Rectified 5-cubes

{{More footnotes|date=March 2026}} {| class="wikitable skin-invert-image" style="float:right; margin-left:1em;" |- align=center |100px 5-cube {{CDD|node_1|4|node|3|node|3|node|3|node}} |100px Rectified 5-cube {{CDD|node|4|node_1|3|node|3|node|3|node}} |rowspan=2|150px Birectified 5-cube Birectified 5-orthoplex {{CDD|node|4|node|3|node_1|3|node|3|node}} |- align=center |100px 5-orthoplex {{CDD|node|4|node|3|node|3|node|3|node_1}} |100px Rectified 5-orthoplex {{CDD|node|4|node|3|node|3|node_1|3|node}} |- !colspan=5|Orthogonal projections in A5 Coxeter plane |} In five-dimensional geometry, a '''rectified 5-cube''' is a convex uniform 5-polytope, being a rectification of the regular 5-cube.

There are 5 degrees of rectifications of a 5-polytope, the zeroth here being the 5-cube, and the 4th and last being the 5-orthoplex. Vertices of the rectified 5-cube are located at the edge-centers of the 5-cube. Vertices of the birectified 5-cube are located in the square face centers of the 5-cube. {{clear}}

== Rectified 5-cube == {{Uniform polyteron db|Uniform polyteron stat table|rin}}

=== Alternate names === * Rectified penteract (acronym: rin) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/rin.htm (o3x3o3o4o - rin)]}}

=== Construction === The rectified 5-cube may be constructed from the 5-cube by truncating its vertices at the midpoints of its edges.

=== Coordinates === The Cartesian coordinates of the vertices of the rectified 5-cube with edge length <math>\sqrt{2}</math> is given by all permutations of: :<math>(0,\ \pm1,\ \pm1,\ \pm1,\ \pm1)</math>

=== Images === {{5-cube Coxeter plane graphs|t1|150}}

== Birectified 5-cube == {{Uniform polyteron db|Uniform polyteron stat table|nit}}

E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr52 as a second rectification of a 5-dimensional cross polytope.

=== Alternate names === * Birectified 5-cube/penteract * Birectified pentacross/5-orthoplex/triacontaditeron * Penteractitriacontaditeron (acronym: nit) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/nit.htm (o3o3x3o4o - nit)]}} * Rectified 5-demicube/demipenteract

=== Construction and coordinates === The birectified 5-cube may be constructed by birectifying the vertices of the 5-cube at <math>\sqrt{2}</math> of the edge length.

The Cartesian coordinates of the vertices of a ''birectified 5-cube'' having edge length 2 are all permutations of:

:<math>\left(0,\ 0,\ \pm1,\ \pm1,\ \pm1\right)</math>

=== Images === {{5-cube Coxeter plane graphs|t2|150}}

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

== Related polytopes == These polytopes are a part of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.

== 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|polytera.htm|5D uniform polytopes (polytera) with acronyms}} o3x3o3o4o - rin, o3o3x3o4o - nit {{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:5-polytopes