Editing 24-cell (section)

{{Short description|Regular object in four dimensional geometry}}
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{{Infobox polychoron
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[Schlegel diagram]]<br>(vertices and edges)
 | Type=[[Convex regular 4-polytope]]
 | Last=[[Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[Rectified 24-cell|23]]
 | Schläfli={3,4,3}<br>r{3,3,4} = <math>\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><br>{3<sup>1,1,1</sup>} = <math>\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math>
 | CD={{CDD|node_1|3|node|4|node|3|node}}<br>{{CDD|node|3|node_1|3|node|4|node}} or {{CDD|node_1|split1|nodes|4a|nodea}}<br>{{CDD|node|3|node_1|split1|nodes}} or {{CDD|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[dodecagon]]
 | Coxeter_Group=[[F4 (mathematics)|F<sub>4</sub>]], [3,4,3], order 1152<br>B<sub>4</sub>, [4,3,3], order 384<br>D<sub>4</sub>, [3<sup>1,1,1</sup>], order 192
 | Vertex_Figure=[[Cube]]
 | Dual=[[Polytope#Self-dual polytopes|Self-dual]]
 | Property_List=[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]], [[isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[Net (polyhedron)|Net]]]]
In [[four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[Platonic solid]]) with [[Schläfli symbol]] {3,4,3}. It is also called '''C<sub>24</sub>''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[Octahedron|octahedral]] [[Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[vertex figure]] is a [[cube]]. The 24-cell is [[self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[polygon]] nor a [[simplex]]. The other two are also 4-polytopes, but not convex: the [[grand stellated 120-cell]] and the [[great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[cuboctahedron]] and its dual the [[rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[tesselate]] four-dimensional space face-to-face, forming the [[24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[Parallelohedron|parallelotope]], the simplest one that is not also a [[zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}