24-Cell

The 24-cell is a finite regular four-dimensional polytope with Schläfli symbol . It is also known as the hyperdiamond or icositetrachoron, and is composed of 24 octahedra, with 3 to an edge. The 24-cell has 24 vertices and 96 edges. It is one of the six regular polychora.

The 24-cell is self-dual, and is the unique regular convex polychoron which has no direct three-dimensional analog.

The vertices of the 24-cell with circumradius and edge length are given by the permutations of Coxeter (1969, p. 404). There are 4 distinct nonzero distances between vertices of the 24-cell in 4-space.

The 96 edges of the 24-cell can be partitioned into three tesseracts, as illustrated above.

The even coefficients of the lattice are 1, 24, 24, 96, ... (OEIS A004011), and the 24 shortest vectors in this lattice form the 24-cell (Coxeter 1973, Conway and Sloane 1993, Sloane and Plouffe 1995).

The skeleton of the 24-cell is an 8-regular graph of girth 3 and diameter 3. It is also an integral graph with graph spectrum (Buekenhout and Parker 1998). The skeleton of the 24-cell is implemented in the Wolfram Language as GraphData["TwentyFourCellGraph"], illustrated above in three projective embeddings and three order-3 LCF embeddings.

The 24-cell has

distinct nets (Buekenhout and Parker 1998). The order of its automorphism group is (Buekenhout and Parker 1998).

One construction for the 24-cell evokes comparison with the rhombic dodecahedron. Given two equal cubes, we construct this dodecahedron by cutting one cube into six congruent square pyramids, and attaching these to the six squares bounding the other cube. Similarly, given two equal tesseracts, the 24-cell can be constructed by cutting one tesseract into eight congruent cubic pyramids, and attaching these to the eight cubes bounding the other tesseract.