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 polyhedra. 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 polyhedra.
Every regular polyhedron has 
axes of symmetry, where 
is the number of polyhedron
edges, and 
planes of symmetry, where 
is the number of sides of the corresponding Petrie
polygon.
