Why Phi? – Jupiter-Venus harmonics and the Fibonacci/Lucas series

Jupiter – the dominant planet in the solar system

The aim here is to show a Lucas number based pattern in five rows of synodic data, then add in a note on Mercury as well.

There’s also a strong Fibonacci number element to this, as shown below.

The results can be linked back to earlier posts on planetary harmonics involving the Lucas and Fibonacci series (use ‘search this site’ box on our home page).

In the interest of clarity, whole numbers have been used below but very slight variations do exist.

The data table in question is on the right.

Lucas series starts: 2,1,3,4,7,11,18,29,47,76 (etc: add any two consecutive numbers for next one).

Fibonacci series starts: 0,1,1,2,3,5,8,13,21,34 (etc: add any two consecutive numbers for next one).

Explanation:
The nearest Lucas number to 92/3 (J-V / J-S) is 29, which is the fixed multiplier.

The first two columns show the progression of the number of conjunctions.
For Jupiter-Saturn synods this follows the Fibonacci series from 3 to 21.

For Jupiter-Venus pattern it’s a formula using the fixed Lucas multiplier (29), a Fibonacci multiplier (from 3 to 21, as per J-S) and a Fibonacci addition (from 5 to 34).

Saturn-Venus is also included, and is the sum of J-S and J-V (by definition).

Examples:
377 = 13 (J-S) * 29
398 J-V = 377 + 21 — both Fibonacci numbers
411 S-V = 377 + 21 + 13 — all Fibonacci numbers

As we saw in an earlier post: Why Phi? – Jupiter, Saturn and the de Vries cycle – the number of Jupiter-Earth synods in the period of 21 Jupiter-Saturn synods is 382. So we have:
13 J-S = 377 + 21 (398) J-V
21 J-S = 377 + 5 (382) J-E

5,13,21 and 377 are all Fibonacci numbers, and the totals are also the closest matches available for those pairs of conjunctions.

Mercury notes:
Jupiter-Mercury conjunctions, which occur nearly 81 times per Jupiter-Saturn conjunction (‘losing’ only 1 J-Merc after 5 J-S), don’t seem to merit their own post along the lines of this one, but can be included in the discussion here.

Since 61 J-V = 100 V-Merc = 161 J-Merc, there’s an obvious Fibonacci/Phi link straight from the numbers.

Reducing J-V and J-Merc by 1 gives a ratio of 60:100:160 which is the same as 3:5:8 in Fibonacci numbers, whereas increasing them by 1 gives 62:100:162 which is almost a ‘golden ratio’ (100/62 = 1.6129 and Phi = 1.618~). Obviously 161.8 is nearly 162.

So the actual numbers sit between those two variations.
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Also from the Talkshop:
Lunar-planetary links to the Lucas sequence – part 3 and summary

Why Phi? – some Earth-Mars orbital harmonics
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Fun with numbers…

In the Fibonacci series, every two alternate numbers sum to the next number in the Lucas series e.g.: 5+13 = 18, 8+21 = 29.

In the Lucas series, every two alternate numbers sum to the ‘next number * 5’ in the Fibonacci series e.g.: 3+7 = 2*5, 4+11 = 3*5.

There are various other possibilities like this as the two series are very closely related, e.g.:
29 (Lucas) could be either 8+21 or 34-5 in Fibonacci terms.