A polyhedron is said to be regular if its Faces and Vertex Figures are Regular (not necessarily Convex) polygons (Coxeter 1973, p. 16). Using this definition, there are a total of nine regular polyhedra, five being the Convex Platonic Solids and four being the Concave (stellated) Kepler-Poinsot Solids. However, the term ``regular polyhedra'' is sometimes used to refer exclusively to the Convex Platonic Solids.

It can be proven that only nine regular solids (in the Coxeter sense) exist by noting that a possible
regular polyhedron must satisfy

Gordon showed that the only solutions to

of the form are the permutations of and . This gives three permutations of (3, 3, 4) and six of (3, 5, ) as possible solutions to the first equation. Plugging back in gives the Schläfli Symbols of possible regular polyhedra as , , , , , , , , and (Coxeter 1973, pp. 107-109). The first five of these are the Platonic Solids and the remaining four the Kepler-Poinsot Solids.

Every regular polyhedron has axes of symmetry, where is the number of Edges, and Planes of symmetry, where is the number of sides of the corresponding Petrie Polygon.

**References**

Coxeter, H. S. M. ``Regular and Semi-Regular Polytopes I.'' *Math. Z.* **46**, 380-407, 1940.

Coxeter, H. S. M. *Regular Polytopes, 3rd ed.* New York: Dover, pp. 1-17, 93, and 107-112, 1973.

Cromwell, P. R. *Polyhedra*. New York: Cambridge University Press, pp. 85-86, 1997.

© 1996-9

1999-05-25