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 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).