https://en.formulasearchengine.com/w/api.php?action=feedcontributions&feedformat=atom&user=173.110.9.209formulasearchengine - User contributions [en]2026-07-03T20:39:50ZUser contributionsMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Cybernetical_physics&diff=25235Cybernetical physics2014-01-26T20:00:32Z<p>173.110.9.209: </p>
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<div>{| class=wikitable align=right width=500<br />
|- align=center<br />
|[[File:5-cube_t0.svg|100px]]<BR>[[5-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}<br />
|[[File:5-cube_t1.svg|100px]]<BR>Rectified 5-cube<BR>{{CDD|node|4|node_1|3|node|3|node|3|node|3|node}}<br />
|[[File:5-cube_t2.svg|100px]]<BR>Birectified 5-cube<BR>{{CDD|node|4|node|3|node_1|3|node|3|node|3|node}}<br />
|[[File:5-cube_t3.svg|100px]]<BR>[[Rectified 5-orthoplex]]<BR>{{CDD|node|4|node|3|node|3|node|3|node_1|3|node}}<br />
|[[File:5-cube_t4.svg|100px]]<BR>[[5-orthoplex]]<BR>{{CDD|node|4|node|3|node|3|node|3|node|3|node_1}}<br />
|-<br />
!colspan=5|[[Orthogonal projection]]s in A<sub>5</sub> [[Coxeter plane]]<br />
|}<br />
In five-dimensional [[geometry]], a '''rectified 5-cube''' is a convex [[uniform 5-polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[5-cube]].<br />
<br />
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-ocube are located in the square face centers of the 5-cube.<br />
<br />
== Rectified 5-cube==<br />
{{Uniform polyteron db|Uniform polyteron stat table|rin}}<br />
=== Alternate names===<br />
* Rectified penteract (acronym: rin) (Jonathan Bowers)<br />
<br />
=== Construction ===<br />
The rectified 5-cube may be constructed from the [[5-cube]] by [[Rectification (geometry)|truncating]] its vertices at the midpoints of its edges.<br />
<br />
=== Coordinates===<br />
The [[Cartesian coordinates]] of the vertices of the rectified 5-cube with edge length <math>\sqrt{2}</math> is given by all permutations of:<br />
:<math>(0,\ \pm1,\ \pm1,\ \pm1,\ \pm1)</math><br />
<br />
=== Images ===<br />
{{5-cube Coxeter plane graphs|t1|150}}<br />
<br />
== Birectified 5-cube==<br />
{{Uniform polyteron db|Uniform polyteron stat table|nit}}<br />
=== Alternate names===<br />
* Birectified 5-cube/penteract<br />
* Birectified pentacross/5-orthoplex/triacontiditeron<br />
* Penteractitriacontiditeron (acronym: nit) (Jonathan Bowers)<br />
* Rectified 5-demicube/demipenteract<br />
<br />
===Construction and coordinates===<br />
The ''birectified 5-cube'' may be constructed by [[Rectification (geometry)|birectifing]] the vertices of the [[5-cube]] at <math>\sqrt{2}</math> of the edge length.<br />
<br />
The [[Cartesian coordinate]]s of the vertices of a ''birectified 5-cube'' having edge length&nbsp;2 are all permutations of:<br />
<br />
:<math>\left(0,\ 0,\ \pm1,\ \pm1,\ \pm1\right)</math><br />
<br />
=== Images===<br />
{{5-cube Coxeter plane graphs|t2|150}}<br />
=== Related polytopes===<br />
{{2-isotopic_uniform_hypercube_polytopes}}<br />
<br />
== Related polytopes==<br />
<br />
Thes polytopes are a part of 31 [[Uniform_polyteron#Uniform_polyteron|uniform polytera]] generated from the regular [[5-cube]] or [[5-orthoplex]].<br />
<br />
{{Penteract family}}<br />
<br />
== Notes==<br />
{{reflist}}<br />
<br />
== References ==<br />
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: <br />
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 <br />
** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', editied 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]<br />
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]<br />
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]<br />
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]<br />
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)<br />
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. <br />
* {{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}} o3x3o3o4o - rin, o3o3x3o4o - nit<br />
<br />
== External links ==<br />
* {{MathWorld|title=Hypercube|urlname=Hypercube}}<br />
*{{GlossaryForHyperspace | anchor=Measure | title=Measure polytope }}<br />
* [http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]<br />
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]<br />
<br />
{{Polytopes}}<br />
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[[Category:5-polytopes]]</div>173.110.9.209