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 c ...
, 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 the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the
tetrahedral
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
cells of the 7-simplex.
Truncated 7-simplex
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 c ...
, a truncated 7-simplex is a convex
uniform 7-polytope, being a
truncation of the regular
7-simplex.
Alternate names
* Truncated octaexon (Acronym: toc) (Jonathan Bowers)
Coordinates
The vertices of the ''truncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,2). This construction is based on
facets of the
truncated 8-orthoplex.
Images
Bitruncated 7-simplex
Alternate names
* Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers)
Coordinates
The vertices of the ''bitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,2). This construction is based on
facets of the
bitruncated 8-orthoplex
In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.
There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the e ...
.
Images
Tritruncated 7-simplex
Alternate names
* Tritruncated octaexon (acronym: tattoc) (Jonathan Bowers)
[Klitizing, (o3o3x3x3o3o3o - tattoc)]
Coordinates
The vertices of the ''tritruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,2). This construction is based on
facets of the
tritruncated 8-orthoplex
In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.
There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the e ...
.
Images
Related polytopes
These three polytopes are from a set of 71
uniform 7-polytopes with A
7 symmetry.
See also
*
List of A7 polytopes
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.
* x3x3o3o3o3o3o - toc, o3x3x3o3o3o3o - roc, o3o3x3x3o3o3o - tattoc
External links
Polytopes of Various Dimensions
{{Polytopes
7-polytopes