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 rectified 7-simplex is a convex
uniform 7-polytope
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.
A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose ...
, 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
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−1(1/ ...
.
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 (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
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 polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw ...
of the
251 honeycomb. It is called 0
5,1 for its branching Coxeter-Dynkin diagram, shown as .
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
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 8-orthoplex
In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.
There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being t ...
.
Images
Birectified 7-simplex
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
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 8-orthoplex
In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope
In Eight-dimensional space, eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ...
.
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, thei ...
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−1(1/ ...
es in
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
configuration.
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 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
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
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
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, thei ...
of two regular 7-simplices in
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
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 geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.
A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose ...
s with A
7 symmetry.
See also
*
List of A7 polytopes
In 7-dimensional geometry, there are 71 uniform polytopes with A7 symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices.
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A7 Coxeter grou ...
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.
* o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he
External links
Polytopes of Various Dimensions
{{Polytopes
7-polytopes