Why Phi? – more Jupiter-Saturn orbital harmonics

We’re now looking for a pattern arising from the Jupiter-Saturn synodic conjunctions and the orbit periods.

Focussing on the numbers of Jupiter orbits that are equal, or nearly equal, to an exact number of Saturn orbits (years), a pattern can be found by first subtracting the number of conjunctions from the number of Saturn orbits.

The second step is to convert the last column of the table, which is essentially the number of retrograde revolutions of the J-S conjunction position in the time period, into a mathematical formula using Fibonacci and Lucas numbers.

The results are in the table on the right.

The Fibonacci numbers follow their series progression, but the Lucas number is fixed.
Example: 11 * 1, +2 = 13.

The tables could be extended, but after the first two rows the rest follow automatically anyway, each column being the sum of the numbers in the two previous rows.

That’s also how the Fibonacci and Lucas series are constructed, each number being the sum of the two previous ones. Both series are closely related to the golden ratio.

Jupiter-Saturn-Earth orbits chart

Finally, we can see that the last row of the table once again corresponds to the data in this chart (on the right) from an earlier Talkshop post:
Why Phi? – Jupiter, Saturn and the de Vries cycle.

The Jupiter-Saturn conjunction position moves about 117.147 degrees between each occurrence.
117.147 * 126 = 14760.522
41 * 360 degrees = 14760 (= 41 retrograde revolutions)

Kepler’s trigon

Hence the 41 in the last row of the tables above.
126 / 3 = 42 trigons.