Rectified 7-orthoplexes

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

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 hecatonicosaoctaexon (Acronym: rez) (Jonathan Bowers) - rectified 128-faceted polyexon

Images

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 2 {\displaystyle {\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

Alternate names

• Birectified heptacross
• Birectified hecatonicosaoctaexon (Acronym: barz) (Jonathan Bowers) - birectified 128-faceted polyexon

Images

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length 2 {\displaystyle {\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

• 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 Ivić Weiss, Wiley-Interscience Publication, 1995, , ISBN 978-0-471-01003-6 (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.
• Klitzing, Richard. . o3x3o3o3o3o4o - rez, o3o3x3o3o3o4o - barz

External links