# Pyramids, Dipyramids, and Trapezohedra

A particularly popular polyhedron is the pyramid. If we restrict ourselves to regular polygons for faces, there are three possible pyramids: the triangle-based tetrahedron, the square pyramid, and the pentagonal pyramid. Being bounded by regular polygons, these last two fall within the class of Johnson solids. One interesting property of pyramids is that like the tetrahedron, their duals are also pyramids. (Incidentally, the Egyptian pyramids have square bases but the triangular side faces are not quite equilateral; they are very close to half a golden rhombus.)

Joining two such pyramids at their bases gives a set of three dipyramids, in which all faces are equilateral triangles: the triangular dipyramid, the octahedron (square dipyramid), and the pentagonal dipyramid. These fall within the class of convex equilateral deltahedra.

There are two geometrically distinct classes of dipyramids which are not always distinguished terminologically. The above could be termed equilateral dipyramids because their faces are equilateral triangles. The other class of dypyramids are the duals to the prisms. These form an infinite set and are not equilateral except for the dual to the cube, the octahedron, which falls in both classes of dipyramids. As an example, here is the compound of the hexagonal prism and its dual hexagonal dipyramid, clearly showing their dual relationship and the fact that the triangles are not equilateral.

Parallel to the prisms and their dual dipyramids, there is an infinite series of antiprisms and their duals, called trapezohedra. This is a confusing nomenclature because the faces of these polyhedra are not trapezoids in the modern sense of the word. The faces are "kite shaped" with two sets of two equal adjacent faces. As an example, here is the hexagonal antiprism, the dual hexagonal trapezohedron, and their compound.

Here is a list of pyramids, dipyramids and trapezohedra based on triangular up to decagonal prisms and antiprisms.