View source for Rectified 5-cubes

{{No footnotes|date=October 2022}}
{| class=wikitable align=right style="margin-left:1em;"
|- align=center
|[[File:5-cube t0.svg|100px]]<BR>[[5-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node|3|node}}
|[[File:5-cube t1.svg|100px]]<BR>Rectified 5-cube<BR>{{CDD|node|4|node_1|3|node|3|node|3|node}}
|rowspan=2|[[File:5-cube t2.svg|150px]]<BR>[[Birectified 5-cube]]<BR>Birectified 5-orthoplex<BR>{{CDD|node|4|node|3|node_1|3|node|3|node}}
|- align=center
|[[File:5-cube t4.svg|100px]]<BR>[[5-orthoplex]]<BR>{{CDD|node|4|node|3|node|3|node|3|node_1}}
|[[File:5-cube t3.svg|100px]]<BR>[[Rectified 5-orthoplex]]<BR>{{CDD|node|4|node|3|node|3|node_1|3|node}}
|-
!colspan=5|[[Orthogonal projection]]s in A<sub>5</sub> [[Coxeter plane]]
|}
In five-dimensional [[geometry]], a '''rectified 5-cube''' is a convex [[uniform 5-polytope]], being a [[Rectification (geometry)|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.
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== Rectified 5-cube==
{{Uniform polyteron db|Uniform polyteron stat table|rin}}
=== Alternate names===
* Rectified penteract (acronym: rin) (Jonathan Bowers)

=== Construction ===
The rectified 5-cube may be constructed from the [[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}}

[[Emanuel Lodewijk Elte|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/triacontiditeron
* Penteractitriacontiditeron (acronym: nit) (Jonathan Bowers)
* Rectified 5-demicube/demipenteract

===Construction and coordinates===
The ''birectified 5-cube'' may be constructed by [[Rectification (geometry)|birectifying]] the vertices of the [[5-cube]] at <math>\sqrt{2}</math> of the edge length.

The [[Cartesian coordinate]]s 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 polyteron#Uniform polyteron|uniform polytera]] generated from the regular [[5-cube]] or [[5-orthoplex]].

{{Penteract family}}

== Notes==
{{reflist}}

== References ==
* [[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 Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (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 (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)}} o3x3o3o4o - rin, o3o3x3o4o - nit

== 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]]