One case after another, the Renaissance scholar identified the ways in which the regular polygons can tile the plane. Four centuries later, his thoroughness might be viewed as routine. And yet his diagrams belie this view – those marvelous diagrams! One after another, yes, but with ever-increasing intricacy, with regular figures regularly arranged, few-sided with many-sided polygons, some with stars.

Johannes Kepler was investigating what he called the congruence of polygons. Those select few polygons that possessed this property formed the basis of his Harmony of the Universe, the remarkable amalgam of geometry, music, and astronomy by which he sought to explain the motions of the heavens. This work contained his masterpiece, the third law of planetary motion. It also asserted that the six (known) planets revolved around the sun in orbits isolated by the shells of the five Platonic solids! It was mysticism, devout religious belief, and great science all rolled up into one. It was fabulous.