24-cell

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.

24-cell
Schlegel diagram (vertices and edges)
Type	Convex regular 4-polytope
Schläfli symbol	{3,4,3} r{3,3,4} = ${\displaystyle \left\{{\begin{array}{l}3\\3,4\end{array}}\right\}}$ {31,1,1} = ${\displaystyle \left\{{\begin{array}{l}3\\3\\3\end{array}}\right\}}$
Coxeter diagram	or or
Cells	24 {3,4}
Faces	96 {3}
Edges	96
Vertices	24
Vertex figure	Cube
Petrie polygon	dodecagon
Coxeter group	F4, [3,4,3], order 1152 B4, [4,3,3], order 384 D4, [31,1,1], order 192
Dual	Self-dual
Properties	convex, isogonal, isotoxal, isohedral
Uniform index	22

The boundary of the 24-cell is composed of 24 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. The 24-cell and the tesseract are the only convex regular 4-polytopes in which the edge length equals the radius.

The 24-cell does not have a regular analogue in 3 dimensions. It is the only one of the six convex regular 4-polytopes which is not the four-dimensional analogue of one of the five regular Platonic solids. It is the unique regular polytope, in any number of dimensions, which has no regular analogue in the adjacent dimension, either below or above. However, it can be seen as the analogue of a pair of irregular solids: the cuboctahedron and its dual the rhombic dodecahedron.

Translated copies of the 24-cell can tile 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 parallelotope, the simplest one that is not also a zonotope.