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 rectified 6-simplex is a convex
uniform 6-polytope
In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.
The complete set of convex uniform 6-polytopes has not been determined, bu ...
, being a
rectification
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Recti ...
of the regular
6-simplex.
There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the ''rectified 6-simplex'' are located at the edge-centers of the ''6-simplex''. Vertices of the ''birectified 6-simplex'' are located in the triangular face centers of the ''6-simplex''.
Rectified 6-simplex
E. L. Elte
Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibor extermination camp, Sobibór)[ Em ...]
identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 0
4,1 for its branching Coxeter-Dynkin diagram, shown as .
Alternate names
* Rectified heptapeton (Acronym: ril) (Jonathan Bowers)
Coordinates
The vertices of the ''rectified 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on
facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
of the
rectified 7-orthoplex
In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.
There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being th ...
.
Images
Birectified 6-simplex
E. L. Elte
Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibor extermination camp, Sobibór)[ Em ...]
identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 0
3,2 for its branching Coxeter-Dynkin diagram, shown as .
Alternate names
* Birectified heptapeton (Acronym: bril) (Jonathan Bowers)
Coordinates
The vertices of the ''birectified 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,1). This construction is based on
facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
of the
birectified 7-orthoplex
In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.
There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being th ...
.
Images
Related uniform 6-polytopes
The rectified 6-simplex polytope is the
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
of the
7-demicube, and the
edge figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
of the uniform
241 polytope.
These polytopes are a part 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 - ril, o3x3o3o3o3o - bril
External links
Polytopes of Various Dimensions
{{Polytopes
6-polytopes