Platonic solids, or regular convex polyhedra, are named after the Greek
philosopher Plato who theorized that the five classical elements (Empedocles’
wind, water, fire, and earth, with an added element for spirit) were actually
comprised of regular polyhedra. They are five in number and named for the
number of faces they exhibit. They are: the tetrahedron, the hexahedron, the
octahedron, the dodecahedron, and the icosahedron. Platonic solids have been
the metaphysical and aesthetic inspiration of geometers for thousands of
years. Johannes Kepler, a 17^th^ century German mathematician, astronomer, and
astrologist, detailed a theory in which the relational distances between the
planetary orbits is given by circumscribing the platonic solids within
spheres. “In Mysterium Cosmographicum, published in 1596, Kepler laid out a
model of the solar system in which the five solids were set inside one another
and separated by a series of inscribed and circumscribed spheres. The six
spheres each corresponded to one of the planets (Mercury, Venus, Earth, Mars,
Jupiter, and Saturn). The solids were ordered with the innermost being the
octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and
finally the cube. In this way the structure of the solar system and the
distance relationships between the planets was dictated by the Platonic
solids. In the end, Kepler’s original idea had to be abandoned, but out of his
research came the recognition that the orbits of planets are ellipses rather
than circles, as well as his two laws of orbital dynamics, changing the
courses of physics and astronomy, plus the discovery of the Kepler solids.”^1^
Plato wrote about these polyhedra in the dialogue Timaeus c.360
B.C. in which he associated each of the four classical elements with a regular
solid. Earth was associated with the cube, air with the octahedron, water with
the icosahedron, and fire with the tetrahedron. There was intuitive
justification for these associations: the heat of fire feels sharp and
stabbing (like little tetrahedra). Air is made of the octahedron; its
minuscule components are so smooth that one can barely feel it. Water, the
icosahedron, flows out of one’s hand when picked up, as if it is made of tiny
little balls. By contrast, a highly un-spherical solid, the hexahedron (cube)
represents earth. These clumsy little solids cause dirt to crumble and break
when picked up, in stark difference to the smooth flow of water. Moreover, the
solidity of the Earth was believed to be due to the fact that the cube is the
only regular solid that tesselates Euclidean space. The fifth Platonic solid,
the dodecahedron, Plato obscurely remarks, “…the god used for arranging the
constellations on the whole heaven”. Aristotle added a fifth element, aithêr
(aether in Latin, “ether” in English) and postulated that the heavens were
made of this element, but he had no interest in matching it with Plato’s fifth solid.
Platonic solids occur frequently in nature. Their forms are the complex crystalizations of minerals and appear as the skeletal remains of several species of …
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