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 is 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.
For 
, 1, 2, 3, the number of 
-dimensional faces of the 16-cell is
 |
giving the face vector 
.
Thus its face numbers are scaled binomial coefficients,
connecting them to Pascal's triangle and the
binomial theorem.
The 16-cell has
 |
distinct nets (Buekenhout and Parker 1998). The order of the automorphism group is 
(Buekenhout and Parker 1998).
Its skeleton is the 16-cell graph.
." Disc. Math. 186, 69-94, 1998.Coxeter,
H. S. M.