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Author Topic:  Are there fibonnaci spirals in the mset  (Read 1287 times)

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Offline Fraktalist

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Are there fibonnaci spirals in the mset
« on: April 15, 2018, 10:13:45 PM »
Quick thought/question

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/are-there-fibonnaci-spirals-in-the-mset/1221/
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Offline pauldelbrot

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Re: Are there fibonnaci spirals in the mset
« Reply #1 on: April 15, 2018, 11:34:16 PM »
Does this answer your question? :)

(-1.76245269873728085 + i0.0106118511934039755)
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Offline Fraktalist

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Re: Are there fibonnaci spirals in the mset
« Reply #2 on: April 16, 2018, 12:49:18 PM »
nope.
maybe my attached image explains better what I mean.
I haven't yet found a spiral that really matches that overlay of the golden spiral. and I wonder if this is because the spirals in the mset are different (ins some mathematical sense that I don't understand) or if I just haven't found the proper location yet.
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Offline lkmitch

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Re: Are there fibonnaci spirals in the mset
« Reply #3 on: April 16, 2018, 05:56:16 PM »
My guess is that you can come arbitrarily close to a Fibonacci spiral, but may not find one exactly. You may wish to look a a family of spirals (say, around Misiurewicz points for disks with fractions 1/n) and see if you can get close.
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Offline pauldelbrot

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Re: Are there fibonnaci spirals in the mset
« Reply #4 on: April 16, 2018, 06:24:30 PM »
That looks pretty damn close to me. Most likely it becomes exact as you zoom into the center of that one.

Also it seems we're using different terminology. To me, "logarithmic spiral" describes what you're apparently looking for, and "Fibonacci spiral" describes the spiral patterns in sunflower centers and cauliflower and stuff. Which also occur in the M-set.
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Offline claude

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Re: Are there fibonnaci spirals in the mset
« Reply #5 on: April 16, 2018, 08:20:42 PM »
What is the formula for the Fibonacci spiral?  A logarithmic spiral has \( r=e^\theta \), but is your spiral that or something else: \( r=\phi^\theta \) perhaps?

For (natural) logarithmic spirals I conjecture the answer is "no", because the multiplier of every Misiurewicz point is an algebraic number, while \( e \) the base of natural logarithms is transcendental.  EDIT actually I'm not so sure, because the multiplier is about features lining up when you zoom and rotate, and the spiral curve need not be aligned to features... need to think some more!

For the other formula I gave, it could probably be reduced to a pair of simultaneous equations to find a Misiurewicz point with a desired multiplier, I may try to figure it out later this week...
« Last Edit: April 16, 2018, 08:49:47 PM by claude, Reason: need to think some more! »
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Offline gerrit

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Re: Are there fibonnaci spirals in the mset
« Reply #6 on: April 16, 2018, 08:43:52 PM »
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Offline pauldelbrot

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Re: Are there fibonnaci spirals in the mset
« Reply #7 on: April 16, 2018, 09:04:36 PM »
To my understanding, a logarithmic spiral is any spiral fitting r = xθ for some x, or (equivalently) r = ekθ for some k (put k = ln x). The spiral around any Misiurewicz point should approach, ever more closely as one zooms toward the point, a logarithmic spiral by that definition.
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Offline claude

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Re: Are there fibonnaci spirals in the mset
« Reply #8 on: April 16, 2018, 09:38:39 PM »
thanks for the link gerrit, which confirms pauldelbrot's statement.

the golden spiral from gerrit's link has \[ r = \left(\phi^\frac{2}{\pi}\right)^\theta \]

I think this works out for multiplier \( m \) of a Misiurewicz point:\[ r + i \theta = \log m \] which is a transcendental equation in \( \theta \) (need to choose the correct complex log branch..., and the domain for theta should be limited to one turn to avoid spurious solutions, maybe?)  Misiurewicz points are countable, so enumerating the equations to try to find ones with solutions may be feasible...

I think it works out as \[ 4 \log \phi \frac{\arg m}{\log |m|} = 2 \pi k ; 0 \ne k \in \mathbb{Z} \]
Running a search now, best results so far (no exact matches with preperiod + period <= 17) are near these coordinates:
Code: [Select]
-0.605248005094052854 + i 0.693916592283719935
-0.287310554310880206 + i 0.834108235487727723
-1.992419648924411035 + i 0
-1.985461174731804057 + i 0.000660454720266961
While these all have a near-golden spiral curve passing through asymptotically self-similar features, none have the desired "appearance", so I haven't posted the code (let me know if you want to see it anyway).

I asked here: https://math.stackexchange.com/questions/2740655/golden-spirals-in-the-mandelbrot-set
« Last Edit: April 17, 2018, 02:30:15 AM by claude, Reason: M.SE question »
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Offline Fraktalist

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Re: Are there fibonnaci spirals in the mset
« Reply #9 on: April 17, 2018, 11:45:16 AM »
awesome! thx a lot for looking into this!

(i have a problem finding any spirals at your locations though - could you add the zoom factor as well?)
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Offline claude

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Re: Are there fibonnaci spirals in the mset
« Reply #10 on: April 17, 2018, 12:28:05 PM »
I tested with KF with zoom factor 1e10 for all of them.  Sometimes the spirals are non-obvious (like inner branch of inner branch...).
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Offline Fraktalist

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Re: Are there fibonnaci spirals in the mset
« Reply #11 on: April 17, 2018, 02:05:58 PM »
okay, those I found at your coordinates are not what I was looking for.
it's not about logarithmic spirals in general, but about the special fibonacci golden spiral:
https://en.wikipedia.org/wiki/Golden_spiral
there are a few formulas as well..
of which I have no understanding, as usual, I do this with only visual input. I use https://www.phimatrix.com/ to overlay and find the closest approcimation of it.

attached is your second coordinate pair. not matching the curvature

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Offline claude

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Re: Are there fibonnaci spirals in the mset
« Reply #12 on: April 17, 2018, 02:10:56 PM »
Quote from: Fraktalist on April 17, 2018, 02:05:58 PM
attached is your second coordinate pair. not matching the curvature
try reflecting it perhaps?  the fit is non-obvious, perhaps it's so wrong it's right ?
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Offline Fraktalist

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Re: Are there fibonnaci spirals in the mset
« Reply #13 on: April 17, 2018, 03:18:45 PM »
? reflecting?
nope, that can't help. the curvature of the golden spiral is at least x2 of the one in the example.
in the section where the golden spiral makes 2 full rounds, the example doesn't even make one, the angle is wrong. and the 'perfect angle' is actually what I was searching  for.

have a look at my first screenshot again, there the curvature of the spirals pretty much match. but not exactly.
see attached, the 7th spiral in the seahorse valley is close to perfect but a bit too little curvature. while the 8th has a bit too much curvature.
(not sure if it really was 7th/8th, but that's the rough region where I was searching visually)

edit: attached a location that is very close to it
it's actually the 10th largest bulb heading into the seahorse valley, so a bit deeper than the one marked in my screenshot.
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Offline pauldelbrot

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Re: Are there fibonnaci spirals in the mset
« Reply #14 on: April 17, 2018, 04:24:32 PM »
Look at the spiral on the smaller bulb between the "a bit too little" and "a bit too much" one. It should be close. Play binary-search with the biggest bulb between two bulbs and it should asymptote to exact.
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