dipyramids

A dipyramid is a convex polyhedron formed by joining two congruent pyramids base to base along a common base polygon. The base has n sides, and the resulting solid is called the n-dipyramid or bipyramid over that base.

In general, a dipyramid has two apex vertices and the n base vertices, for a total of V = n + 2. The edges number E = 3n (the n base edges plus n edges from each apex to all base vertices), and the faces are F = 2n triangular faces. If the two apexes lie symmetrically above and below the base plane, the shape has a vertical axis of symmetry and the symmetry group depends on the base polygon, typically denoted by D_nd or D_nd.

Examples help illustrate the family. For n = 3, the triangular dipyramid can be formed by gluing two regular tetrahedra along a common triangular face, yielding a 6-face polyhedron. For n = 4, the square bipyramid is the octahedron, with 8 triangular faces. For n = 5, the pentagonal dipyramid has 10 triangular faces. In general, when the base is a regular polygon and the apex distances are chosen so that the resulting edges are congruent, the dipyramid forms a deltahedron for certain n.

Relation to prisms: a dipyramid is the dual polyhedron of the corresponding prism; the dual of an n-prism is the n-dipyramid. The term bipyramid is often used interchangeably with dipyramid. In chemistry and crystallography, dipyramidal describes two apex ligands arranged about a central atom.