Kepler discovered these laws while trying to achieve the Pythagorean? purpose of finding the harmony of the [celestial spheres]?. In his cosmovision, it was not a coincidence that the number of perfect polyhedra was equal to the number of known planets. Having embraced the [Copernican system]?, he set out to prove that the distances from the planets to the sun where given by spheres inside perfect polyedra inside spheres. He thereby identified the five platonic solids with the five planets - Mercury, Venus, Mars, Jupiter, Saturn and the five classical elements. (The Earth, moon and sun were not considered to be planets).
To work his model out, he relied on the meticulous observations of Tycho Brahe.
In 1596 Kepler published The Cosmic Mystery . Here is an selection explaining the relation between the planets and the platonic solids:
To emphasize his theory, Kepler envisaged an impressive model of the universe which shows a cube, inside a sphere, with a tetrahedron inscribed in it, another sphere inside it with a dodecahedron inscribed, a sphere with an icosahedron inscribed inside, and finally a sphere with an octahedron inscribed. Each of these celestial spheres had a planet embedded within them, and thus defined the planet's orbit.
In his 1619 book, Harmonice Mundi, as well as the treatise Misterium Cosmographicum, he also made an association between the Platonic solids with the classical conception of the elements: The tetrahedron was the form of fire, the octahedron was that of air, the cube was earth, the icosahedron was water, and the dodecahedron was the cosmos as a whole or ether. There is some evidence this association was of ancient origin, as Plato relates one Timaeus of Locri who thought of the Universe as being enveloped by a gigantic dodecahedron while the other four solids represent the "elements" of fire, air, earth, and water.
Since he was the first to recognize the non-convex regular solids (such as the stellated dodecahdrons), they are named Kepler solids in his honor.
He also made fundamental investigations into combinatorics, geometrical optimization, and natural phenomena such as snowflake?s, always with an emphasis on form and design. He was also notable for defining antiprisms.