Regular Polyhedron
A polyhedron is said to be regular if its Faces and Vertex Figures areRegular (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 fourbeing the Concave (stellated) Kepler-Poinsot Solids. However, the term ``regularpolyhedra'' 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 possibleregular 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 tothe 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 theremaining 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.
- Hardy-Littlewood k-Tuple Conjecture
- Hardy-Littlewood Tauberian Theorem
- Hardy-Ramanujan Number
- Hardy-Ramanujan Theorem
- Hardy's Inequality
- Hardy's Rule
- Harmonic Addition Theorem
- Harmonic Analysis
- Harmonic Brick
- Harmonic Conjugate Function
- Harmonic Conjugate Points
- Harmonic Coordinates
- Harmonic Decomposition
- Harmonic Divisor Number
- Harmonic Equation
- Harmonic Function
- Harmonic-Geometric Mean
- Harmonic Homology
- Harmonic Logarithm
- Harmonic Map
- Harmonic Mean
- Harmonic Mean Index
- Harmonic Number
- Harmonic Progression
- Harmonic Range