Bipyramid

A bipyramid is a polyhedron formed by joining two congruent pyramids along a common base polygon. The two apexes lie on opposite sides of the base plane and are connected to every vertex of the base. When the base is a regular polygon, the shape is often called an n-gonal bipyramid or dipyramid.

If the base has n sides, the bipyramid has V = n + 2 vertices, E = 3n edges, and F = 2n triangular faces. All faces are triangles, and the base polygon itself is not a face of the solid. The polyhedron is convex when the apexes are on opposite sides of the base and properly positioned relative to the base.

A regular bipyramid is the dual polyhedron of an n-gonal prism; conversely, the prism is the dual of the bipyramid. The symmetry of a regular n-gonal bipyramid is the dihedral group corresponding to the base, with an axis through the two apexes and reflections in planes containing the axis.

Examples include the triangular bipyramid, which is the regular octahedron: it has eight triangular faces and all edges equal when the construction allows. For n greater than 3, it is not possible to have all side faces equilateral triangles with two congruent apexes; the side triangles are typically isosceles. Nonetheless, such shapes remain standard examples of bipyramids and are used to illustrate dual relationships with prisms and to study properties of triangular faces in polyhedra.