Kepler’s First Attempt

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When it comes to orbits, Johannes Kepler knew his stuff. He’s the one who in 1602 realized that planets orbit in ellipses rather than circles, which became the first of his Three Planetary Laws. But no one is perfect, and these were not his first attempts at describing the motions of the Heavens. In 1596 he published Mysterium Cosmographicum (The Mysteries of the Cosmos), in which he proposed the following model for the solar system:

In this model, the six known planets were envisioned as traveling in circles, along the equators of six giant spheres. The six giant spheres were separated by the five platonic solids. Saturn and Jupiter were separated by a giant cube, and Jupiter and Mars by a giant tetrahedron. It’s harder to see the interior planets in the drawing above, so here’s a close up:

Mars and Earth were separated by a giant dodecahedron, Earth and Venus by a giant icosahedron, and, finally, Venus and Mercury by a giant octahedron. And then, in center of all the orbits, was the Sun.

Let’s see how accurate this model is. If you start with a giant platonic solid, like a cube, you can circumscribe a sphere on the outside and inscribe a sphere on the inside, and then compare the ratio of the radii of the two spheres. It turns out to be √3≈1.73. And lo, if you look at the average radius of Saturn’s orbit (9.021 Astronomical Units) and divide it by the average radius of Jupiter’s orbit (5.20336 AU), it rounds to 1.73. Let’s see how the other ratios match up:

Giant Polyhedron Ratio of spheres in model Ratio of Actual Planet Orbits
Saturn to Jupiter cube 1.73 1.73
Jupiter to Mars tetrahedron 3.00 3.42
Mars to Earth dodecahedron 1.26 1.52
Earth to Venus icosahedron 1.26 1.38
Venus to Mercury octahedron 1.73 1.87

Not too shabby! Plus, as a bonus, you can see that the cube and the octahedron, which are dual polyhedra, have the same ratios of the radii of the circumscribed and inscribed spheres (√3≈1.73); likewise, the dodecahedron and the icosahedron (which are also duals of each other) have the same ratio of the radii of circumscribed and inscribed sphere (\frac{3\sqrt{10+2\sqrt{5}}}{3\sqrt{3}+\sqrt{15}}≈1.26). And unlike the Titius-Bode law, the big gap between Jupiter and Mars didn’t really cause any problems since the tetrahedron fit nicely in there. But a few years later Kepler realized it was wrong, and Uranus’s discovery later would have sealed the deal in any case. Poor Kepler. But it’s still an impressive idea, and was deemed important enough even recently to put on a 2002 commemorative 10-Euro coin in Austria (designed by Thomas Pesendorfer).

Yeah Kepler!

The planet data came from NASA; the data on the radii of circumscribed and inscribed spheres came from Wolfram MathWorld. It’s not clear if the coin is copyrighted or even copyrightable or not; it seems to fall under fair-use guidelines, however. You can find the coin at the Austrian Mint.

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3 Responses to “Kepler’s First Attempt”

  1. Krithiga Says:

    The somewhat fictionalized book, Kepler (by John Banville) recounts this story and if I remember right, Kepler first hypothesized this model of spheres while lecturing a class!

  2. Ξ Says:

    Neat — I haven’t read that. I did find reference to his having initially tried to separate them with regular polygons (a square instead of a cube, for example) but the numbers didn’t work out right and then he realized that he needed to look at three-dimensional figures.

  3. kid Says:

    this is cool

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