Regular Polyhedron

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

$$\cos^2\left({\pi\over p}\right)+\cos^2\left({\pi\over q}\right)+\cos^2\left({\pi\over r}\right)=1.$$

Gordon showed that the only solutions to

$$1+\cos\phi_1+\cos\phi_2+\cos\phi_3=0$$

of the form $\phi_i = \pi m_i/n_i$ are the permutations of $({\textstyle{2\over 3}} \pi, {\textstyle{2\over 3}} \pi, {\textstyle{1\over 3}}\pi)$ and $({\textstyle{2\over 3}}\pi, {\textstyle{2\over 5}}\pi, {\textstyle{4\over 5}}\pi)$. This gives three permutations of (3, 3, 4) and six of (3, 5, ${\textstyle{5\over 3}}$) as possible solutions to the first equation. Plugging back in gives the Schläfli Symbols of possible regular polyhedra as $\{3, 3\}$, $\{3, 4\}$, $\{4, 3\}$, $\{3, 5\}$, $\{5, 3\}$, $\{3, {\textstyle{5\over 2}}\}$, $\{{\textstyle{5\over 2}}, 3\}$, $\{5,{\textstyle{5\over 2}}\}$, and $\{{\textstyle{5\over 2}}, 5\}$ (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 $e+1$ axes of symmetry, where $e$ is the number of Edges, and $3h/2$ Planes of symmetry, where $h$ is the number of sides of the corresponding Petrie Polygon.

See also Convex Polyhedron, Kepler-Poinsot Solid, Petrie Polygon, Platonic Solid, Polyhedron, Polyhedron Compound, Sponge, Vertex Figure

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.