Classification of Convex Polyhedra whose Faces are Regular Polygons
Theorem
The convex polyhedra whose faces are all regular polygons are as follows:
- The $5$ Platonic solids
- The square-sided prisms and regular antiprisms, countably infinite in number
- The $13$ Archimedean polyhedra
- The $92$ Johnson polyhedra.
Proof
The Platonic solids are the convex polyhedra all of whose faces are congruent and whose vertices are regular.
From Five Platonic Solids, there are $5$ of these.
The square-sided prisms are made from two regular polygons of an arbitrary number of sides forming the bases, separated by as many squares as there are sides of the two bases, forming the lateral faces.
From Square-Sided Prisms are Countably Infinite, they form a countably infinite set.
Note that when the bases are themselves squares, the square-sided prism is then a cube, and so has already been counted amongst the Platonic solids.
The regular antiprisms are made from two regular polygons of an arbitrary number of sides forming the bases, separated by twice as many equilateral triangles as there are sides of the two bases, forming the lateral faces.
From Regular Antiprisms are Countably Infinite, they form a countably infinite set.
Note that when the bases are themselves equilateral triangles, the regular antiprism is then a regular octahedron, and so has already been counted amongst the Platonic solids.
The Archimedean polyhedra consist of all isogonal convex polyhedra whose faces are all regular polygons, excluding:
- the Platonic solids
- the square-sided prisms
- the regular antiprisms.
From Thirteen Archimedean Polyhedra, there are $13$ of these.
The Johnson polyhedra are all the other convex polyhedra whose faces are all regular polygons.
From 92 Johnson Polyhedra, there are $92$ of these.
$\blacksquare$
Historical Note
The final step of this result was demonstrated by Norman Woodason Johnson, who first enumerated what are now known as the Johnson polyhedra in $1966$.
He conjectured, but did not prove, that there were no more than $92$ of them.
In $1969$ Victor Abramovich Zalgaller proved that Johnson's list was complete.
Sources
- 1966: Norman W. Johnson: Convex polyhedra with regular faces (Canad. J. Math. Vol. 18: pp. 169 – 200)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $13$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $13$