24-cell

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

In 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, icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid,octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.

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. In fact, the 24-cell is the unique convex self-dual regular Euclidean polytope which is neither a polygon nor a simplex. Due to this singular property, it does not have a good analogue in 3 dimensions.

A 24-cell is given as the convex hull of its vertices. The vertices of a 24-cell centered at the origin of 4-space, with edges of length 1, can be given as follows: 8 vertices obtained by permuting

and 16 vertices of the form

The first 8 vertices are the vertices of a regular 16-cell and the other 16 are the vertices of the dual tesseract. This gives a construction equivalent to cutting a tesseract into 8 cubical pyramids, and then attaching them to the facets of a second tesseract. This is equivalent to the dual of a rectified 16-cell. The analogous construction in 3-space gives the rhombic dodecahedron which, however, is not regular.

We can further divide the last 16 vertices into two groups: those with an even number of minus (−) signs and those with an odd number. Each of groups of 8 vertices also define a regular 16-cell. The vertices of the 24-cell can then be grouped into three sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.