In seven-dimensional

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, a hexicated 7-simplex is a convex

uniform 7-polytope In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose ...

, including 6th-order truncations (hexication) from the regular

7-simplex In 7-dimensional geometry, a 7- simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/ ...

. 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 Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansio ...

operation applied to the regular

. 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

, a hexicated 7-simplex is a convex

, a hexication (6th order truncation) of the regular

, or alternately can be seen as an

operation. Ammann-Beenker_tiling_example

Root vectors

Its 56 vertices represent the root vectors of the

simple Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...

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 A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...

of the hexicated 8-orthoplex, . A second construction in 8-space, from the center of a

rectified 8-orthoplex In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex. There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being th ...

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

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

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

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

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

of the hexiruncitruncated 8-orthoplex, .

Images

Hexiruncicantellated 7-simplex

In seven-dimensional

, a hexiruncicantellated 7-simplex is a

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

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

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

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

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

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

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

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

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

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

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

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

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

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 In mathematics, the permutohedron of order ''n'' is an (''n'' − 1)-dimensional polytope embedded in an ''n''-dimensional space. Its vertex coordinates (labels) are the permutations of the first ''n'' natural numbers. The edges ident ...

of order 8. The omnitruncated 7-simplex is a

zonotope In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments i ...

, the

Minkowski sum In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski ...

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 A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...

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)Klitzing, (x3x3x3x3x3x3x - guph)

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

of the hexipentisteriruncicantitruncated 8-orthoplex, t_{0,1,2,3,4,5,6}, .

Images

Related polytopes

These polytope are a part of 71

s 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,
wiley.com
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. 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

External links

Polytopes of Various Dimensions

{{Polytopes 7-polytopes