Rectified 7-orthoplexes

{| class="wikitable skin-invert-image" style="float:right; margin-left:8px; width:400px" |- align=center |100px 7-orthoplex {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|4|node}} |100px Rectified 7-orthoplex {{CDD|node|3|node_1|3|node|3|node|3|node|3|node|4|node}} |100px Birectified 7-orthoplex {{CDD|node|3|node|3|node_1|3|node|3|node|3|node|4|node}} |100px Trirectified 7-orthoplex {{CDD|node|3|node|3|node|3|node_1|3|node|3|node|4|node}} |- align=center |100px Birectified 7-cube {{CDD|node|3|node|3|node|3|node|3|node_1|3|node|4|node}} |100px Rectified 7-cube {{CDD|node|3|node|3|node|3|node|3|node|3|node_1|4|node}} |100px 7-cube {{CDD|node|3|node|3|node|3|node|3|node|3|node|4|node_1}} |- !colspan=4|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. {{clear}}

== Rectified 7-orthoplex == {| class="wikitable" style="float:right; margin-left:10px; width:250px" !style="background:#e7dcc3" colspan=2|Rectified 7-orthoplex |- |style="background:#e7dcc3"|Type||uniform 7-polytope |- |style="background:#e7dcc3"|Schläfli symbol|| r{3,3,3,3,3,4} |- |style="background:#e7dcc3"|Coxeter-Dynkin diagrams||{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|4|node}} {{CDD|node|3|node_1|3|node|3|node|3|node|split1|nodes}} |- |style="background:#e7dcc3"|6-faces||142 |- |style="background:#e7dcc3"|5-faces||1344 |- |style="background:#e7dcc3"|4-faces||3360 |- |style="background:#e7dcc3"|Cells||3920 |- |style="background:#e7dcc3"|Faces||2520 |- |style="background:#e7dcc3"|Edges||840 |- |style="background:#e7dcc3"|Vertices||84 |- |style="background:#e7dcc3"|Vertex figure||5-orthoplex prism |- |style="background:#e7dcc3"|Coxeter groups||B7, [3,3,3,3,3,4] D7, [34,1,1] |- |style="background:#e7dcc3"|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. :{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|split1|nodes}} or {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|4|node}}

=== Alternate names === * rectified heptacross * rectified hecatonicosaoctaexon (Acronym: rez) (Jonathan Bowers) - rectified 128-faceted polyexon{{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/rez.htm (o3x3o3o3o3o4o - rez)]}}

=== Images === {{7-cube Coxeter plane graphs|t5|120}}

=== 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 <math> \sqrt{2}\ </math> 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 == {| class="wikitable" style="float:right; margin-left:10px; width:250px" !style="background:#e7dcc3" colspan=2|Birectified 7-orthoplex |- |style="background:#e7dcc3"|Type||uniform 7-polytope |- |style="background:#e7dcc3"|Schläfli symbol|| 2r{3,3,3,3,3,4} |- |style="background:#e7dcc3"|Coxeter-Dynkin diagrams||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|4|node}} {{CDD|node|3|node|3|node_1|3|node|3|node|split1|nodes}} |- |style="background:#e7dcc3"|6-faces||142 |- |style="background:#e7dcc3"|5-faces||1428 |- |style="background:#e7dcc3"|4-faces||6048 |- |style="background:#e7dcc3"|Cells||10640 |- |style="background:#e7dcc3"|Faces||8960 |- |style="background:#e7dcc3"|Edges||3360 |- |style="background:#e7dcc3"|Vertices||280 |- |style="background:#e7dcc3"|Vertex figure||{3}×{3,3,4} |- |style="background:#e7dcc3"|Coxeter groups||B7, [3,3,3,3,3,4] D7, [34,1,1] |- |style="background:#e7dcc3"|Properties||convex |}

=== Alternate names === * Birectified heptacross * Birectified hecatonicosaoctaexon (Acronym: barz) (Jonathan Bowers) - birectified 128-faceted polyexon{{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/barz.htm (o3o3x3o3o3o4o - barz)]}}

=== Images === {{7-cube Coxeter plane graphs|t4|120}}

=== Cartesian coordinates === Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length <math> \sqrt{2}\ </math> 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 == {{reflist}}

== 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''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{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. * {{KlitzingPolytopes|polyexa.htm|7D uniform polytopes (polyexa) with acronyms}} o3x3o3o3o3o4o - rez, o3o3x3o3o3o4o - barz {{sfn whitelist| CITEREFKlitzing}}

== External links == * [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions] * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

Category:7-polytopes