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 (for the same column).
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 (126 – 85) in the last row of the tables above.
126 / 3 = 42 trigons.
42 – 41 = 1 complete cycle, i.e. return to the starting position.
Tallbloke's Talkshop 
It seems strange that the spacings between the occurrences of both planets having whole numbers of orbits isn’t regular. What is the criterion or ‘error limit’? Is that arbitrary?
Reblogged this on Climate Collections.
TB: S/J = 2.4823782
2.4823782 * 2 = 4.9647564 (~99.3% match to 5)
2.4823782 * 27 = 67.024211 (~99.96% match to 67)
The rest are derived by adding the previous two together, like the Fibonacci series for example. We could go up by multiples of 2 but the % match would keep declining (relative to our table).
I’m fairly confident the 41 in the last row is the right answer for exact multiples of 360 degrees, at least without going to much bigger numbers.
Unless the NASA data is faulty…
https://ssd.jpl.nasa.gov/?planet_phys_par
– – –
2.4823782 * 85 = 211.00214 (~99.99898% match to 211)
Did you run? Did you vote? what is happening in the Old country?
Looks like the same method works for Earth-Mercury as well. Post pending.
Update: Saturn-Mars and Venus-Earth patterns are there too.
Venus-Earth is simple – replace multiples of 8 Earth orbits with multiples of 7 and add the Fibonacci number:
7*1, +1 = 8E = 13*1V
7*2, +2 = 16E = 13*2V
7*3, +3 = 24E = 13*3V
etc.
No post needed for that.
Going back to the basics of the Lucas-Fibonacci link, one way of framing it is to start with the Lucas-Fib. number pair that produces another Fib. no, e.g. (Lucas first):
3*1 = 3
4*2 = 8
7*3 = 21
11*5 = 55
18*8 = 144
etc.
Taking 4*2 = 8, we see:
4*2, +0 = 8
4*3, +1 = 13
4*5, +1 = 21
4*8, +2 = 34
4*13, +3 = 55
etc.
So the Lucas number is fixed, and the multiplier and addition go up in a Fibonacci progression, with the result being the next Fib. no. after the line before it.
The same format can be used with any of the pairs in the list but some may go negative e.g. 7*3:
7*3, -0 = 21
7*5, -1 = 34
7*8, -1 = 55
7*13, -2 = 89
7*21, -3 = 144
etc.
This is the type of construct the table in the blog post is based on, as a way of interpreting the planetary numbers. Once the first two rows of the table are set up, each column in every other row is the sum of the two numbers above it.
Excellent work OB. You’re getting towards a generalised scheme.
Thanks TB. One thing to notice is the way the synodic period of 2 planets relates to the orbit periods.
For example, the Earth-Mars and Earth-Venus synodic periods are both greater than the two orbit periods they relate to, as are Neptune-Uranus and Neptune-Pluto. The other ‘neighbours’ have a synodic period in between the two orbit periods.
So that’s a factor in these Lucas-Fibonacci tables.
Neptune-Uranus worked out:
647 U-N takes a long time (~110,900 years) but the accuracy is very high.
Lucas no. (7 here) is fixed, and Fib. nos. follow the correct sequence (given their start no.).
Full Fib. series starts: 0,1,1,2,3,5,8,13…etc.
To check it, multiply the synods number by 374.4507 (degrees) then divide by the Neptune orbits. The result should be close to 360 degrees.
About three Neptune orbits are ‘dropped’ in the period, since 26 * 26 = 676, but there are only 673 N orbits.
The corresponding number for Uranus is 6. (51*26, -1320).
An alternative J-S version.
An advantage with this one is a similar pattern formula for all the columns.
Saturn: 7 * Lucas series (from 1), + Fib series (from 1) — as shown (adding Fib. nos. 1,2,3,5,8).
J-S: 11 * Lucas series (from 1), + Fib series (from 1) — 11*1, +1 = 12 etc. (adding 1,1,2,3,5).
Jupiter: 18 * Lucas series (from 1), + Fib series (from 2) — 18*1, +2 = 20 etc. (adding 2,3,5,8,13).
It can be extended to the J-S revolutions (i.e. J-S minus S).
Then the pattern is: 3 * Lucas series (from 1), + Fib series (from 1) — 3*1, +1 = 4 etc. (adding Fib. nos. 1,2,3,5,8).
The Jupiter numbers are always the sum of Saturn and J-S.