In six-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 6-simplex is a convex
uniform 6-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
6-simplex.
There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the triangular faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the
tetrahedral cells of the 6-simplex.
Truncated 6-simplex
Alternate names
* Truncated heptapeton (Acronym: til) (Jonathan Bowers)
Coordinates
The vertices of the ''truncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,2). This construction is based on
facets of the
truncated 7-orthoplex.
Images
Bitruncated 6-simplex
Alternate names
* Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers)
Coordinates
The vertices of the ''bitruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,2). This construction is based on
facets of the
bitruncated 7-orthoplex.
Images
Tritruncated 6-simplex
The tritruncated 6-simplex is an
isotopic uniform polytope, with 14 identical
bitruncated 5-simplex facets.
The tritruncated 6-simplex is the
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of two
6-simplexes in dual configuration: and .
Alternate names
* Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers)
[Klitzing, (o3o3x3x3o3o - fe)]
Coordinates
The vertices of the ''tritruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,2). This construction is based on
facets of the
bitruncated 7-orthoplex. Alternately it can be centered on the origin as permutations of (-1,-1,-1,0,1,1,1).
Images
Related polytopes
Related uniform 6-polytopes
The truncated 6-simplex is one of 35
uniform 6-polytopes based on the
,3,3,3,3 Coxeter group, all shown here in A
6 Coxeter plane orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...
s.
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.
* o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe
External links
Polytopes of Various Dimensions
{{Polytopes
6-polytopes