16-Cell

The 16-cell is the finite regular four-dimensional cross polytope with Schläfli symbol . It is also known as the hyperoctahedron (Buekenhout and Parker 1998) or hexadecachoron, and its composed of 16 tetrahedra, with 4 to an edge. It has 8 vertices, 24 edges, and 32 faces. It is one of the six regular polychora.

The 16-cell is a four-dimensional dipyramid based on the three-dimensional square dipyramid with its two apices in opposite directions along the fourth dimension (Coxeter 1973, p. 121).

The 16-cell is the dual of the tesseract.

The vertices of the 16-cell with circumradius 1 and edge length are the permutations of (, 0, 0, 0) (Coxeter 1969, p. 403). There are 2 distinct nonzero distances between vertices of the 16-cell in 4-space.

The skeleton of the 16-cell, illustrated above in a number of embeddings, is isomorphic to the 4-cocktail party graph, circulant graph , and complete 4-partite graph . It is a 6-regular graph of girth 3 and diameter 2. It is a 6-regular graph of girth 3 and diameter 2. It has graph spectrum , and so is an integral graph. The 16-cell graph has cycle polynomial

(OEIS A167982).

The 16-cell graph is one of the four smallest simple graphs for which , where is the rectilinear crossing number and is the (unrestricted) crossing number. Minimal crossing and rectilinear crossing embeddings are illustrated above.

The skeleton of the 16-cell is the graph square of the cubical graph and graph cube of the cycle graph .

The skeleton of the 16-cell is implemented in the Wolfram Language as GraphData["SixteenCellGraph"]. When embedded in three-space, the 16-cell skeleton is a cube with an "X" connecting diagonally opposite vertices on each face (and therefore could be considered a "crossed cube" graph).

The 16-cell has

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