Plato held the view that mathematical objects really existed so that they are discovered by mathematicians (in the same way that new continents are discovered by explorers) rather than invented. Plato believed that mathematics provided the best training for thinking about science and philosophy. The five regular solids are named Platonic Solids today after Plato.
Of the 5 solids, the tetrahedron has the smallest volume for its surface area and the icosahedron the largest; they therefore show the properties of dryness and wetness respectively and so correspond to Fire and Water. The cube, standing firmly on its base, corresponds to the stable Earth but the octahedron which rotates freely when held by two opposite vertices, corresponds to the mobile Air. The dodecahedron corresponds to the Universe because the zodiac has 12 signs (the constellations of stars that the sun passes through in the course of one year) corresponding to the 12 faces of the dodecahedron.
“To earth, then, let us assign the cubic form, for earth is the most immovable of the four and the most plastic of all bodies, and that which has the most stable bases must of necessity be of such a nature. Now, of the triangles which we assumed at first, that which has two equal sides is by nature more firmly based than that which has unequal sides, and of the compound figures which are formed out of either, the plane equilateral quadrangle has necessarily a more stable basis than the equilateral triangle, both in the whole and in the parts. Wherefore, in assigning this figure to earth, we adhere to probability, and to water we assign that one of the remaining forms which is the least movable, and the most movable of them to fire, and to air that which is intermediate. Also we assign the smallest body to fire, and the greatest to water, and the intermediate in size to air, and, again, the acutest body to fire, and the next in acuteness to air, and the third to water. Of all these elements, that which has the fewest bases must necessarily be the most movable, for it must be the acutest and most penetrating in every way, and also the lightest as being composed of the smallest number of similar particles, and the second body has similar properties in a second degree, and the third body, in the third degree. Let it be agreed, then, both according to strict reason and according to probability, that the pyramid is the solid which is the original element and seed of fire, and let us assign the element which was next in the order of generation to air, and the third to water.We must imagine all these to be so small that no single particle of any of the four kinds is seen by us on account of their smallness, but when many of them are collected together, their aggregates are seen. And the ratios of their numbers, motions, and other properties, everywhere God, as far as necessity allowed or gave consent, has exactly perfected and harmonized in due proportion.“
