Rectified 7-orthoplexes

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 Orthogonal projections in B7 Coxeter plane 7-orthoplex Rectified 7-orthoplex Birectified 7-orthoplex Trirectified 7-orthoplex Birectified 7-cube Rectified 7-cube 7-cube

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
Type uniform 7-polytope
Schläfli symbol r{3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces 142
5-faces 1344
4-faces 3360
Cells 3920
Faces 2520
Edges 840
Vertices 84
Vertex figure 5-orthoplex prism
Coxeter groups C7, [3,3,3,3,3,4]
D7, [34,1,1]
Properties convex

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

• rectified heptacross
• rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon[1]

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
Type uniform 7-polytope
Schläfli symbol 2r{3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3}x{3,3,4}
Coxeter groups C7, [3,3,3,3,3,4]
D7, [34,1,1]
Properties convex

Alternate names

• Birectified heptacross
• Birectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - birectified 128-faceted polyexon[2]

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

Trirectified 7-orthoplex
Type uniform 7-polytope
Schläfli symbol 3r{3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3}x{3,4}
Coxeter groups C7, [3,3,3,3,3,4]
D7, [34,1,1]
Properties convex

Alternate names

• Trirectified heptacross
• Trirectified hecatonicosoctaexon (trirectified 128-faceted polyexon)

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,±1,0,0,0)

Notes

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

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, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Richard Klitzing, 7D, uniform polytopes (polyexa) o3o3x3o3o3o4o - rez, o3o3x3o3o3o4o - barz