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 exactly two minimally curvy graphs of graph order 8. It has , where is the rectilinear crossing number and is the unrestricted graph crossing number. Minimal crossing and rectilinear crossing embeddings are illustrated above.

Its projective plane crossing number is 4.

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).