# Johnson Solids

An interesting problem is to find the set of all convex polyhedra with regular faces. This includes the Platonic solids, the Archimedean solids, the prisms and anti-prisms as special symmetric cases. It also includes equilateral deltahedra and dipyramids and any irregular convex solid you can make by sticking together triangles, squares, pentagons, etc., in any haphazard manner which happens to close. The ones which are not Platonic, Archimedean, prisms or antiprisms are called the Johnson Solids. A rather difficult problem is how to find them all and be sure not to miss any or include any twice. It turns out there are 92.

Some of the simpler Johnson solids are assemblages of pyramids, prisms, and antiprisms, e.g., a gyrobifastigium is just two triangular prisms glued together with a twist.  Others are fragments of Archimedean solids, e.g., a pentagonal rotunda is half an icosidodecahedron and pentagonal cupola is a part of a rhombicosidodecahedron.  These can be assembled in various convex ways, e.g., to make a pentagonal orthocupolarontunda or a pentagonal gyrocupolarotunda.  They can also be stretched out with a prism to make an elongated pentagonal orthocupolarotunda or with an antiprism to make a gyroelongated pentagonal cupolarotunda.  Most of the Johnson solids can be understood as assemblages of simple pieces along these lines.

Several Johnson solids are elementary, in the sense that they can not be made as assemblages of fragments of well-known polyhedra.  The snub disphenoid is one example. Some of these may be somewhat surprising and one would expect they would be overlooked in a casual attempt to enumerate all these polyhedra, e.g., the

They certainly have some great names which one can try to drop in a casual conversation!  From the models, you should be able to determine what rotunda, cupola, elongated- and gyroelongated-, augmented- and diminished- each mean.  On the remaining terminology, Johnson writes:

If we define a lune as a complex of two triangles attached to opposite sides of a square, the prefix spheno- refers to a wedgelike complex formed by two adjacent lunes.  The prefix dispheno- denotes two such complexes, while hebespheno- indicates a blunter complex of two lunes separated by a third lune.  The suffix -corona refers to a crownlike complex of eight triangles, and -megacorona, to a larger such complex of 12 triangles.  The suffix -cingulum indicates a belt of 12 triangles.
Explore the list of all 92 Johnson solids. If you are going to go through the whole set of 92 to try to get to know them, you might find it helpful to think about one or more of the following three questions as you look at each:

Exercise: Look through the entire set of Johnson solids and determine which of them are made from only a single type of polygon.

Exercise: Look through the entire set of Johnson solids and determine which of them are chiral.

Answer: Surprisingly few are chiral: Only these.

Exercise: Look through the entire set of Johnson solids and determine which of them have more than one axis of symmetry.