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 circulant graph , which
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
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The skeleton of the 16-cell is implemented in Mathematica 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.
The 16-cell has
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distinct nets (Buekenhout and Parker 1998). The order of the automorphism group is
(Buekenhout and Parker 1998).