Short answers, pulled from the story.
The five Platonic solids are the tetrahedron (four triangular faces), the cube (six square faces), the octahedron (eight triangular faces), the dodecahedron (twelve pentagonal faces), and the icosahedron (twenty triangular faces). Each is a convex, regular polyhedron in which all faces are congruent regular polygons and the same number of faces meet at every vertex.
The solids are named for the ancient Greek philosopher Plato, who wrote about them in his dialogue Timaeus around 360 B.C. Plato associated each of the four classical elements with one of the solids and proposed that the fifth, the dodecahedron, was used by the god for arranging the constellations on the whole heaven.
Only five convex regular polyhedra exist because the interior angles at every vertex must sum to less than 360 degrees for the shape to close into a solid rather than tile a flat plane. Triangular faces allow three, four, or five per vertex; square faces allow only three; pentagonal faces allow only three. Hexagonal and larger faces cannot form a solid vertex at all. These constraints yield exactly five valid combinations.
Theaetetus, a contemporary of Plato, is credited with the first known proof that no other convex regular polyhedra exist and with mathematical descriptions of all five solids. Euclid later formalized the construction and proof in Book XIII of the Elements, and Proposition 18 of that book argues that the five are exhaustive.
In Mysterium Cosmographicum, published in 1596, Kepler proposed that the five Platonic solids nested inside one another could explain the orbital distances of the six planets then known. Each adjacent pair of planetary spheres was separated by one of the solids, ordered from innermost to outermost as octahedron, icosahedron, dodecahedron, tetrahedron, and cube. Though the model was abandoned, the research led Kepler to his three laws of orbital dynamics.
Yes. The tetrahedron, cube, and octahedron occur in crystal structures. Allotropes of boron and compounds such as boron carbide incorporate discrete icosahedral clusters of twelve boron atoms. Many viruses, including the herpes virus, take the icosahedral form because it can be assembled from identical protein subunits. Dan Shechtman discovered icosahedral structure in aluminum in 1984, work recognized with the Nobel Prize in Chemistry in 2011.