Math Mutation 26: That Krazy Kepler You've probably heard of 17th-century astronomer Johannes Kepler, one of the key scientific figures of his time. He is most famous for having determined that planetary orbits are actually elliptical rather than circular, as part of his general laws of planetary motion. He also made some significant contributions to the developing field of optics. But before he made his famous contributions, he published an earlier book called the Mysterium Cosmographicum, or "The Sacred Mystery of the Cosmos", proposing the rather absurd theory that all planetary orbits were determined by the five platonic solids. The platonic solids, as you may remember from earlier podcasts, are the five three-dimensional convex solids in which every face and every angle are congruent. There are five of these: the 4-faced tetrahedron, the 6-faced cube, the 8-faced octahedron, the 12-faced dodecahedron, and the 20-faced icosahedron. Kepler noticed the odd coincidence that there were exactly five of these solids, and also exactly six planets known at the time. Determined that this must be part of God's plan for the universe rather than a simple coincidence, he decided to relate the two numbers. How did these numbers relate to each other? Kepler began by expanding the orbits of each planet into a sphere. (When he wrote this book, he had not yet determined that planetary orbits are elliptical.) Then he figured out that he could tightly fit a platonic solid between each pair of planetary spheres, at least to the degree of accuracy with which the orbits were known at that time. To make this work, he had to choose a rather strange order for the solids: the innermost one was the octahedron, followed by the icosahedron, the dodecahedron, the tetrahedron, and the cube. To get from each planet's orbit to the next orbit, you would draw a platonic solid circumscribed around that orbital sphere, or the minimal-sized version of that solid that could completely contain the orbit. Then you would draw a circumscribed sphere outside that solid, and that sphere would coincide with the next planet's orbit. It's pretty easy to laugh at this theory today. To start with, of course, there are the additional planets Uranus and Neptune discovered after Kepler's time, which cannot be accounted for since, as we proved in an earlier podcast, there can be no more than five platonic solids. But as a more basic question, why the heck should the orbits of planets be affected by the fact that we could inscribe giant polyhedrons between them? And even if we play devil's advocate for the moment and say that the theory is reasonable, why are the solids used in the arbitrary order that Kepler had to choose? The polyhedrons are not sorted by vertex angle, number of faces, or face type. Strangely, Kepler never completely abandoned his theory even after his later discoveries, and pubished a revised version of the Mysterium in 1621, presumably reconciling its earlier conclusions with his more recent, and more accurate, discoveries. But we shouldn't be too hard on poor Kepler. Back then, the boundaries between physics, math, and religion were all kind of fuzzy, and scientists like him believed they were uncovering God's plan for the universe. And if you accept that everything around us was created manually by an anthropomorphic God, why shouldn't He insert mysterious mathematical connections that would otherwise not be needed, just to satisfy His desire for an orderly universe? The other important factor to keep in mind is that his willingness to entertain off-the-wall theories, ideas that might be considered ridiculous by his colleagues, is what made Kepler such a great mathematician and scientist. To many of his contemporaries, the theory that planetary orbits were determined by the platonic solids was probably a lot more reasonable the idea that they were ellipses rather than circles! And this has been your math mutation for today. References:
  • Kepler on Wikipedia