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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 cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex. There are unique 6 degrees of cantellation for the 7-simplex, including truncations.


Cantellated 7-simplex


Alternate names

* Small rhombated octaexon (acronym: saro) (Jonathan Bowers)


Coordinates

The vertices of the ''cantellated 7-simplex'' can be most simply positioned in 8-space as
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
s of (0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 8-orthoplex.


Images


Bicantellated 7-simplex


Alternate names

* Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)


Coordinates

The vertices of the ''bicantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 8-orthoplex.


Images


Tricantellated 7-simplex


Alternate names

* Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)


Coordinates

The vertices of the ''tricantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). This construction is based on facets of the tricantellated 8-orthoplex.


Images


Cantitruncated 7-simplex


Alternate names

* Great rhombated octaexon (acronym: garo) (Jonathan Bowers)


Coordinates

The vertices of the ''cantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 8-orthoplex.


Images


Bicantitruncated 7-simplex


Alternate names

* Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)


Coordinates

The vertices of the ''bicantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 8-orthoplex.


Images


Tricantitruncated 7-simplex


Alternate names

* Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)Klitizing, (o3o3x3x3x3o3o - gatroh)


Coordinates

The vertices of the ''tricantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the tricantitruncated 8-orthoplex.


Images


Related polytopes

This polytope is one of 71 uniform 7-polytopes with A7 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. * x3o3x3o3o3o3o - saro, o3x3o3x3o3o3o - sabro, o3o3x3o3x3o3o - stiroh, x3x3x3o3o3o3o - garo, o3x3x3x3o3o3o - gabro, o3o3x3x3x3o3o - gatroh


External links


Polytopes of Various Dimensions


{{Polytopes 7-polytopes