Rectified 7-orthoplexes


7-orthoplex

Rectified 7-orthoplex

Birectified 7-orthoplex

Trirectified 7-orthoplex

Birectified 7-cube

Rectified 7-cube

7-cube
Orthogonal projections in B7 Coxeter plane

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 the 7-cube. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in the tetrahedral cell centers of the 7-orthoplex.

Rectified 7-orthoplex

Rectified 7-orthoplex
Typeuniform 7-polytope
Schläfli symbol r{3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces142
5-faces1344
4-faces3360
Cells3920
Faces2520
Edges840
Vertices84
Vertex figure5-orthoplex prism
Coxeter groupsC7, [3,3,3,3,3,4]
D7, [34,1,1]
Propertiesconvex

The rectified 7-orthoplex is the vertex figure for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb.

or

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Construction

There are two Coxeter groups associated with the rectified heptacross, one with the C7 or [4,3,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D7 or [34,1,1] Coxeter group.

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified heptacross, centered at the origin, edge length {\sqrt {2}}\ are all permutations of:

(±1,±1,0,0,0,0,0)

Root vectors

Its 84 vertices represent the root vectors of the simple Lie group D7. The vertices can be seen in 3 hyperplanes, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an expanded 6-simplex passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B7 and C7 simple Lie groups.

Birectified 7-orthoplex

Birectified 7-orthoplex
Typeuniform 7-polytope
Schläfli symbol 2r{3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces142
5-faces1428
4-faces6048
Cells10640
Faces8960
Edges3360
Vertices280
Vertex figure{3}×{3,3,4}
Coxeter groupsC7, [3,3,3,3,3,4]
D7, [34,1,1]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length {\sqrt {2}}\ are all permutations of:

(±1,±1,±1,0,0,0,0)

Trirectified 7-orthoplex

A trirectified 7-orthoplex is the same as a trirectified 7-cube.

Notes

  1. Klitzing, (o3o3x3o3o3o4o - rez)
  2. Klitzing, (o3o3x3o3o3o4o - barz)

References

External links

This article is issued from Wikipedia - version of the Sunday, October 26, 2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.