1. J OHANNES K EPLER 1571 to 1630 http://kepler.nasa.gov/johannes

2. Johannes Kepler–The Phenomenologist How are things happening? Mathematical explanation Reality is the human explanation Copernicus did not think his model represented reality Major Works: Harmonices Mundi (1619) Rudolphian Tables (1612) Astronomia Nova Dioptrice Johannes Kepler (1571–1630)

3. Euclidean Regular Figures A regular figure is a closed linear figure with every side and every angle equal to each other. For example, an equilateral triangle, a square, an equilateral pentagon, hexagon, and so forth. There is no limit to the number of regular figures with different numbers of sides.

4. In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex. Moreover, all its edges are congruent, as are its vertices and angles.geometryconvex polyhedronregular regular polygoncongruent vertex

5. The Platonic Solids Unlike regular figures, their number is not unlimited. There are actually only five possibilities: – Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron This was discussed by Plato. They are traditionally called the “Platonic Solids.” That there could only be five of them was proved by Euclid in the last proposition of the last book of The Elements.

6. Coincidence 5 planets- Mercury, Venus, Mars, Jupiter, Saturn 5 Platonic Solids Gibbs and Kepler do not believe in coincidences

7. J OHANNES K EPLER Kepler tried to fit planetary orbits into a nested system based upon the five perfect geometric solids ( By permission Sternwarte Kremsmünster)

8. Music of the Worlds Harmonica Mundi

9. Conic Sections Kepler was the man!

11. The orbits of the planets are ellipses, with the Sun at one focus of the ellipse.

13. It’s the Law!

14. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse.

16. The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes:

17. It’s the Law! P 2 = a 3

18. Planet a (AU) a 3/2 P (yr) Mercury 0.38 0.24 0.24 Venus 0.72 0.61 0.61 Earth 1.00 1.00 1.00 Mars 1.52 1.88 1.88 Jupiter 5.2 11.8 11.8 Saturn 9.6 29.5 29.5

19. Why? Kepler didn’t care why. He had found mathematical descriptions for the motion of the planets. Newton supplied the why or perhaps just additional how information.