The five (and only five!) Platonic solids are "perfect" since they are regular solids (or volumes) that are made of identical faces and that have identical vertices (or corners). The faces themselves are regular, i.e. they have edges of identical length which form identical angles. The vertices of the Platonic solids touch a sphere (the circumscribed sphere) and inside they hold a smaller sphere that touches each of the faces from within (the inscribed sphere).
So: the tetrahedron is made of 4 identical regular triangles, the cube (which could also be called the hexahedron) is made of 6 identical squares, the octahedron of 8 identical regular triangles, the dodecahedron of 12 identical regular pentagons and the icosahedron of 20 identical regular triangles.
Because of their perfectness, Plato was very fascinated by them (Exhibit Nr. 19). For him, they represented the perfect ideas that formed reality. He associated these forms to the four elements: air, water, fire and earth. The fact that there were five perfect forms and only four elements was a bit troublesome (see Cat. Nr. 104). Aristotle talked about the ether as the fifth element that pervades the Universe, but he was no longer interested to associate the elements with perfect geometrical forms.
Much later Johannes Kepler (1571 - 1630) would still try to match the orbits of the then known planets (Mercury, Venus, Mars, Jupiter and Saturnus) with circumscribed or inscribed spheres of the five Platonic solids nested into one another. He eventually gave up, and in fact concluded, rightly, that the orbits of the planets were not circles after all, but ellipses!
Seeing the Platonic solids, seeing their perfectness or even a clumsy attempt to build them, still stir awe. They still give a feeling that, somehow, they might have something to do with the interconnectedness of all things.
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