Kepler's first try

It reported that Kepler first tried to model the gaps between the planetary orbits in two dimensions. Each planet's oval orbit was fit to be enclosed in a ring, and the rings separated by regular polygons.

In fact, this was the subject of his lecture in Graz on July 9, 1595, when the 3D model (with the Platonic solids) burst upon his mind. He saw in a flash that the 2D model could not work, and here is why. See (Koestler, 1960; p. 45).

The equilateral triangle

Consider the equilateral triangle enclosed in two circles, as shown here. The outer circle is to be the inner boundary of the ring containing the oval orbit of a planet. The inner circle is to the the outer boundary of the ring containing the oval orbit of another planet. The ratio of the radius of the larger circle to that of the smaller circle is determined by the geometry of the equilateral triangle. We call this ratio the harmonic ratio of the equilateral triangle.

The harmonic ratios of the regular polygons

Kepler wanted to fit a regular polygon into each of the five gaps between the orbits of the six planets. But the harmonic ratios were known to Euclid, which Kepler had carefully studied::
  • equilateral triangle, 2
  • square, sqrt(2) ~= 1.414
  • pentagon, ~1.236
And as Archimedes knew, these ratios get smaller and smaller as the number of sides of the regular polygon increases. Hence, beyond the first two or three rings defined in this way, none will fit to the gaps in the planetary orbits. The sequence: 2, 1.414, 1.236,... decreases too rapidly. Kepler saw all this in a flash in Graz, on July 9, 1595.
Rev'd 17 jan 2002 by ralph abraham