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16-Cell

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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.

16CellGraphs

The skeleton of the 16-cell, illustrated above in a number of embeddings, is isomorphic to the circulant graph Ci_8(1,2,3), 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 (-2)^30^46^1, and so is an integral graph. The 16-cell graph has cycle polynomial

 C(x)=744x^8+960x^7+640x^6+288x^5+102x^4+32x^3.

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

 2^5(2^73^3+1+3^2)=110912

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

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