In eight-dimensional
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a truncated 8-simplex is a convex
uniform 8-polytope, being a
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 ...
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 angl ...
.
There are four unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex are located on the triangular faces of the 8-simplex. Vertices of the tritruncated 8-simplex are located inside the
tetrahedral
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
cells of the 8-simplex.
Truncated 8-simplex
Alternate names
* Truncated enneazetton (Acronym: tene) (Jonathan Bowers)
Coordinates
The
Cartesian coordinate
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
s of the vertices of the ''truncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,2). This construction is based on
facets of the
truncated 9-orthoplex
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 ...
.
Images
Bitruncated 8-simplex
Alternate names
* Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers)
Coordinates
The
Cartesian coordinate
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
s of the vertices of the ''bitruncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,2). This construction is based on
facets of the
bitruncated 9-orthoplex
In geometry, a bitruncation is an operation on regular polytopes. The original edges are lost completely and the original faces remain as smaller copies of themselves.
Bitruncated regular polytopes can be represented by an extended Schläfli s ...
.
Images
Tritruncated 8-simplex
Alternate names
* Tritruncated enneazetton (Acronym: tatene) (Jonathan Bowers)
Coordinates
The
Cartesian coordinate
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
s of the vertices of the ''tritruncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,2,2). This construction is based on
facets of the
tritruncated 9-orthoplex.
Images
Quadritruncated 8-simplex
The quadritruncated 8-simplex an
isotopic polytope, constructed from 18
tritruncated 7-simplex
In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.
There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of t ...
facets.
Alternate names
* Octadecazetton (18-facetted 8-polytope) (Acronym: be) (Jonathan Bowers)
[Klitizing, (o3o3o3x3x3o3o3o - be)]
Coordinates
The
Cartesian coordinate
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
s of the vertices of the ''quadritruncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,2,2,2). This construction is based on
facets of the
quadritruncated 9-orthoplex.
Images
Related polytopes
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.
* x3x3o3o3o3o3o3o - tene, o3x3x3o3o3o3o3o - batene, o3o3x3x3o3o3o3o - tatene, o3o3o3x3x3o3o3o - be
External links
Polytopes of Various Dimensions
{{Polytopes
8-polytopes