In seven-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 7-simplex is a convex
uniform 7-polytope
In seven-dimensional space, seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope Ridge (geometry), ridge being shared by exactly two 6-polytope Facet (mathematics), facets.
A uniform 7-polytope is ...
, being a
rectification 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 ...
.
There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the ''rectified 7-simplex'' are located at the edge-centers of the ''7-simplex''. Vertices of the ''birectified 7-simplex'' are located in the triangular face centers of the ''7-simplex''. Vertices of the ''trirectified 7-simplex'' are located in the
tetrahedral
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
cell centers of the ''7-simplex''.
Rectified 7-simplex
The rectified 7-simplex is 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
251 honeycomb. It is called 0
5,1 for its branching Coxeter-Dynkin diagram, shown as .
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S.
Alternate names
* Rectified octaexon (Acronym: roc) (Jonathan Bowers)
Coordinates
The vertices of the ''rectified 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This construction is based on
facets of the
rectified 8-orthoplex.
Images
Birectified 7-simplex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 0
4,2 for its branching Coxeter-Dynkin diagram, shown as .
Alternate names
* Birectified octaexon (Acronym: broc) (Jonathan Bowers)
Coordinates
The vertices of the ''birectified 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1). This construction is based on
facets of the
birectified 8-orthoplex.
Images
Trirectified 7-simplex
The ''trirectified 7-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 ...
of two 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 ...
es in
dual configuration.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S.
This 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
133 honeycomb. It is called 0
3,3 for its branching Coxeter-Dynkin diagram, shown as .
Alternate names
* Hexadecaexon (Acronym: he) (Jonathan Bowers)
Coordinates
The vertices of the ''trirectified 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1). This construction is based on
facets of the
trirectified 8-orthoplex.
The ''trirectified 7-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 ...
of two regular 7-simplices in
dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space: the 70 distinct permutations of (1,1,1,1,−1,−1,−1,-1).
Images
Related polytopes
Related polytopes
These polytopes are three of 71
uniform 7-polytope
In seven-dimensional space, seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope Ridge (geometry), ridge being shared by exactly two 6-polytope Facet (mathematics), facets.
A uniform 7-polytope is ...
s with A
7 symmetry.
See also
*
List of A7 polytopes
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.
* o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - broc, o3o3o3x3o3o3o - he
External links
Polytopes of Various Dimensions
{{Polytopes
7-polytopes