Tallbloke: The Why Phi? Pi Slice

Posted: September 11, 2013 by tallbloke in Celestial Mechanics, Phi, solar system dynamics

sunfleavesgridI 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 AnotatedPineConeSize450Fibonacci 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.

  1. Ian Wilson says:


    A wonderful tale worth telling! Just wondering whether the following might be of some use:

    Take the reciprocal of PHI (.618) and cube it. Multiply that by 6 and divide the result by ten. Add to this 3 and you have PI to four decimal places = 3.1416.

  2. Ian Wilson says:


    The above is from a web page by Glenn R. Smith:


  3. oldbrew says:

    ‘When we divide the 360 degrees of the circle by the number phi, we get the angles 222.4922… and 137.5077…’

    TB knows this but just to be clear – the reverse of that is:
    222.4922 / 137.5077 = phi

    Another way of explaining the preference for 13/8:
    137.5 x 13/8 = 223.4375 – within 1 of the 222.5 ‘target’
    137.5 x 8/5 = 220 – exactly 2.5 from the target.

  4. tallbloke says:

    Ian: thanks, nice result. So you’re saying phi and triangular numbers 1,3,6,10 are connected to the circle? You’d be dead right.

    OB: your facility with numbers runs rings round me. 😉

  5. Roger Andrews says:


    I was writing a post on this important subject but now you’ve posted this I’m going to have to rewrite it.

    In the meantime here are some more flower pics for you to look at. Let me know if you find the golden ratio, phi, or the famous circularity constant Pi in any of them. 🙂

  6. Ulric Lyons says:

    (561*7) / 1250, or (231*17) / 1250 = 3.1416.

    231 is t21, and 561 is t33.

    If the units are tenths of Megalithic yards, then a 1250 diameter circle is 340 Imperial feet, as in the Avebury circles, with an error of ~0.03 inches on the diameter. And of course the old method was to divide the circle by 7.

    [Mod note] List of Triangular numbers

  7. tallbloke says:

    Roger A:

    Daffodils at dawn you cad!

  8. Roger Andrews says:

    TB: You missed your chance to win the cement bicycle. The circularity constant Pi is in fact detectable in all six pics.

    How about buttercups at breakfast?

  9. tallbloke says:

    Roger A: I say Foo to your flowers and balls to your buttercups. We’ll feed on Phi fractal broccoli and I hope you don’t choke on it. 😉

  10. Roger Andrews says:


  11. oldbrew says:

    Roger Andrews says:
    ‘In the meantime here are some more flower pics for you to look at. Let me know if you find the golden ratio, phi, or the famous circularity constant Pi in any of them.’

    Four petals = 2 + 2
    Six petals = 3 + 3

    ‘…the number of spirals in the centers of sunflowers, and in the centers of flowers in general, correspond to a Fibonacci number’

    Any pics with 7 petals? 😉

  12. Fascinating stuff

    Here l divided the plant into sections by layer
    This plant is
    6 sections inner circle
    10 sections outer circle
    10 : 6 = 1.666

    Both 10 and 6 are triangle numbers


  13. Brian H says:

    OT, have you seen any pix of flowers shot in UV? What the bees see….

  14. Brian H says:

    Anent which, dark skin reflects UV, white skin absorbs it. In a UV camera, negroid skin shines silver, Nordic skin is black. I’d guess Oriental skin is a ‘tweener.