A couple of additional curious properties of the Platonic solids and their faces and vertices:
has 20 faces and 12 vertices.
has 12 faces and 20 vertices.
has 8 faces and 6 vertices.
has 6 faces and 8 vertices.
has 4 faces and 4 vertices.
Each of the pairs above are dual polyhedra, i.e. if one connects the centers of each face of one polyhedron, the other polyhedron of the pair results. Doing this with a tetrahedron yields another tetrahedron.
Imagine a line drawn from any vertex to the center of the solid. If you were to 'slice off' the vertex on a plane perpendicular to the imagined line, without slicing past any of the other vertices, the new face would be the shape of a face of the next solid 'down' in the sequence.
The triangle-faced icosahedron
would yield a pentagon
The pentagon-faced dodecahedron
would yield a triangle
The triangle-faced octahedron
would yield a square
The square-faced cube
would yield a triangle.
The triangle-faced tetrahedron
, oddball that it is, would yield a triangle, since it is the last in the line and sufficient unto itself
Plato associated the tetrahedron, cube, octahedron and icosahedron with fire, earth, air, and water. He linked the dodecahedron with the universe in its entirety, as that "which god used for embroidering the constellations on the whole heaven." Plato's followers were not comfortable with the disjunction between the four elements and the five solids. Aristotle proposed a fifth element (quintessence), the aether (to associate with the dodecahedron) which filled the cosmos and
served as a medium for change and motion (this idea held, in various forms, until the Michelson Morley Experiment in 1887.)
Plato also had a system for explaining the interactions between the four elements by performing various arithmetical operations using the number of faces of the associated solids. For example, heating water produces two 'particles' of air and one of fire, or 20= (2x8)+4 (1 icosahedron=2 octahedra + 1 tetrahedron). Considering that water is H2O, this particular example is oddly prescient.
Information about association of solids with elements from: The Golden Ratio, Mario Livio, Broadway Books, 2002