Login
Login

Rectified 5-cubes#Rectified 5-cube

{{No footnotes|date=October 2022}}

class=wikitable align=right style="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)

= 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 \sqrt{2} is given by all permutations of:

:(0,\ \pm1,\ \pm1,\ \pm1,\ \pm1)

= 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/triacontiditeron
  • Penteractitriacontiditeron (acronym: nit) (Jonathan Bowers)
  • Rectified 5-demicube/demipenteract

=Construction and coordinates=

The birectified 5-cube may be constructed by birectifying the vertices of the 5-cube at \sqrt{2} of the edge length.

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

:\left(0,\ 0,\ \pm1,\ \pm1,\ \pm1\right)

= 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.

{{Penteract family}}

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 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 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