Rectified 5-cubes: Difference between revisions

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Orthogonal projections in A₅ Coxeter plane
5-cube	Rectified 5-cube	Birectified 5-cube Birectified 5-orthoplex
5-orthoplex	Rectified 5-orthoplex	Birectified 5-cube Birectified 5-orthoplex

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.

Rectified 5-cube

Rectified 5-cube rectified penteract (rin)
Type	uniform 5-polytope
Schläfli symbol	r{4,3,3,3}
Coxeter diagram	=
4-faces	42	10 32
Cells	200	40 160
Faces	400	80 320
Edges	320
Vertices	80
Vertex figure	Tetrahedral prism
Coxeter group	B₅, [4,3³], order 3840
Dual
Base point	(0,1,1,1,1,1)√2
Circumradius	sqrt(2) = 1.414214
Properties	convex, isogonal

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,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph
Dihedral symmetry	[4]	[4]

Birectified 5-cube

Birectified 5-cube birectified penteract (nit)
Type	uniform 5-polytope
Schläfli symbol	2r{4,3,3,3}
Coxeter diagram	=
4-faces	42	10 32
Cells	280	40 160 80
Faces	640	320 320
Edges	480
Vertices	80
Vertex figure	{3}×{4}
Coxeter group	B₅, [4,3³], order 3840 D₅, [3^2,1,1], order 1920
Dual
Base point	(0,0,1,1,1,1)√2
Circumradius	sqrt(3/2) = 1.224745
Properties	convex, isogonal

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,\ \pm 1,\ \pm 1,\ \pm 1\right)

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph
Dihedral symmetry	[4]	[4]

Related polytopes

2-isotopic hypercubes
Dim.	2	3	4	5	6	7	8	n
Name	t{4}	r{4,3}	2t{4,3,3}	2r{4,3,3,3}	3t{4,3,3,3,3}	3r{4,3,3,3,3,3}	4t{4,3,3,3,3,3,3}	...
Coxeter diagram
Images
Facets		{3} {4}	t{3,3} t{3,4}	r{3,3,3} r{3,3,4}	2t{3,3,3,3} 2t{3,3,3,4}	2r{3,3,3,3,3} 2r{3,3,3,3,4}	3t{3,3,3,3,3,3} 3t{3,3,3,3,3,4}
Vertex figure	( )v( )	{ }×{ }	{ }v{ }	{3}×{4}	{3}v{4}	{3,3}×{3,4}	{3,3}v{3,4}

Related polytopes

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

Notes

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 [1]
  - (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.
Klitzing, Richard. "5D uniform polytopes (polytera)". o3x3o3o4o - rin, o3o3x3o4o - nit

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

@@ Line 1: / Line 1: @@
+{{No footnotes|date=October 2022}}
-{| class="wikitable" align="right" style="margin-left:10px" width="250"
+{| class=wikitable align=right style="margin-left:1em;"
-!bgcolor=#e7dcc3 colspan=2|Rectified 5-cube
+|- 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}}
 |-
-|bgcolor=#ffffff align=center colspan=2|[[File:Rectified 5-cube.png|280px]]<BR>[[Orthogonal projection]]<BR>inside [[Petrie polygon]]
+!colspan=5|[[Orthogonal projection]]s in A<sub>5</sub> [[Coxeter plane]]
-|-
-|bgcolor=#e7dcc3|Type||[[uniform polyteron]]
-|-
-|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>1</sub>{4,3,3,3}
-|-
-|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||[[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW ring.png]][[Image:CDW 3b.png]][[Image:CDW dot.png]][[Image:CDW 3b.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]]
-|-
-|bgcolor=#e7dcc3|4-faces||42
-|-
-|bgcolor=#e7dcc3|Cells||200
-|-
-|bgcolor=#e7dcc3|Faces||400
-|-
-|bgcolor=#e7dcc3|Edges||320
-|-
-|bgcolor=#e7dcc3|Vertices||80
-|-
-|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Rectified 5-cube verf.png|40px]]<BR>5-cell prism
-|-
-|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Decagon]]
-|-
-|bgcolor=#e7dcc3|[[Coxeter group]]s||C<sub>5</sub>, [3,3,3,4]
-|-
-|bgcolor=#e7dcc3|Dual||?
-|-
-|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
 |}
+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.
-In [[fifth dimension|five-dimensional]] [[geometry]], a '''rectified 5-cube''' is a [[5-polytope|polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[5-cube]].
+{{clear}}
-== Construction ==
+== Rectified 5-cube==
+{{Uniform polyteron db|Uniform polyteron stat table|rin}}
-There are two [[Coxeter group]]s associated with the ''rectified 5-cube'', one with the C<sub>5</sub> or [4,3,3,3] Coxeter group, and a lower symmetry with the D<sub>5</sub> or [3<sup>2,1,1</sup>] Coxeter group.
+=== Alternate names===
+* Rectified penteract (acronym: rin) (Jonathan Bowers)
-== See also ==
+=== Construction ===
+The rectified 5-cube may be constructed from the [[5-cube]] by [[Rectification (geometry)|truncating]] its vertices at the midpoints of its edges.
-* Other [[5-polytope]]s:
-** [[5-simplex]] - {3,3,3,3}
+=== Coordinates===
-** [[5-cube]] (penteract) - {4,3,3,3}
+The [[Cartesian coordinates]] of the vertices of the rectified 5-cube with edge length <math>\sqrt{2}</math> is given by all permutations of:
-** [[5-demicube]] (demipenteract) - {3<sup>1,2,1</sup>}
+:<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]
-* {{MathWorld|title=Hypercube|urlname=Hypercube}}
-*{{GlossaryForHyperspace | anchor=Measure | title=Measure polytope }}
-* [http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
 * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
-* [[Richard Klitzing]] 5D quasiregulars, (multi)prisms, non-prismatic Wythoffian polyterons [http://ogre.nu/klitzing/dimensions/polytera.htm o3x3o3o4o - rin]
+{{Polytopes}}
-[[Category:5-polytopes]]
+[[Category:5-polytopes]]
-{{Geometry-stub}}

B5 polytopes
β₅	t₁β₅	t₂γ₅	t₁γ₅	γ₅	t_0,1β₅	t_0,2β₅	t_1,2β₅
t_0,3β₅	t_1,3γ₅	t_1,2γ₅	t_0,4γ₅	t_0,3γ₅	t_0,2γ₅	t_0,1γ₅	t_0,1,2β₅
t_0,1,3β₅	t_0,2,3β₅	t_1,2,3γ₅	t_0,1,4β₅	t_0,2,4γ₅	t_0,2,3γ₅	t_0,1,4γ₅	t_0,1,3γ₅
t_0,1,2γ₅	t_0,1,2,3β₅	t_0,1,2,4β₅	t_0,1,3,4γ₅	t_0,1,2,4γ₅	t_0,1,2,3γ₅	t_0,1,2,3,4γ₅