View source for Rectified 24-cell

{| class="wikitable" align="right" style="margin-left:10px" width="320"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Rectified 24-cell'''
|-
|bgcolor=#ffffff align=center colspan=3|[[File:Schlegel half-solid cantellated 16-cell.png|280px]]<BR>[[Schlegel diagram]]<BR>8 of 24 cuboctahedral cells shown
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s
|colspan=2|r{3,4,3} = <math>\left\{\begin{array}{l}3\\4,3\end{array}\right\}</math><BR>rr{3,3,4}=<math>r\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><BR>r{3<sup>1,1,1</sup>} = <math>r\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s
|colspan=2|{{CDD|node|3|node_1|4|node|3|node}}<BR>{{CDD|node_1|3|node|3|node_1|4|node}}<BR>{{CDD|node_1|3|node|split1|nodes_11}} or {{CDD|node|splitsplit1|branch3_11|node_1}}
|-
|bgcolor=#e7dcc3|Cells
|48
|24 [[Cuboctahedron|''3.4.3.4'']] [[File:Cuboctahedron.png|20px]]<BR>24 [[cube|''4.4.4'']] [[File:Hexahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|240
|96 [[triangle|{3}]]<br>144 [[square (geometry)|{4}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|288
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|96
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:rectified 24-cell verf.png|50px]][[File:Cantellated 16-cell verf.png|50px]][[File:Runcicantellated demitesseract verf.png|50px]]<BR>[[Triangular prism]]
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry groups]]
|colspan=2|F<sub>4</sub> [3,4,3], order 1152<BR>B<sub>4</sub> [3,3,4], order 384<BR>D<sub>4</sub> [3<sup>1,1,1</sup>], order 192
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[edge-transitive]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[24-cell|22]]'' 23 ''[[truncated 24-cell|24]]''
|}
[[File:Rectified icositetrachoron net.png|thumb|right|[[Net (polyhedron)|Net]]]]
In [[geometry]], the '''rectified 24-cell''' or '''rectified icositetrachoron''' is a uniform 4-dimensional polytope (or [[uniform 4-polytope]]), which is bounded by 48 [[cell (mathematics)|cells]]: 24 [[cube]]s, and 24 [[cuboctahedron|cuboctahedra]]. It can be obtained by [[Rectification (geometry)|rectification]] of the 24-cell, reducing its octahedral cells to cubes and cuboctahedra.{{Sfn|Coxeter|1973|p=154|loc=§8.4}}

[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as tC<sub>24</sub>.

It can also be considered a '''cantellated 16-cell''' with the lower symmetries B<sub>4</sub> = [3,3,4]. B<sub>4</sub> would lead to a bicoloring of the [[cuboctahedron|cuboctahedral]] cells into 8 and 16 each. It is also called a '''runcicantellated demitesseract''' in a D<sub>4</sub> symmetry, giving 3 colors of cells, 8 for each.

== Construction ==

The rectified 24-cell can be derived from the 24-cell by the process of [[rectification (geometry)|rectification]]: the 24-cell is truncated at the midpoints. The vertices become [[cube]]s, while the [[octahedron|octahedra]] become [[cuboctahedron|cuboctahedra]].

== Cartesian coordinates ==

A rectified 24-cell having an edge length of {{radic|2}} has vertices given by all permutations and sign permutations of the following [[Cartesian coordinate]]s:
: (0,1,1,2) [4!/2!×2<sup>3</sup> = 96 vertices]

The dual configuration with edge length 2 has all coordinate and sign permutations of:
: (0,2,2,2) [4×2<sup>3</sup> = 32 vertices]
: (1,1,1,3) [4×2<sup>4</sup> = 64 vertices]

== Images ==
{{24-cell_4-cube_Coxeter_plane_graphs|t1|100|t2}}

{| class="wikitable" width=360
![[Stereographic projection]]
|-
|colspan=2 align=center|[[File:Rectified 24cell.png|360px]]<BR>
|-
|Center of [[stereographic projection]]<BR>with 96 triangular faces blue
|}

== Symmetry constructions ==

There are three different symmetry constructions of this polytope. The lowest <math>{D}_4</math> construction can be doubled into <math>{C}_4</math> by adding a mirror that maps the bifurcating nodes onto each other. <math>{D}_4</math> can be mapped up to <math>{F}_4</math> symmetry by adding two mirror that map all three end nodes together.

The [[vertex figure]] is a [[triangular prism]], containing two cubes and three cuboctahedra. The three symmetries can be seen with 3 colored cuboctahedra in the lowest <math>{D}_4</math> construction, and two colors (1:2 ratio) in <math>{C}_4</math>, and all identical cuboctahedra in <math>{F}_4</math>.

{| class='wikitable' 
![[Coxeter group]]
!<math>{F}_4</math> = [3,4,3]
!<math>{C}_4</math> = [4,3,3]
!<math>{D}_4</math> = [3,3<sup>1,1</sup>]
|- align=center
!Order
|1152
|384
|192
|- align=center
!Full<BR>symmetry<BR>group
|[3,4,3]
|[4,3,3]
|<[3,3<sup>1,1</sup>]> = [4,3,3]<BR>[3[3<sup>1,1,1</sup>]] = [3,4,3]
|- align=center
![[Coxeter diagram]]
|{{CDD|node|3|node_1|4|node|3|node}}
|{{CDD|node|4|node_1|3|node|3|node_1}}
|{{CDD|nodes_11|split2|node|3|node_1}}
|-
![[Facet (geometry)|Facets]]
|'''3:''' {{CDD|node|3|node_1|4|node}}<BR>'''2:''' {{CDD|node_1|4|node|3|node}}
|'''2,2:''' {{CDD|node_1|3|node|3|node_1}}<BR>'''2:''' {{CDD|node|4|node_1|2|node_1}}
|'''1,1,1:''' {{CDD|node_1|3|node|3|node_1}}<BR>'''2:''' {{CDD|node_1|2|node_1|2|node_1}}
|- align=center
![[Vertex figure]]
|[[File:Rectified 24-cell verf.png|80px]]
|[[File:Cantellated 16-cell verf.png|80px]]
|[[File:Runcicantellated demitesseract verf.png|80px]]
|}

==Alternate names==
* Rectified 24-cell, Cantellated 16-cell ([[Norman Johnson (mathematician)|Norman Johnson]])
* Rectified icositetrachoron (Acronym rico) (George Olshevsky, Jonathan Bowers)
** Cantellated hexadecachoron
* Disicositetrachoron
* Amboicositetrachoron ([[John Horton Conway|Neil Sloane & John Horton Conway]])

== Related polytopes ==
The convex hull of the rectified 24-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 192 cells: 48 [[cube]]s, 144 [[square antiprism]]s, and 192 vertices. Its vertex figure is a [[triangular bifrustum]].

== Related uniform polytopes ==
{{Demitesseract family}}

{{24-cell_family}}

The ''rectified 24-cell'' can also be derived as a ''cantellated 16-cell'':
{{Tesseract family}}

== Citations ==
{{RefList}}

== References ==
* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
* {{Cite book | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=Regular Polytopes (book) }}
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;409: Hemicubes: 1<sub>n1</sub>)
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
* {{PolyCell | urlname = section2.html| title = 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 23}}
** {{PolyCell | urlname = section3.html| title = 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 23}}
** {{PolyCell | urlname = section7.html| title = 7. Uniform polychora derived from glomeric tetrahedron B4 - Model 23 }}
* {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|o3x4o3o - rico}}

{{Polytopes}}

[[Category:Uniform 4-polytopes]]