Rectified 8-simplex
Encyclopedia
8-simplex |
Rectified 8-simplex |
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Birectified 8-simplex |
Trirectified 8-simplex |
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| Orthogonal projections in A8 Coxeter plane | |||
|---|---|---|---|
In eight-dimensional geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification
Rectification (geometry)
In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...
of the regular 8-simplex.
There are unique 3 degrees of rectifications. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedral
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
cell centers of the 8-simplex.
Rectified 8-simplex
| Rectified 8-simplex | |
|---|---|
| Type | uniform polyzetton |
| Schläfli symbol | t1{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... s |
|
| 7-faces | 18 |
| 6-faces | 108 |
| 5-faces | 336 |
| 4-faces | 630 |
| Cells | 576 |
| Faces | 588 |
| Edges | 252 |
| Vertices | 36 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
7-simplex prism |
| Petrie polygon Petrie polygon In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets... |
enneagon Enneagon In geometry, a nonagon is a nine-sided polygon.The name "nonagon" is a prefix hybrid formation, from Latin , used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century... |
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
A8, [37], order 362880 |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
Coordinates
The Cartesian coordinates of the vertices of the rectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 9-orthoplexRectified 9-orthoplex
In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex.There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified...
.
Rectified 8-simplex
| Birectified 8-simplex | |
|---|---|
| Type | uniform polyzetton |
| Schläfli symbol | t2{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... s |
|
| 7-faces | 18 |
| 6-faces | 144 |
| 5-faces | 588 |
| 4-faces | 1386 |
| Cells | 2016 |
| Faces | 1764 |
| Edges | 756 |
| Vertices | 84 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
{3}x{3,3,3,3} |
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
A8, [37], order 362880 |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
Coordinates
The Cartesian coordinates of the vertices of the birectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 9-orthoplex.Trirectified 8-simplex
| Trirectified 8-simplex | |
|---|---|
| Type | uniform polyzetton |
| Schläfli symbol | t3{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... s |
|
| 7-faces | 18 |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1260 |
| Vertices | 126 |
| Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... |
{3,3}x{3,3,3} |
| Petrie polygon Petrie polygon In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets... |
enneagon Enneagon In geometry, a nonagon is a nine-sided polygon.The name "nonagon" is a prefix hybrid formation, from Latin , used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century... |
| Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... |
A7, [37], order 362880 |
| Properties | convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... |
Coordinates
The Cartesian coordinates of the vertices of the trirectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 9-orthoplex.Related polytopes
This polytope is the vertex figureVertex figure
In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...
of the 9-demicube, and the edge figure of the uniform 261 honeycomb.
It is also one of 135 uniform 8-polytopes with A8 symmetry.