- 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
- sphenocorona_(J86)
- sphenomegacorona_(J88)
- hebesphenomegacorona_(J89)
- disphenocingulum (J90)
- bilunabirotunda (J91)
- triangular hebesphenorotunda (J92) (illustrated at right)

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 aExplore the list of allluneas a complex of two triangles attached to opposite sides of a square, the prefixspheno-refers to a wedgelike complex formed by two adjacent lunes. The prefixdispheno-denotes two such complexes, whilehebespheno-indicates a blunter complex of two lunes separated by a third lune. The suffix-coronarefers to a crownlike complex of eight triangles, and-megacorona, to a larger such complex of 12 triangles. The suffix-cingulumindicates a belt of 12 triangles.

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

**Answer: ***Only the ones listed on this
page.*

**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.*

**Answer: **By my count, these.

References: Norman Johnson [1966] published a complete listing of these solids, gave them the names used here, and conjectured that there were no others. Zalgaller [1969] gives a proof that this list is indeed complete. The paper by Martin Berman [1971] gives nets and photographs for making paper models of all 92, and the book by Pugh [1976] has a chapter describing them.