Tallbloke: The Why Phi? Pi Slice

I made an interesting discovery last night about the relationship between the golden ratio, phi, and the famous circularity constant, Pi, which might have a bearing on our investigation of the reasons why the Fibonacci series is exhibited throughout the solar system. It’s well known that many plant species throw off new leaf stems at phi related angles as they grow. It’s thought that this maximises the light the leaves receive. Since Roger Andrews doesn’t believe this actually shows up in plants much, I’ve included the photo of a growing sunflower, as seen on the right. Sure enough, those leaves are popping out at phi angles. You need to follow anticlockwise to see the regularity of the 222.5 degree increments.

It’s Phi ‘o’ Clock. Time for a break from the planetary data and for a look at some phi basics. When we divide the 360 degrees of the circle by the number phi, we get the angles 222.4922… and 137.5077…

In the solar system, given the abundant sunshine, and tiny planet shadows, the more likely reason for the manifestation of phi and the Fibonacci series is that it is less likely destructive resonances will arise, threatening the stability of orbits, as I’ll show below.

First though, we’ll take a tour of some of the interesting properties of the two leaf arrangement shown on the right above, and gather a few facts which will come in handy later.

Kepler’s second law tells us A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. What is the relationship of the area of the smaller sector to the larger in the arrangement above?

The area ‘A’ of a sector of a circle is given by the simple formula: A=angle/360*Pi*R²
For a Radius ‘R’ of 1 and angle 137.507764 this is simply 137.5/360*Pi = 1.19998
The area of the whole circle is simply Pi, since R² = 1
The ratio of Pi to 1.19998 is phi²
The ratio of the smaller sector to the larger is Pi-1.19998:1.19998 which is simply phi itself.

This demonstrates that in orbital mechanics where phi and Fibonacci ratios are involved, orbital radii, angular separations and the areas swept by orbits are intimately connected by phi and the powers of phi. We’ll have more to say about that in a later ‘Why Phi?’ post on the Galilean moons.

The discovery I made last night is this:

If we step round the circle with 222.49223… degree increments, it takes a lot of steps before we get back close to the start point. On examining the data I observed a pattern to this tendency for the phi angle not to return to its start point. Compare the angles found by adding the phi angle to itself and multiples of 360 degrees; for brevity I’ve included only the step numbers where we get a ‘near miss’, and the error angle rounded to the nearest integer.

Rotation	Phi angle	Step number	Error	Fibonacci	Error
0	222	0	-138	144	8
360	445	1	+85	89	4
720	668	2	-52	55	3
1080	1112	3	+32	34	2
1800	1779	5	-21	21	0
2880	2892	8	+12	13	1
4680	4672	13	-8	8	0
7560	7564	21	+4	5	1
12240	12237	34	-3	3	0

As you can see, we get two opposing Fibonacci series! As the step number arrives at each successively higher Fibonacci number, the error in degrees descends as another approximate Fibonacci series. Just as the small fractions in the Fibonacci series alernately deviate above and below phi, the error in degrees alternates above and below the whole rotation number. The error numbers aren’t identical to the fibonacci series, but look at the error of the error, so to speak; It’s yet another approximate Fibonacci series!  This curious ability of the Fibonacci series to manifest itself and converge towards the Golden ratio again via an unexpected ‘side shoot’ no matter what manipulation you subject it to has shown up repeatedly in our investigations of the ‘Why Phi?’ question. I hope people’s own investigations will help dispel any suspicion that we are engaged in ‘numerological cherry picking’. Another thing worth noting in this table is that the point where the error and the step number ‘cross over’ is at the 13/8 ratio, where the ‘error of the error’ becomes small. Maybe this is why the 13/8 configuration of opposing spirals is common in seed heads  and pinecones. Notice also, that leaf 3 is directly above leaf 16 in the photo of the sunflower at the top of the post. 16-3=13, 15-2=13 14-1=13.

With this preamble out of the way, we can get down to looking at some real celestial mechanics. In the next post I’ll start from the Sun and work outwards. Mercury and Venus are the first planet pair I’ll deal with, followed by Venus and Earth, and then Jupiter and Saturn. There are several surprises in store.