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 width=500
+{| 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|3|node}}
+|[[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|3|node}}
+|[[File:5-cube t1.svg|100px]]<BR>Rectified 5-cube<BR>{{CDD|node|4|node_1|3|node|3|node|3|node}}
-|[[File:5-cube_t2.svg|100px]]<BR>Birectified 5-cube<BR>{{CDD|node|4|node|3|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_t3.svg|100px]]<BR>[[Rectified 5-orthoplex]]<BR>{{CDD|node|4|node|3|node|3|node|3|node_1|3|node}}
-|[[File:5-cube_t4.svg|100px]]<BR>[[5-orthoplex]]<BR>{{CDD|node|4|node|3|node|3|node|3|node|3|node_1}}
+|[[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]]
@@ Line 11: / Line 13: @@
 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-ocube are located in the square face centers of the 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==
@@ Line 30: / Line 33: @@
 == 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
@@ Line 37: / Line 43: @@
 ===Construction and coordinates===
-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.
+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:
@@ Line 45: / Line 51: @@
 === Images===
 {{5-cube Coxeter plane graphs|t2|150}}
-=== Related polytopes==
+=== Related polytopes===
 {{2-isotopic_uniform_hypercube_polytopes}}
 == Related polytopes==
-Thes polytopes are a part of 31 [[Uniform_polyteron#Uniform_polyteron|uniform polytera]] generated from the regular [[5-cube]] or [[5-orthoplex]].
+These polytopes are a part of 31 [[Uniform polyteron#Uniform polyteron|uniform polytera]] generated from the regular [[5-cube]] or [[5-orthoplex]].
 {{Penteract family}}
@@ Line 60: / Line 66: @@
 * [[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''', 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]
+** '''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]
@@ Line 69: / Line 75: @@
 == 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]

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