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The Rhythm of Numbers · Physics · Kepler · added

Kepler’s Figures

In 1595 Kepler marked on the circle of the zodiac the places where Jupiter and Saturn meet. Three meetings made a triangle, and the triangle slowly turned. Pick any two bodies and watch their meetings draw a figure. The figure is a fraction from the ladder of a continued fraction, and the next term of the ladder says how slowly it turns.

How to use it

  • Choose a pair from the buttons, or type two periods of your own. Then press Play, or step through the meetings one at a time.
  • Watch the figure close. Jupiter and Saturn draw a triangle; Venus and the Earth draw a five-pointed star every 8 years.
  • Pick a rung in the table under the picture to color the figure by its rounds: the current round is gold, earlier ones green.
  • Switch to ellipses to see the meetings with Kepler’s own laws in place of steady circles. Sound is off until you turn it on.

The meetings, seen from the centre

The ladder behind the figure

fractionfigureone roundturn per roundrounds to turn by one pointnext term

Each meeting falls a fixed step further round the circle. The fractions closest to that step are the rungs of its continued fraction. A fraction p/q is a figure with q points, each joined to the one p places on. Choose a rung to colour the picture by its rounds: the current round is gold, earlier ones green. The number of rounds before the figure has turned by one of its own points always lies between the next term and the next term plus two.

What is known and what is ours. The cycles are ancient: 8 years for Venus, 59 for Saturn, 79 for Mars and 83 for Jupiter were used in Babylon, and Kepler drew the Jupiter and Saturn triangle himself. He put its full turn at 794 years, which is 40 meetings; with the long-term mean motions used here it is 883 years, the same as the period of the slow swing of Jupiter and Saturn that astronomers call the great inequality. That meetings step round the circle by 1/(r − 1) of a turn, where r is the ratio of the periods, is elementary. Reading the figure as a rung of the continued fraction, and its slow turning as the next term, is this project's addition. Years shown are for meetings seen from the Sun, from mean orbits, and are approximate.

Read more

Large and small, in plain words →Butterfly of a Number →The paper, Chapter 18 →

Notes and sources

  • The instrument and its text are Teddy Rook’s. It goes with Section 18.5 of the paper, where Proposition 18.6 states the rule and gives its proof.
  • Periods of the planets are long-term mean values from JPL (Standish and Williams, valid 3000 BC to 3000 AD). Pluto and the moons are from NASA’s planetary fact sheets, and TRAPPIST-1 from Agol and others (2021).
  • Years shown are for meetings seen from the Sun, from mean orbits, and are approximate. Real orbits are not circles, so real meetings wander a few degrees about these places.
  • Everything is drawn and played in your browser. Nothing you do here is sent anywhere.

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