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 runcinated 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 ...
with 3rd order
truncations (
runcination
In geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges, and vertices, creating new facets in place of the original face, edge, and vertex centers.
It is a higher order truncatio ...
) 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 eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The ''triruncinated 8-simplex'' and ''triruncicantitruncated 8-simplex'' have a doubled symmetry, showing
8order reflectional symmetry in the A
8 Coxeter plane
In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
Definitions
Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...
.
Runcinated 8-simplex
Alternate names
* Runcinated enneazetton
* Small prismated enneazetton (Acronym: spene) (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 ''runcinated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,2). 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
runcinated 9-orthoplex.
Images
Biruncinated 8-simplex
Alternate names
* Biruncinated enneazetton
* Small biprismated enneazetton (Acronym: sabpene) (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 ''biruncinated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,1,1,1,2,2). 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
biruncinated 9-orthoplex.
Images
Triruncinated 8-simplex
Alternate names
* Triruncinated enneazetton
* Small triprismated enneazetton (Acronym: satpeb) (Jonathan Bowers)
[Klitzing (o3o3x3o3o3x3o3o - satpeb)]
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 ''triruncinated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,1,1,1,2,2,2). 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
triruncinated 9-orthoplex.
Images
Runcitruncated 8-simplex
Images
Biruncitruncated 8-simplex
Images
Triruncitruncated 8-simplex
Images
Runcicantellated 8-simplex
Images
Biruncicantellated 8-simplex
Images
Runcicantitruncated 8-simplex
Images
Biruncicantitruncated 8-simplex
Images
Triruncicantitruncated 8-simplex
Images
Related polytopes
This polytope is one of 135
uniform 8-polytopes with A
8 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.
* x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb
External links
Polytopes of Various Dimensions
{{Polytopes
8-polytopes