In eight-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 c ...

, a cantellated 8-simplex is a convex

uniform 8-polytope In Eight-dimensional space, eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope Ridge (geometry), ridge being shared by exactly two 7-polytope Facet (mathematics), fa ...

, being a

cantellation In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tiling ...

of the regular

8-simplex In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is co ...

. There are six unique cantellations for the 8-simplex, including

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...

s of

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

Cantellated 8-simplex

Alternate names

* Small rhombated enneazetton (acronym: srene) (Jonathan Bowers)

Coordinates

The

Cartesian coordinate A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

s of the vertices of the ''cantellated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,2). 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 cantellated 9-orthoplex.

Images

Bicantellated 8-simplex

Alternate names

* Small birhombated enneazetton (acronym: sabrene) (Jonathan Bowers)

Coordinates

The

s of the vertices of the ''bicantellated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on

of the bicantellated 9-orthoplex.

Images

Tricantellated 8-simplex

Alternate names

* Small trirhombihexadecaexon (acronym: satrene) (Jonathan Bowers)

Coordinates

The

s of the vertices of the ''tricantellated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on

of the tricantellated 9-orthoplex.

Images

Cantitruncated 8-simplex

Alternate names

* Great rhombated enneazetton (acronym: grene) (Jonathan Bowers)

Coordinates

The

s of the vertices of the ''cantitruncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,3). This construction is based on

of the bicantitruncated 9-orthoplex.

Images

Bicantitruncated 8-simplex

Alternate names

* Great birhombated enneazetton (acronym: gabrene) (Jonathan Bowers)

Coordinates

The

s of the vertices of the ''bicantitruncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,3,3). This construction is based on

of the bicantitruncated 9-orthoplex.

Images

Tricantitruncated 8-simplex

* Great trirhombated enneazetton (acronym: gatrene) (Jonathan Bowers)Klitizing, (o3o3x3x3x3o3o3o - gatrene)

Coordinates

The

s of the vertices of the ''tricantitruncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,3,3,3). This construction is based on

of the bicantitruncated 9-orthoplex.

Images

Related polytopes

This polytope is one of 135 uniform 8-polytopes with A₈ symmetry.

Notes

References

H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

: ** 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,

*** (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'', Ph.D. * x3o3x3o3o3o3o3o - srene, o3x3o3x3o3o3o3o - sabrene, o3o3x3o3x3o3o3o - satrene, x3x3x3o3o3o3o3o - grene, o3x3x3x3o3o3o3o - gabrene, o3o3x3x3x3o3o3o - gatrene

External links

Polytopes of Various Dimensions

{{Polytopes 8-polytopes