On Johnson solids: three-dimensional geometric objects with regular faces
Johnson solids are polyhedra bounded exclusively by regular polygons that do not fall into the categories of Platonic solids, Archimedean solids, prisms, or antiprisms. There are exactly 92 distinct Johnson solids. Those were first described by the American mathematician Norman Johnson in 1966. In 1969, Victor Zalgaller proved that this list was complete, meaning no other polyhedra possess this specific set of properties. The proof is complicated and is based on an exhaustive examination of all possible ways to assemble regular polygons into convex polyhedra.
Johnson solids are designated by the letter J followed by a number, J1 through J92. Johnson solids are always convex, many of them possess only low symmetry. There is no general construction rule for these shapes.
Johnson solids can be divided into two groups: elementary polyhedra and composite polyhedra. Composite polyhedra can be decomposed into elementary polyhedra. Elementary polyhedra cannot be further subdivided into polyhedra that are bounded solely by regular polygons. Examples of elementary Johnson solids include the first six forms, Johnson pyramids, cupolae, and rotundae, as well as complex forms found later in the list, such as the sphenocorona and the disphenocingulum. Composite solids include elongated pyramids and elongated bipyramids, which can be assembled from Johnson pyramids and prisms. An antiprism structure appears, for instance, in the gyroelongated square dipyramid.
Another subgroup consists of deltahedra, polyhedra bounded exclusively by equilateral triangles, such as the gyroelongated square bipyramid and the snub disphenoid. Other subgroups can also be identified, such as solids possessing a circumsphere, like the gyroelongated pentagonal pyramid J11.