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 cantellated 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 ...
, 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
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 unique 6 degrees of cantellation for the 7-simplex, including
truncations.
Cantellated 7-simplex
Alternate names
* Small rhombated octaexon (acronym: saro) (Jonathan Bowers)
Coordinates
The vertices of the ''cantellated 7-simplex'' can be most simply positioned in 8-space as
permutations of (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 8-orthoplex
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-transitiv ...
.
Images
Bicantellated 7-simplex
Alternate names
* Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)
Coordinates
The vertices of the ''bicantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). 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
bicantellated 8-orthoplex.
Images
Tricantellated 7-simplex
Alternate names
* Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)
Coordinates
The vertices of the ''tricantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). 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
tricantellated 8-orthoplex.
Images
Cantitruncated 7-simplex
Alternate names
* Great rhombated octaexon (acronym: garo) (Jonathan Bowers)
Coordinates
The vertices of the ''cantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). 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
cantitruncated 8-orthoplex.
Images
Bicantitruncated 7-simplex
Alternate names
* Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)
Coordinates
The vertices of the ''bicantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). 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
bicantitruncated 8-orthoplex.
Images
Tricantitruncated 7-simplex
Alternate names
* Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)
[Klitizing, (o3o3x3x3x3o3o - gatroh)]
Coordinates
The vertices of the ''tricantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). 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
tricantitruncated 8-orthoplex.
Images
Related polytopes
This polytope is one of 71
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 ...
s with A
7 symmetry.
See also
*
List of A7 polytopes
In 7-dimensional geometry, there are 71 uniform 7-polytope, uniform polytopes with A7 symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices.
Each can be visualized as symmetric orthographic projections in Coxeter planes of ...
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.
* x3o3x3o3o3o3o - saro, o3x3o3x3o3o3o - sabro, o3o3x3o3x3o3o - stiroh, x3x3x3o3o3o3o - garo, o3x3x3x3o3o3o - gabro, o3o3x3x3x3o3o - gatroh
External links
Polytopes of Various Dimensions{{Polytopes
7-polytopes