24-cell

{{Infobox polychoron | Name=24-cell | Image_File=Schlegel wireframe 24-cell.png | Image_Caption=Schlegel diagram(vertices and edges) | Type=Convex regular 4-polytope | Last=21 | Index=22 | Next=23 | Schläfli={3,4,3}r{3,3,4} = \left\{\begin{array}{l}3\\3,4\end{array}\right\}{31,1,1} = \left\{\begin{array}{l}3\\3\\3\end{array}\right\} | CD= or or | Cell_List=24 {3,4} 20px | Face_List=96 {3} | Edge_Count=96 | Vertex_Count= 24 | Petrie_Polygon=dodecagon | Coxeter_Group=F4, [3,4,3], order 1152B4, [4,3,3], order 384D4, [31,1,1], order 192 | Vertex_Figure=Cube | Dual=Self-dual | Property_List=convex, isogonal, isotoxal, isohedral }} In four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.

== Geometric description == The 24-cell is a convex four-dimensional polytope boundary, an analogy of nearly the cuboctahedron and its dual the rhombic dodecahedron in four dimensions. It is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together, it has 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. Like other four-dimensional regular polytope, 5-cell, the 24-cell is self-dual. The 24-cell and the tesseract are the only convex regular 4-polytopes in which the edge length equals the radius. It has Schläfli symbol \{3,4,3\}.