Some questions about cardoids, circles and the area

Mircode

Alien

Posts: 37

Some questions about cardoids, circles and the area

« on: November 23, 2013, 12:37:24 AM »

Hi there!

Recently I was digging a little deeper into the Mandelbrot science and some questions came up. I think they are related so I don't open a new thread for each of them.

1) It is often stated that the circles are not perfect except for the big one. Is that somehow proven or obvious? I have never encountered a noticeably deformed circle. I suspect that this might just be an impression when seeing the distorted minibrots at the elephant valley.

2) Same thing with the cardoids. There was a very nice overlay of the main cardoid with one inside the elephant valley. I can't find the post anymore, but they looked identical, just the circles around were distributed with a little shift so the whole thing looked bent, although the basic shapes retained their shape.

3) I know that the circles and the cardoids are positioned at the roots of the iterated sequence. Is there any easy way to tell whether a root is the center of a circle or of a cardoid?

4) Is it possible to compute the radius of circles (or if they are not circles, the area) around such a root? Is it possible for cardoids? Also to compute their orientation?

If the whole thing just consisted out of perfect circles and cardoids and we knew their sizes, the area should be computable not just by pixel counting...


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Mircode

Alien

Posts: 37

Re: Some questions about cardoids, circles and the area

« Reply #1 on: November 25, 2013, 02:08:25 PM »

This may be interesting, didn't have time to study it yet.

http://commons.wikimedia.org/wiki/File:Mandelbrot_set_Components.jpg


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lkmitch

Fractal Lover

Posts: 238

Re: Some questions about cardoids, circles and the area

« Reply #2 on: November 25, 2013, 04:43:41 PM »

1) One way to see that is to look analytically at the components' boundaries. When you do that, the main cardioid boundary falls out for fixed-point iterations, and the circular disc centered at -1/0 for period-two oscillation. When you look at the same analysis for period-three, you don't get the equation for a circle. Alternatively, you can place a circular disc mask inside the disc and see that the boundary doesn't quite match.

2) In a quick study I did, it seemed like the cardioids themselves were not distorted, but rather, the arrangement of discs around them. This was done with images (rather than analysis), so I don't have any proof of that.

3) There probably is a way to distinguish cardioid centers from disc centers, but I don't know what it is.

4) There is a proxy for a circle radius that I've used before, it's the distance from the root to the tangent point between the disc and the cardioid. For the main cardioid, Mandelbrot worked out an approximate model of disc size. I looked at that and found that the errors between the model and the actual Mandelbrot set appear to be fractally distributed.


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Adam Majewski

Fractal Lover

Posts: 221

Re: Some questions about cardoids, circles and the area

« Reply #3 on: November 25, 2013, 09:29:49 PM »

method of distinguish cardioids from pseudocircles is described in : Universal Mandelbrot Set by A. Dolotin


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claude

Fractal Bachius

Posts: 563

Re: Some questions about cardoids, circles and the area

« Reply #4 on: December 01, 2013, 04:13:04 PM »

Quote from: Mircode on November 23, 2013, 12:37:24 AM

4) Is it possible to compute the radius of circles (or if they are not circles, the area) around such a root? Is it possible for cardoids? Also to compute their orientation?

There's some notes on size/orientation estimates here:

http://www.ibiblio.org/e-notes/MSet/Windows.htm
http://www.ibiblio.org/e-notes/MSet/Scaling.htm

I implemented it in Haskell:

Code:

-- size-estimate.hs
import Data.Complex (Complex((:+)), polar)
import System.Environment (getArgs)

lambdabeta :: Int -> Complex Double -> (Complex Double, Complex Double)
lambdabeta period c = (lambda, beta)
  where
    zs = take period . iterate (\z -> z * z + c) $ 0
    lambdas = drop 1 . map (2 *) $ zs
    lambda = product lambdas
    beta = sum . map recip . scanl (*) 1 $ lambdas

sizeorient :: Int -> Complex Double -> (Double, Double)
sizeorient period c = polar . recip $ beta * lambda * lambda
  where
    (lambda, beta) = lambdabeta period c

main :: IO ()
main = do
  [p, x, y] <- getArgs
  let period = read p
      re     = read x
      im     = read y
      (size, orient) = sizeorient period (re :+ im)
  putStrLn $ "period = " ++ show period
  putStrLn $ "re     = " ++ show re
  putStrLn $ "im     = " ++ show im
  putStrLn $ "size   = " ++ show size
  putStrLn $ "orient = " ++ show orient

Example:

Code:

$ ./size-estimate 4 -0.15652016683 1.0322471089
period = 4
re     = -0.15652016683
im     = 1.0322471089
size   = 8.482858703709158e-3
orient = -0.6171988505484176

You need to know the period and nucleus of the atom you're trying to describe.


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claude

Fractal Bachius

Posts: 563

Re: Some questions about cardoids, circles and the area

« Reply #5 on: December 01, 2013, 06:06:34 PM »

Quote from: Mircode on November 23, 2013, 12:37:24 AM

3) I know that the circles and the cardoids are positioned at the roots of the iterated sequence. Is there any easy way to tell whether a root is the center of a circle or of a cardoid?

As Adam suggested, there's a book and paper, I used this version: http://arxiv.org/abs/hep-th/0701234

The relevant part is section 5, in particular equation 5.8. In the paper $\gamma$ is defined implicitly in equation 5.1, and it took me quite some time to figure out that it's the same as $\frac{\partial}{\partial z} f(z, c)$ . Equation 5.8 then becomes:

$<br />\epsilon = - \frac{1}{ \frac{\partial}{\partial c}F \frac{\partial}{\partial z}F} \left(\frac{\frac{\partial}{\partial c}\frac{\partial}{\partial c}F}{2 \frac{\partial}{\partial c}F} + \frac{\frac{\partial}{\partial c}\frac{\partial}{\partial z}F}{\frac{\partial}{\partial z}F}\right)<br />$

When $\epsilon$ is near 0 then you have a cardioid, when $\epsilon$ is near 1 then you have a circle.


« Last Edit: December 01, 2013, 06:08:16 PM by claude, Reason: fix typo »	Logged

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