There are 4 main groups of solids, they are as follows, The Platonic Solids, The Kepler Poinsot, The Archimedean Solids, and The Archimedean Duals (Catalan Solids).
In the Platonic group there are 5 shapes, these shapes were discovered by the Ancient Greeks, in their love for geometry. The Ancient Greeks believied that the five platonic solids were related/connected to earth, air, fire, water and aether (mysiterious substance, the poetic personification of the clear upper air breathed by the Olympians). The way in which they were believed to be connected, we do not know, but the Ancient Greeks called them the atoms of the universe.
In the Kepler Poinsot group there are 4 shapes, these shapes were discovered by Kepler Poinsot an Austrian mathematician and astronomer. The Kepler Poinsot soilds are stellations of the Platonic Solids. Click Here for more informaton on stellating a solid.
In the Archimedean group there are 13 shapes, these shapes were discovered by Archimedes an Greek mathematition, born in Syracuse, Sicly 287 BC. Archimedes was killed by a Roman soldier at the age of 75, in the Second Punic War 212 BC. Apparently he did not notice the pressence of the soldier, as he was so absorded in a theoretical problem. He merely glanced up, and told the solier not to disturb his diagrams, on which he was killed by means of a sword. He will always be regarded as one of the greatest mathemations ever.
In the Archimedean Duals group there are 13 shapes, these shapes were first published by a Belgium mathematition in 1862. In honour of this they were also named the Catalan Solids.
So what is a dual? Every shape has its own dual polyhedra, where they share the same edges but have opposite faces and verticies. For example the dual of a cube is the octahedron, the cube having 6 faces, 8 verticies and 12 edges, and the octahedron having 8 faces, 6 verticies and 12 edges. Some shapes don't actually have a dual, for example the tetrahedron, these are called self duals. Duals can sometimes be known as reciprocal shapes, and taking the dual is called reciprocation.
There are also other groups of Polyhedra they are as follows: Prisms and Anti-Prisms, Quasi-Regular Polyhedra, Johnson Solids, Pyramids, Dipyramids, Convex Deltahedra, Zonohedra and Uniform Polyhedra.
Edward Holmes, SMC @ CCMS, email@example.com