Kepler's Lenten Pretzel

A 400th Anniversary. One of the figures in the book by Owen Gingerich reviewed last time is Johannes Kepler's diagram, from his Astronomia Nova (1609) which he likened to a "panis quadragesimalis" or Lenten Pretzel. It shows the path, relative to the Earth, taken by the planet Mars during the years 1580 to 1596, based on the observations of the astronomer Tycho Brahe. "Mars repeatedly approaches the Earth, makes a backward loop, only to recede and repeat the process roughly two years later at a spot in the zodiac about fifty degrees to the east of the previous loop. The loops themselves trace round the entire sky in approximately seventeen years during which time Mars itself circumnavigates the sky eight times." Ptolemy and Copernicus (relying on less accurate data) had tried to explain this convoluted orbit as a combination of circular motions in various ways. Kepler resolved the problem by showing that the answer was to use elliptical orbits.