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
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
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(1/7) ...
.
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 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
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
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(1/7) ...
.
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
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
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.
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.
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
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
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