View source for Rectified 5-cubes

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

There are 5 degrees of rectifications of a 5-polytope, the zeroth here being the [[5-cube|5-cube]], and the 4th and last being the [[5-orthoplex|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|5-cube]] by [[Rectification (geometry)|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 Cr<sub>5</sub><sup>2</sup> 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 [[Rectification (geometry)|birectifying]] the vertices of the [[5-cube|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&nbsp;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|5-cube]] or [[5-orthoplex|5-orthoplex]].

{{Penteract family}}

== Notes ==
{{reflist}}

== References ==
* [[Biography:Harold Scott MacDonald Coxeter|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]
* [[Biography:Norman Johnson (mathematician)|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]

{{Polytopes}}

[[Category:5-polytopes]]

{{Sourceattribution|Rectified 5-cubes}}