24-cell

In mathematics, the 24-cell is the 4-dimensional convex regular polytope with 24 facets. Its Schläfli symbol is {3,4,3}.

The 24 facets are octahedral, and six meet at each vertex. The number of vertices is also 24, as the 24-cell is self-dual.

Vertices of a 24-cell centred at the origin of 4-space, with edges of length 1, can be given as follows: 16 vertices of the form (±½,±½,±½,±½), and 8 vertices obtained from (0,0,0,±1) by permuting coordinates. (Note that the first 16 vertices are the vertices of a tesseract, and the other 8 are the vertices of the dual of the tesseract. An analogous construction in 3-space gives the rhombic dodecahedron, which, however, is not regular.) These 24 vectors generate a lattice in R⁴. If we interpret the vectors as quaternions, then the lattice is closed under multiplication and is therefore a ring.

The symmetry group of the 24-cell is the Weyl group of F₄. This is a solvable group of order 1152.