Kepler claimed to have had an epiphany…
July 1595 CE
Kepler claimed to have had an epiphany on July 19, 1595, while teaching in Graz, demonstrating the periodic conjunction of Saturn and Jupiter in the zodiac; he realized that regular polygons bound one inscribed and one circumscribed circle at definite ratios, which, he reasons, might be the geometrical basis of the universe.
After failing to find a unique arrangement of polygons that fits known astronomical observations (even with extra planets added to the system), Kepler has begun experimenting with three-dimensional polyhedra.
He finds that each of the five Platonic solids could be uniquely inscribed and circumscribed by spherical orbs; nesting these solids, each encased in a sphere, within one another would produce six layers, corresponding to the six known planets—Mercury, Venus, Earth, Mars, Jupiter, and Saturn.
By ordering the solids correctly—octahedron, icosahedron, dodecahedron, tetrahedron, cube—Kepler finds that the spheres can be placed at intervals corresponding (within the accuracy limits of available astronomical observations) to the relative sizes of each planet’s path, assuming the planets circle the Sun.
Kepler also finds a formula relating the size of each planet’s orb to the length of its orbital period: from inner to outer planets, the ratio of increase in orbital period is twice the difference in orb radius.
However, Kepler will later reject this formula, deeming it insufficiently precise.