In six-dimensional
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a rectified 6-simplex is a convex
uniform 6-polytope, being a
rectification 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 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 of the
rectified 7-orthoplex.
Images
Birectified 6-simplex
E. L. Elte 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 of the
birectified 7-orthoplex.
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 general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
of the
7-demicube, and the
edge figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines acr ...
of the uniform
241 polytope.
These polytopes are a part of 35
uniform 6-polytopes based on the
,3,3,3,3Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
, all shown here in A
6 Coxeter plane
In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which hav ...
orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Plane (mathematics), two dimensions. Orthographic projection is a form of parallel projection in ...
s.
Notes
References
*
H.S.M. Coxeter:
** 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
wiley.com
*** (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, o3o3x3o3o3o - bril
External links
Polytopes of Various Dimensions
{{Polytopes
6-polytopes