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 geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.
A uniform 8-polytope is one which is vertex-transitive ...
, 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 c ...
.
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 p ...
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 i ...
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 of the
cantellated 9-orthoplex.
Images
Bicantellated 8-simplex
Alternate names
* Small birhombated enneazetton (acronym: sabrene) (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 i ...
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
facets of the
bicantellated 9-orthoplex.
Images
Tricantellated 8-simplex
Alternate names
* Small trirhombihexadecaexon (acronym: satrene) (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 i ...
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
facets of the
tricantellated 9-orthoplex.
Images
Cantitruncated 8-simplex
Alternate names
* Great rhombated enneazetton (acronym: grene) (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 i ...
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
facets of the
bicantitruncated 9-orthoplex.
Images
Bicantitruncated 8-simplex
Alternate names
* Great birhombated enneazetton (acronym: gabrene) (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 i ...
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
facets of the
bicantitruncated 9-orthoplex.
Images
Tricantitruncated 8-simplex
* Great trirhombated enneazetton (acronym: gatrene) (Jonathan Bowers)
[Klitizing, (o3o3x3x3x3o3o3o - gatrene)]
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 i ...
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
facets of the
bicantitruncated 9-orthoplex.
Images
Related polytopes
This polytope is one of 135
uniform 8-polytopes with A
8 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,
*** (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