Fibonacci and the Mandelbrot set

[greentexas]

This location in the M-set is tied to the Fibonacci sequence. I found it using Kalles Fraktaler.

Real: -0.39056035661
Imag: -0.587129633128
Zoom depth: 1048576
Iterations: 40000

If you are using Kalles Fraktaler, set the color division to a large number. If I made no mistakes, then the period of this bulb is 144, a Fibonacci number. I am not certain whether there is an algorithm to generate more precise numbers for very high periods.  However, I believe the distance from complex zero, for the point which has period "infinity" but was generated with the Fibonacci sequence, is close to half of the square root of two.

The reason this is related to Fibonacci is because it is located between the period 3 and period 5 bulb on the Mandelbrot set.
The largest bulb between these two is a period 8 bulb.
Between the period 5 and period 8 bulb, the largest bulb has a period of 13.
Between the period 8 and 13 bulbs, the next largest bulb has period 21.
etc.

[claude]

The sequence of internal angles is determined by Farey addition:

1/2, 2/3, 3/5, 5/8, 8/13, ... , a/b, c/d, (a+c)/(b+d), ...
--> 1/phi = (sqrt(5)-1)/2 by the property of Fibonacci numbers

the bond point between the period 1 cardioid and the period q bulb with internal angle p/q is  c=u(1-u) where u = exp(2 pi i p/q) / 2
the size of the bulb is approximately  sin(pi p/q) / q^2

1/phi is irrational, so there is no bulb there, but the corresponding coordinates are approximately: c= -0.39054087021839995-0.5867879073469687i

the absolute value of this last c is within 0.32% of sqrt(2)/2

[greentexas]

Thanks for the information! I may try to zoom around there at some point. I know that when you go in between bulbs, the iterations tend to get large somewhat quickly. I'm guessing your location would require at least 500,000 iterations for a very nice render. I may learn more about the iterations behind it soon enough.

[Chillheimer]

Yep, fascinating stuff that the golden mean is so present in the mset.
I'm currently working on a 5 minute youtube-documentary for fractalogy.org on the connection between the fibonacci-sequence and the mandelbrot set.
one thing we discovered during research is that not only is the fibonacci sequence in there, but also it's less famous brother, the Lucas-sequence. (2,1,3,4,7,11,18....)
same concept as how to find the Fibonacci sequence, but on the right side of the period 3 bulbs.
we didn't find any source mentioning this.

another thing I noticed, but I'm not sure if this is really true or my measurements are accurate enough:
the mainbuld is  phi*2 the size of the period 2 bulb, the period 2 bulb is phi*2 the size of the period 3 bulb.. but below thet, the value drifts away.

If anyone knows of any other occurences of the golden ratio or Fibonacci in the Mandelbrot-set, please let us know!

while I'm at it: This is by far the best  and most scientific source for anything Phi-related: http://www.goldennumber.net/
They debunk many of the widespread myths but also debunk some of the 'debunkers' that go to far and dismiss important facts.

[greentexas]

I may actually start zooming into the Lucas series location. Seems pretty interesting to me! I wonder whether there's a formula for that. Also, even though the period 2 bulb is phi*2 times the size of the period, this size divider converges near the Seahorse/Elephant Valley. A period 2000 bulb is about the same size as a period 2001. You are 100% correct.

[Chillheimer]

wow!
so this means phi IS directly observable in the Mandelbrot-Set. I was looking for that, but didn't want to make the same mistake that is made so often with people seing Phi everywhere where it is not, just drawing the golden spiral on whatever object.

hmm.. this means (kind of) that Phi is in the Mset, but it converges "in reverse". not by zooming deeper, but when you look at the whole, on the lowest, most basic iterationlevel it is there.
the more fascinating question would be: WHY?! How can this be explained?!
Claude, you seem to know your stuff any thoughts on this?

[claude]

https://en.wikipedia.org/wiki/Cardioid the area of the period 1 cardioid with a = 0.25 is 3 pi / 8. The area of the period 2 circle with r = 0.25 is pi / 16.  The ratio is 6, not sure how you got phi out of it...  maybe you mean something other than area by size, I calculated (with wolfram alpha) the half-diameter of the cardioid as 3 sqrt(3)/8 = 0.6495... so no phi there either...

[Chillheimer]

hi claude!
hmmm. I did this visually, using the program phi-matrix. I'm very bad at math, maybe it's a misunderstanding due to mathematical language barrier?
I've attached a screenshot to show what I mean.
The radius of the circles inside of each other are all in a phi-relationship to each other
I made the largest circle match the cardiod and then moved the hole set over to the left to the period 2 bulb and found it matched with the third circle in there. I hope my screenshot explains better than my words.

Do you think this is wrong and just visual close by, and would't withstand a better measurement? (I can't, limited screen resolution, only visual.. and my math isn't enough to actually calculate this)
It's just so close, I find it hard to dismiss as coincidence.

[claude]

if the circles are ratio of phi apart then you need phi^2 for two circle steps, not 2 * phi... anyway, 0.25 * phi^2 = 0.6545... so it's close - 0.77% from the true value - but not exact by any means.  (the 0.25 is the radius of the period 2 bulb)

[Chillheimer]

damnit.. looked too good to be true, which is why we didn't intend to include it in the short documentary.
thanks for clearing things up.

[Chillheimer]

hm.. before I discard this as mistake too soon:
what did you use as radius for the cardiod? I used the attachment points of the period-3bulbs, so probably sth like 0,649... ahh.. and this is clearly not 0,6545..
I'm answering my own question here.

hm. but is 0.7% not close "enough"?
as the difference becomes larger for higher number-pairs couldn't we say that at period 2 it nearly reaches the golden mean, and it would, but you can't continue the sequence, so if you zoom out the most you actually reach phi?
if we compared the difference of each number pair like the bulbs 34/55, 21/34, 13/21, 8/13.... we could extrapolate were the final difference (for period 1 bulb) would have to be from the previous values.
If my little thought experiment was right it should tend towards 0% difference.
is that understandable? does it make sense?

so the next question is: how do I find out the radius of the smaller bulbs to compare their relations?

[DarkBeam]

...

The reason this is related to Fibonacci is because it is located between the period 3 and period 5 bulb on the Mandelbrot set.
The largest bulb between these two is a period 8 bulb.
Between the period 5 and period 8 bulb, the largest bulb has a period of 13.
Between the period 8 and 13 bulbs, the next largest bulb has period 21.
etc.

Illuminati confirmed

[claude]

what did you use as radius for the cardiod?

I calculated (with the help of Wolfram Alpha) the parametric y(t) value at which dy/dt = 0, which occurs at a maximum.  So the "northernmost point" of the cardioid.

hm. but is 0.7% not close "enough"?

If something can be determined exactly, and it doesn't quite match - I don't think it's close enough... either be exact or go home

so the next question is: how do I find out the radius of the smaller bulbs to compare their relations?

Hmm, isn't trivial, as they aren't exact circles (apart from period 2, see http://math.stackexchange.com/questions/1857237/perfect-circles-in-the-mandelbrot-set )

I'd probably use size estimate formula:
https://code.mathr.co.uk/mandelbrot-numerics/blob/HEAD:/c/lib/m_d_size.c
http://ibiblio.org/e-notes/MSet/windows.htm <-- where I got the formula from


edit
I think Phi does occur in this series after all!  As the seuqence of bulbs gets higher period, they converge on a fixed non-zero angle so the sin factor tends to cancel out in the ratio of neighbouring bulbs.  As child bulbs have approximate size proportional to sin(pi*p/q) / q^2, neighbouring bulbs have a size ratio of F_n² / F_n+1² which tends to phi^2 by the property of Fibonacci numbers.

[claude]

I think Phi does occur in this series after all!

verified:

Code:

angle           size                            sqrt(previous size / size)
1/2             5.000000000000000000e-01        -nan
2/3             1.889138370331852612e-01        1.626870999918952787e+00
3/5             7.758028645290857006e-02        1.560472771525585722e+00
5/8             3.063336907762340480e-02        1.591396167039477616e+00
8/13            1.212357732508995492e-02        1.589578530291053671e+00
13/21           4.718818581607213403e-03        1.602871669687447653e+00
21/34           1.825331666867260330e-03        1.607850800967384641e+00
34/55           7.018010107021663106e-04        1.612738298419523764e+00
55/89           2.690908505279404356e-04        1.614944229416002353e+00
89/144          1.029844324700121419e-04        1.616455211270165782e+00
144/233         3.937888926335102711e-05        1.617163978047241768e+00
233/377         1.504959059330040950e-05        1.617593481829286306e+00
377/610         5.750111619544143409e-06        1.617797668592526605e+00
610/987         2.196670525181071280e-06        1.617914906008529607e+00
987/1597        8.391188351414709879e-07        1.617970975463269179e+00
1597/2584       3.205274213144294171e-07        1.618002319861711724e+00
2584/4181       1.224330979379497081e-07        1.618017354629392113e+00
4181/6765       4.676576459621554058e-08        1.618025641244231538e+00
6765/10946      1.786302894386842545e-08        1.618029623782582194e+00
10946/17711     6.823088424645143419e-09        1.618031793995375889e+00
17711/28657     2.606191512380136503e-09        1.618032858067842028e+00
28657/46368     9.954773064984584093e-10        1.618033395516129236e+00
46368/75025     3.802386414640876738e-10        1.618033679365561195e+00
75025/121393    1.452382645241693747e-10        1.618033836564569894e+00

calculated by https://code.mathr.co.uk/mandelbrot-numerics/blob/HEAD:/c/bin/m-fibonacci-phi.c

EDIT added a zoom by factor of phi^4 each frame, so taht features align

[Chillheimer]

wow. thank you for looking into this and providing proof

to make sure I understand correctly:

the size-ratio of bulbs that are fibonacci neighbours do converge to phi^2 the higher the fibonacci period value gets?

so this is why i found this to be true approximately for the obvious largest bulbs but it was not exact enough.

I LOVE that fact. the mandelbrot-set never stops to amaze me.
I know that this might not be very interesting for many people, but I think this is quite a discovery.
So you can not only mathematically approximate Pi with the MSet but also the Golden Mean.

I bet Mandelbrot would have liked that discovery, too!

I usually don't do this but: High five, claude!!