View source for Cantellated 24-cells

{| class=wikitable align=right width=450
|- align=center valign=top
|[[File:24-cell t0 F4.svg|150px]]<BR>[[24-cell]]<BR>{{CDD|node_1|3|node|4|node|3|node}}
|[[File:24-cell t02 F4.svg|150px]]<BR>Cantellated 24-cell<BR>{{CDD|node_1|3|node|4|node_1|3|node}}
|[[File:24-cell t012 F4.svg|150px]]<BR>Cantitruncated 24-cell<BR>{{CDD|node_1|3|node_1|4|node_1|3|node}}
|-
!colspan=3|[[Orthogonal projection]]s in F<sub>4</sub> [[Coxeter plane]]
|}
In four-dimensional [[geometry]], a '''cantellated 24-cell''' is a convex [[uniform 4-polytope]], being a [[cantellation]] (a 2nd order truncation) of the regular [[24-cell]].

There are 2 unique degrees of cantellations of the 24-cell including permutations with truncations.

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==Cantellated 24-cell==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
!bgcolor=#e7dcc3 colspan=3|Cantellated 24-cell
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2| rr{3,4,3}<BR>s<sub>2</sub>{3,4,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]
|colspan=2|{{CDD|node_1|3|node|4|node_1|3|node}}<BR>{{CDD|node_h|3|node_h|4|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells
|144
|24 [[Image:small rhombicuboctahedron.png|20px]][[small rhombicuboctahedron|(3.4.4.4)]]<br>24 [[Image:Cuboctahedron.png|20px]][[Cuboctahedron|(3.4.3.4)]]<br>
96 [[Image:triangular prism.png|20px]][[triangular prism|(3.4.4)]]
|-
|bgcolor=#e7dcc3|Faces
|720
|288 [[triangle]]s<br>432 [[Square (geometry)|square]]s
|-
|bgcolor=#e7dcc3|Edges
|colspan=2| 864
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2| 288
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[Image:Cantellated 24-cell verf.png|80px]]<BR>Wedge
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
|colspan=2|F<sub>4</sub>, [3,4,3], order 1152
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[truncated 24-cell|24]]'' 25 ''[[runcinated 24-cell|26]]''
|}
[[File:Small rhombated icositetrachoron net.png|thumb|right|[[Net (polyhedron)|Net]]]]
The '''cantellated 24-cell''' or '''small rhombated icositetrachoron''' is a [[uniform 4-polytope]].

The boundary of the cantellated 24-cell is composed of 24 [[Truncated octahedron|truncated octahedral]] cells, 24 [[Cuboctahedron|cuboctahedral]] cells and 96 [[triangular prism]]s. Together they have 288 triangular faces, 432 square faces, 864 edges, and 288 vertices.

===Construction===
When the [[Cantellation (geometry)|cantellation]] process is applied to [[24-cell]],
each of the 24 [[octahedron|octahedra]] becomes a [[small rhombicuboctahedron]].
In addition however, since each [[octahedron|octahedra]]'s edge was previously shared with two
other [[octahedron|octahedra]], the separating edges form the three parallel edges of a 
[[triangular prism]] - 96 triangular prisms, since the [[24-cell]] contains 96 edges.
Further, since each vertex was previously shared with 12 faces,
the vertex would split into 12 (24*12=288) new vertices.
Each group of 12 new vertices forms a [[cuboctahedron]].

===Coordinates===
The [[Cartesian coordinate]]s of the vertices of the cantellated 24-cell having edge length&nbsp;2 are all permutations of coordinates and sign of:

: (0, {{radic|2}}, {{radic|2}}, 2+2{{radic|2}})
: (1, 1+{{radic|2}}, 1+{{radic|2}}, 1+2{{radic|2}})

The permutations of the second set of coordinates coincide with the vertices of an inscribed [[runcitruncated tesseract]].

The dual configuration has all permutations and signs of:
:(0,2,2+{{radic|2}},2+{{radic|2}})
:(1,1,1+{{radic|2}},3+{{radic|2}})

===Structure===
The 24 [[small rhombicuboctahedron|small rhombicuboctahedra]] are joined to each other via their triangular faces, to the [[cuboctahedron|cuboctahedra]] via their axial square faces, and to the [[triangular prism]]s via their off-axial square faces. The cuboctahedra are joined to the triangular prisms via their triangular faces. Each triangular prism is joined to two cuboctahedra at its two ends.

=== Cantic snub 24-cell ===

A half-symmetry construction of the cantellated 24-cell, also called a '''cantic snub 24-cell''', as {{CDD|node_h|3|node_h|4|node_1|3|node}}, has an identical geometry, but its triangular faces are further subdivided. The cantellated 24-cell has 2 positions of triangular faces in ratio of 96 and 192, while the cantic snub 24-cell has 3 positions of 96 triangles.

The difference can be seen in the vertex figures, with edges representing faces in the 4-polytope:
{| class=wikitable
|- align=center
|[[File:Cantellated 24-cell verf.png|160px]]<BR>{{CDD|node_1|3|node|4|node_1|3|node}}
|[[File:Cantic snub 24-cell verf.png|160px]]<BR>{{CDD|node_h|3|node_h|4|node_1|3|node}}
|}

===Images===
{{24-cell_Coxeter_plane_graphs|t02|200|t13}}

{| class=wikitable
|+ [[Schlegel diagram]]s
|-
|align=center|[[Image:Cantel_24cell1.png]]<BR>[[Schlegel diagram]]
|align=center|[[Image:Cantel_24cell2.png|240px]]<BR>Showing 24 [[cuboctahedron|cuboctahedra]].
|align=center|[[Image:Cantel_24cell3.png|240px]]<BR>Showing 96 [[triangular prism]]s.
|}

=== Related polytopes ===
The convex hull of two cantellated 24-cells in opposite positions is a nonuniform polychoron composed of 864 cells: 48 [[cuboctahedron|cuboctahedra]], 144 [[square antiprism]]s, 384 [[octahedron|octahedra]] (as triangular antipodiums), 288 [[tetrahedron|tetrahedra]] (as tetragonal disphenoids), and 576 vertices. Its vertex figure is a shape topologically equivalent to a [[cube]] with a [[triangular prism]] attached to one of its square faces.

==Cantitruncated 24-cell==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
!bgcolor=#e7dcc3 colspan=3|Cantitruncated 24-cell
|-
|colspan=3 align=center|[[File:Cantitruncated 24-cell schlegel halfsolid.png|250px]]<BR>[[Schlegel diagram]], centered on truncated cuboctahedron
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2| tr{3,4,3}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]
|colspan=2|{{CDD|node_1|3|node_1|4|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells
|144
|24 [[truncated cuboctahedron|4.6.8]] [[Image:Great rhombicuboctahedron.png|20px]]<BR>96 [[Triangular prism|4.4.3]] [[Image:Triangular prism.png|20px]]<BR>24 [[Truncated cube|3.8.8]] [[Image:Truncated hexahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|720
|192{3}<BR>288{4}<BR>96{6}<BR>144{8}
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|1152
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|576
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[Image:cantitruncated 24-cell verf.png|80px]]<BR>[[Sphenoid (geometry)|sphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
|colspan=2|F<sub>4</sub>, [3,4,3], order 1152
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[bitruncated 24-cell|27]]'' 28 ''[[Runcitruncated 24-cell|29]]''
|}
[[File:Great rhombated icositetrachoron net.png|thumb|right|[[Net (polyhedron)|Net]]]]
The '''cantitruncated 24-cell''' or '''great rhombated icositetrachoron''' is a [[uniform 4-polytope]] derived from the [[24-cell]]. It is bounded by 24 [[truncated cuboctahedron|truncated cuboctahedra]] corresponding with the cells of a 24-cell, 24 [[truncated cube]]s corresponding with the cells of the dual 24-cell, and 96 [[triangular prism]]s corresponding with the edges of the first 24-cell.

===Coordinates===
The [[Cartesian coordinate]]s of a cantitruncated 24-cell having edge length&nbsp;2 are all permutations of coordinates and sign of:

: (1,1+{{radic|2}},1+2{{radic|2}},3+3{{radic|2}})
: (0,2+{{radic|2}},2+2{{radic|2}},2+3{{radic|2}})

The dual configuration has coordinates as all permutations and signs of:
: (1,1+{{radic|2}},1+{{radic|2}},5+2{{radic|2}})
: (1,3+{{radic|2}},3+{{radic|2}},3+2{{radic|2}})
: (2,2+{{radic|2}},2+{{radic|2}},4+2{{radic|2}})

===Projections===

{{24-cell_Coxeter_plane_graphs|t123|240|t0123}}

{| class=wikitable
|+[[Stereographic projection]]
|[[Image:Cantitruncated_24_cell.png|200px]]
|}

== Related polytopes==
{{24-cell_family}}

== References ==
* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: 
** Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}}, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** '''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]
* [[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. 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 = section3.html| title = 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 24, 25}}
* {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes (polychora)}} x3o4x3o - srico, o3x4x3o - grico

{{Polytopes}}
[[Category:Uniform 4-polytopes]]