Rectified 7-cubes

{| class="wikitable skin-invert-image" style="float:right; margin-left:8px; width:500px" |- align=center |120px 7-cube {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}} |120px Rectified 7-cube {{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node}} |120px Birectified 7-cube {{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node}} |120px Trirectified 7-cube {{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node}} |- align=center |120px Birectified 7-orthoplex {{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node}} |120px Rectified 7-orthoplex {{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node}} |120px 7-orthoplex {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node_1}} |- !colspan=4|Orthogonal projections in B7 Coxeter plane |} In seven-dimensional geometry, a '''rectified 7-cube''' is a convex uniform 7-polytope, being a rectification of the regular 7-cube.

There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the edge-centers of the 7-ocube. Vertices of the birectified 7-cube are located in the square face centers of the 7-cube. Vertices of the trirectified 7-cube are located in the cube cell centers of the 7-cube. {{-}}

== Rectified 7-cube == {| class="wikitable" style="float:right; margin-left:10px; width:250px" !style="background:#e7dcc3" colspan=2|Rectified 7-cube |- |style="background:#e7dcc3"|Type||uniform 7-polytope |- |style="background:#e7dcc3"|Schläfli symbol|| r{4,3,3,3,3,3} |- |style="background:#e7dcc3"|Coxeter-Dynkin diagrams||{{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node}} {{CDD|nodes_11|split2|node|3|node|3|node|3|node|3|node}} |- |style="background:#e7dcc3"|6-faces|| 128 + 14 |- |style="background:#e7dcc3"|5-faces|| 896 + 84 |- |style="background:#e7dcc3"|4-faces|| 2688 + 280 |- |style="background:#e7dcc3"|Cells|| 4480 + 560 |- |style="background:#e7dcc3"|Faces|| 4480 + 672 |- |style="background:#e7dcc3"|Edges|| 2688 |- |style="background:#e7dcc3"|Vertices||448 |- |style="background:#e7dcc3"|Vertex figure||5-simplex prism |- |style="background:#e7dcc3"|Coxeter groups||B7, [3,3,3,3,3,4] |- |style="background:#e7dcc3"|Properties||convex |}

=== Alternate names === * rectified hepteract (acronym: rasa) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/rasa.htm (o3o3o3o3o3x4o - rasa)]}}

=== Images === {{7-cube Coxeter plane graphs|t1|120}}

=== Cartesian coordinates === Cartesian coordinates for the vertices of a rectified 7-cube, centered at the origin, edge length <math> \sqrt{2}\ </math> are all permutations of: : (±1,±1,±1,±1,±1,±1,0)

== Birectified 7-cube == {| class="wikitable" style="float:right; margin-left:10px; width:250px" !style="background:#e7dcc3" colspan=2|Birectified 7-cube |- |style="background:#e7dcc3"|Type||uniform 7-polytope |- |style="background:#e7dcc3"|Coxeter symbol|| 0411 |- |style="background:#e7dcc3"|Schläfli symbol|| 2r{4,3,3,3,3,3} |- |style="background:#e7dcc3"|Coxeter-Dynkin diagrams||{{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node}} {{CDD|nodes|split2|node_1|3|node|3|node|3|node|3|node}} |- |style="background:#e7dcc3"|6-faces|| 128 + 14 |- |style="background:#e7dcc3"|5-faces|| 448 + 896 + 84 |- |style="background:#e7dcc3"|4-faces|| 2688 + 2688 + 280 |- |style="background:#e7dcc3"|Cells|| 6720 + 4480 + 560 |- |style="background:#e7dcc3"|Faces|| 8960 + 4480 |- |style="background:#e7dcc3"|Edges|| 6720 |- |style="background:#e7dcc3"|Vertices||672 |- |style="background:#e7dcc3"|Vertex figure||{3}x{3,3,3} |- |style="background:#e7dcc3"|Coxeter groups||B7, [3,3,3,3,3,4] |- |style="background:#e7dcc3"|Properties||convex |}

=== Alternate names === * Birectified hepteract (acronym: bersa) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/bersa.htm (o3o3o3o3x3o4o - bersa)]}}

=== Images === {{7-cube Coxeter plane graphs|t2|120}}

=== Cartesian coordinates === Cartesian coordinates for the vertices of a birectified 7-cube, centered at the origin, edge length <math> \sqrt{2}\ </math> are all permutations of: : (±1,±1,±1,±1,±1,0,0)

== Trirectified 7-cube == {| class="wikitable" style="float:right; margin-left:10px; width:250px" !style="background:#e7dcc3" colspan=2|Trirectified 7-cube |- |style="background:#e7dcc3"|Type||uniform 7-polytope |- |style="background:#e7dcc3"|Schläfli symbol|| 3r{4,3,3,3,3,3} |- |style="background:#e7dcc3"|Coxeter-Dynkin diagrams||{{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node}} {{CDD|nodes|split2|node|3|node_1|3|node|3|node|3|node}} |- |style="background:#e7dcc3"|6-faces||128 + 14 |- |style="background:#e7dcc3"|5-faces|| 448 + 896 + 84 |- |style="background:#e7dcc3"|4-faces|| 672 + 2688 + 2688 + 280 |- |style="background:#e7dcc3"|Cells|| 3360 + 6720 + 4480 |- |style="background:#e7dcc3"|Faces|| 6720 + 8960 |- |style="background:#e7dcc3"|Edges|| 6720 |- |style="background:#e7dcc3"|Vertices|| 560 |- |style="background:#e7dcc3"|Vertex figure||{3,3}x{3,3} |- |style="background:#e7dcc3"|Coxeter groups||B7, [3,3,3,3,3,4] |- |style="background:#e7dcc3"|Properties||convex |}

=== Alternate names === * Trirectified hepteract * Trirectified 7-orthoplex * Trirectified heptacross (acronym: sez) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/sez.htm (o3o3o3x3o3o4o - sez)]}}

=== Images === {{7-cube Coxeter plane graphs|t3|120}}

=== Cartesian coordinates === Cartesian coordinates for the vertices of a trirectified 7-cube, centered at the origin, edge length <math> \sqrt{2}\ </math> are all permutations of: : (±1,±1,±1,±1,0,0,0)

=== Related polytopes === {{2-isotopic uniform hypercube polytopes}}

== 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}} o3o3o3o3o3x4o - rasa, o3o3o3o3x3o4o - bersa, o3o3o3x3o3o4o - sez {{sfn whitelist| CITEREFKlitzing}}

== External links == * [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions] * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

Category:7-polytopes