Kepler's Mysterium Cosmographicum
Johannes Kepler, in his major astronomical work, Mysterium Cosmographicum (The Cosmographic Mystery), published in 1595, speculated that the orbits of the six planets known at the time—Mercury, Venus, Earth, Mars, Jupiter and Saturn—could be arranged in spheres nested around the five Platonic solids: octahedron, icosahedron, dodecahedron, tetrahedron and cube. For the Platonic polyhedra arranged in this order, coinciding circumspheres for a given polyhedron and inspheres for the next polyhedron gave a fair approximation for the relative sizes of planetary orbits around the Sun. Kepler later rejected this model as insufficiently accurate, but it remains as an amusing exercise in solid geometry. The predicted orbits are expressed in astronomical units (AU) equal to the average radius of the Earth's orbit. Choose "polyhedra" to display the two planets whose orbits are contained in the circumsphere and insphere of the polyhedron. Choosing "planets" lets you zoom in and out to reveal the Keplerian structure within the orbital sphere of a given planet.
Wikipedia article: Kepler
Suggested by work of Jeff Bryant and Stephen Wolfram.