,4
File:7-simplex t234.svg t2,3,4
File:7-simplex t015.svg t0,1,5
File:7-simplex t025.svg t0,2,5
File:7-simplex t125.svg t1,2,5
File:7-simplex t035.svg t0,3,5
File:7-simplex t135.svg t1,3,5
File:7-simplex t045.svg t0,4,5
File:7-simplex t016.svg t0,1,6
File:7-simplex t026.svg t0,2,6
File:7-simplex t036.svg t0,3,6
File:7-simplex t0123.svg t0,1,2,3
File:7-simplex t0124.svg t0,1,2,4
File:7-simplex t0134.svg t0,1,3,4
File:7-simplex t0234.svg t0,2,3,4
File:7-simplex t1234.svg t1,2,3,4
File:7-simplex t0125.svg t0,1,2,5
File:7-simplex t0135.svg t0,1,3,5
File:7-simplex t0235.svg t0,2,3,5
File:7-simplex t1235.svg t1,2,3,5
File:7-simplex t0145.svg t0,1,4,5
File:7-simplex t0245.svg t0,2,4,5
File:7-simplex t1245.svg t1,2,4,5
File:7-simplex t0345.svg t0,3,4,5
File:7-simplex t0126.svg t0,1,2,6
File:7-simplex t0136.svg t0,1,3,6
File:7-simplex t0236.svg t0,2,3,6
File:7-simplex t0146.svg t0,1,4,6
File:7-simplex t0246.svg t0,2,4,6
File:7-simplex t0156.svg t0,1,5,6
File:7-simplex t01234.svg t0,1,2,3,4
File:7-simplex t01235.svg t0,1,2,3,5
File:7-simplex t01245.svg t0,1,2,4,5
File:7-simplex t01345.svg t0,1,3,4,5
File:7-simplex t02345.svg t0,2,3,4,5
File:7-simplex t12345.svg t1,2,3,4,5
File:7-simplex t01236.svg t0,1,2,3,6
File:7-simplex t01246.svg t0,1,2,4,6
File:7-simplex t01346.svg t0,1,3,4,6
File:7-simplex t02346.svg t0,2,3,4,6
File:7-simplex t01256.svg t0,1,2,5,6
File:7-simplex t01356.svg t0,1,3,5,6
File:7-simplex t012345.svg t0,1,2,3,4,5
File:7-simplex t012346.svg t0,1,2,3,4,6
File:7-simplex t012356.svg t0,1,2,3,5,6
File:7-simplex t012456.svg t0,1,2,4,5,6
File:7-simplex t0123456.svg t0,1,2,3,4,5,6
^ 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)
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 , Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (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) Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
