dipyramid

A dipyramid, also called an n-gonal bipyramid, is a polyhedron formed by joining two congruent pyramids along a common base. The base is an n-gon; the two apexes lie on opposite sides of the base plane. The surface consists of 2n triangular faces.

Combinatorially, a dipyramid has V = n + 2 vertices, E = 3n edges, and F = 2n faces. The two apices have degree n, and the n base vertices each have degree 4. When the base is a square (n = 4) and the edge lengths are chosen appropriately, the dipyramid becomes the regular octahedron, a Platonic solid. In general, a dipyramid is not regular unless specific height conditions are met; the triangular dipyramid (n = 3) is a simple convex deltahedron that can be formed with six triangular faces.

The n-gonal dipyramid is the dual polyhedron of the n-gonal prism. Construction can be viewed as placing vertices above and below a regular n-gon and connecting each apex to all base vertices, producing a symmetrical, two-pitched shape.

In summary, a dipyramid is the two-pyramid composite over an n-gonal base, yielding a family of solids with 2n triangular faces, governed by the base polygon’s number of sides and the relative height of the apices. It is a fundamental example in the study of bipyramids and dual polyhedra.