The 16-cell beta_4 is the finite regular four-dimensional cross polytope with Schläfli symbol {3,3,4}. 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 sqrt(2) are the permutations of (+/-1, 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 Ci_8(1,2,3), and complete 4-partite graph K_(4×2). 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 (-2)^30^46^1, and so is an integral graph. The 16-cell graph has cycle polynomial


(OEIS A167982).

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.

The 16-cell has


distinct nets (Buekenhout and Parker 1998). The order of the automorphism group is |Aut(G)|=2^7·3=384 (Buekenhout and Parker 1998).

See also

11-Cell, 24-Cell, 57-Cell, 120-Cell, 600-Cell, Cell, Cross Polytope, Hypercube, Pentatope, Polychoron, Polytope, Tesseract

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Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension <=4." Disc. Math. 186, 69-94, 1998.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 121-122, 156, 158, and 243, 1973.Sloane, N. J. A. Sequence A167982 in "The On-Line Encyclopedia of Integer Sequences."Weimholt, A. "16-Cell Foldout.", D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991.

Cite this as:

Weisstein, Eric W. "16-Cell." From MathWorld--A Wolfram Web Resource.

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