Johnson solid

The Johnson solids are a set of polyhedra that are strictly convex (that is, all of their dihedral angles are less than 180°) and regular-faced, but that lack the symmetries of the regular or uniform polyhedra. A Johnson solid can be face-transitive, like the triangular bipyramid, though. Johnson solids can only have triangles, squares, pentagons, hexagons, octagons, or decagons as faces, and it has been proven that only 92 of them exist.

A pyramid is formed by connecting point and an n-gon with triangles. The triangular pyramid is a tetrahedron, and is too symmetric to be a Johnson solid. A cupola is formed by connecting an n-gon and a 2n-gon with a band of alternating triangles and squares. Some solids can be "elongated" by augmenting a prism to one of their faces. "Gyroelongation" adds an antiprism to said face instead of a prism. The triangular pyramid cannot be gyroelongated because its faces would be coplanar to those of the antiprism added. Two pyramids, cupolas, or rotundas can be joined together by their "base" polygon. (Joining them by another common face would create a nonconvex polyhedron.) The square bipyramid is not included because it is an octahedron, which is too symmetric to be a Johnson solid.

Bicupolas and birotundas, and the pentagonal cupola-rotunda compound can be aligned in two different ways, with the "top" faces (the ones parallel to the "base") either aligned with one another or 180° out of alignment. When aligned, the compound is called "ortho-," and when out of alignment, the compound is called "gyro-." The gyrobifastigium can be thought of as a "linear gyrobicupola," with the "linear cupola" being analogous to a triangular pyramid turned on its side. The triangular gyrobicupola and the pentagonal gyrobirotunda are not included, because they are the cuboctahedron and the icosidodecahedron, respectively. Elongation and gyroelongation of these compounds places the prism or antiprism in between the two halves. The elongated square orthobicupola is not included because it is a small rhombicuboctahedron. The elongated square gyrobicupola is the only Johnson solid that has only one vertex figure, although it is not vertex-transitive. As before, the triangular bipyramid cannot be gyroelongated because some adjacent faces would be have a 180° dihedral angle. The gyorelongated pentagonal bipyramid is not included because it is a regular icosahedron.