Revision as of 17:47, 20 December 2015

Here we will give a few algorithms and show the corresponding results.

The Mandelbrot set

Certainly, Wikipedia's page about this set in any language should be a very good introduction to this set. Here I give a very quick definition/reminder:

The Mandelbrot set, denoted M, is the set of complex numbers $c$ such that the critical point $z=0$ of the polynomial $P(z)=z^2+c$ has an orbit that is not attracted to infinity. If you do not know any of the italicized words, go and look on the Internet.

It is interesting for two reasons:

• The Julia set of $P$ is connected if and only if $c\in M$.
• The dynamical system $z\mapsto P(z)$ is stable under a perturbation of $P$ if and only if $c\in \partial M$, where $\partial $ is a notation for the topological boundary.

Drawing algorithms

All the algorithms I will present here are scanline methods.

Basic algorithm

The most basic is the following, it is based on the following theorem:

Theorem: The orbit 0 tends to infinity if and only if at some point it has modulus >2.

This theorem is specific to $z\mapsto z^2+c$, but can be adapted to other families of polynomials by changing the threshold $2$ to another one. Here the threshold does not depend $c$ but in other families it may.

Now here is the algorithm:

Choose a maximal iteration number N
For each pixel p of the image:
  Let c be the complex number represented by p
  Let z be a complex variable
  Set z to 0
  Do the following N times:    
    If |z|>2 then color the pixel white, end this loop prematurely, go to the next pixel
    Otherwise replace z by z*z+c
  When the loop above reached its natural end: color the pixel p in black
  Go to the next pixel

I am not going to use the syntax above again, since it is too detailed. Let us see what it gives in a Python-like style:

for p in allpixels: # replace here by your own loop or pair of loops to scan all pixels
  c = p.affix # here you may replace by your code computing c (complex nb)
  z = 0j
  color = black # this color will be assigned to p at the end, black is a temporary value
  for n in range(0,N):
    if squared_modulus(z)>4: 
      color = white
      break: # this will break the innermost for loop and jump just after
    z = z*z+c
  p.color = color # so it will be white unless we ran into the line color=black

Here in a C++ like style: (std::complex<double> simplified into complex)

for(int j=0; j<height; j++) {
  for(int i=0; i<width; i++) {
    complex c = some formula of i and j;
    complex z = 0.;
    for(int n=0; n<N; n++) {
      if(squared_modulus(z)>4) {
        image[i][j]=black;
        goto label;
      }
      z = z*z+c;
    }
    image[i][j]=white;
    label: {}
  }
}

Let us now look at the kind of pictures we get with that. We an image size of 241x201 pixels, with a mathematical width of 3.0 and so that the center pixel has affix -0.75. The only varying parameter between them is the maximal number of iterations N.

• 

  N=10

• 

  N=20

• 

  N=100

• 

  N=1 000 000

All images except the last one took a fraction of a second to compute on a modern laptop. Back then in the 1980's it was different.

Ideally the maximal number of iterations N should be infinity, but then the computation would never stop. The idea is then that the bigger N is, the more accurate the picture should be.

The N=one million image took 45s to compute on the same laptop, which is pretty long given today's computers power. What happens? Every black pixel requires $10^6$ iterations, and since there are 9 771 of them, we do at least ~$10^10$ times the computation z*z+c. To accelerate this, one should find a way to detect that we are in M so that we use less than the maximal number of iterations.

But that is not the only problem with increasing N. Not only it took much time, but the difference with N=100 is pretty small. Worse: it seems that by increasing N we are losing parts of the picture. What happens is that we are testing pixel centers or corners, and there are strands of M that are thin and wiggle between the pixel centers, so that as soon as N is big enough, the pixel gets colored white.

Here are enlarged versions of parts of these images, so that pixels are visible.

N=10

N=20

N=100

N=10^6

Let us look at a parts of a higher resolution image, computed with a big value of N: notice the small islands. The set M is connected in fact, but the strands connecting the different black parts are invisible, except some horizontal line that appears because, by coincidence, we are testing there complex numbers that belong to the real line: $M\cap \mathbb{R} = [-2,0.5]$.

The next image has been computed using a 4801x4001 px image and then downscaling by a factor of 6 (in both directions) to get an antialiased grayscale image. It also has been cropped down a little bit, to fit in this column without further rescaling.

Last, an enlargment to see the pixels of the above image:

Escape time based coloring

There is a simple and surprisingly efficient modification of the above algorithm.

We started from z=0 and iterated the substitution $z\mapsto z^2+c$ until either |z|>2 (the orbit escapes) or the number of iterations became too big. Instead of coloring white the pixels for which the orbit escapes, we assign a color that depends on n, the first iteration number for which |z|>2.

def f(n): # this function returns a color depending on an integer n
  return ... # put here your custom code
for p in allpixels:
  c = p.affix
  z = 0j
  color = black # this color will be assigned to p at the end, black is a temporary value
  for n in range(0,N):
    if squared_modulus(z)>4: 
      color = f(n)
      break: # this will break the innermost for loop and jump just after
    z = z*z+c
  p.color = color

For the coloring, you are free to take your preferred function of n, I will not develop on that here.

Here is an example of result:

max iterations: 100, size: 481x401 px

The strands became visible again.

Here is another one, on a 4801x4001 px grid, downscaled by a factor 5, with N=10000 (which is probably a bit excessive).

With a nicely chosen color function, one can get pretty results:

click for a higher resolution version

The following was popular in printed books:

click for a higher resolution version (still down-sampled from the original, though)

The potential

You may replace the test |z|>2 by |z|>R, provided that R>2. In fact the picture looks better that way. It has an explanation in terms of the potential.

R=1000, image downsampled by a factor of 8 in linear color space

Physics digression: In very informal terms, in a 2D euclidean universe, put an electrostatic charge on some conducting bounded connected set K, the rest of 2D space being void (dielectric). You will put charge=1 and assume there are so many particles (electrons) that the charge can be considered as a continuum. Then let the charges spontaneously move to minimize the energy. The resulting distribution has the property that the potential is constant on K. The potential function $V$ satisfies the Laplace equation $\Delta V=0$ outside K and $V(c)\sim \rho \log|c| $ when $|c|\longrightarrow+\infty$.

There is however no need to invoke electrostatics and one can define directly the following function and decide to call it the potential: $$V(c)=\lim_{n\to+\infty} \frac{\log_+|P_c^n(0)|}{2^n}$$ where $\log_+(x)$ is defined as $0$ if $x<1$ and $\log(x)$ otherwise. The exponent n in $P_c^n$ refers to composition, not exponentiation: $P^n(z)$ is P applied n times to z. The function V takes value 0 exactly on M. If x>0, the sets of equation V=x are called equipotentials of M. They foliate the outside of M into smooth simple closed curves that encircle it. The potential also finds an importance in the mathematical study of M, as holomorphic functions are involved, but that is not a the topic to be discussed here.

The fact that the formula converges and that its limit has the stated properties is not supposed to be obvious: it is a theorem too. One good thing about the formula is that it converges quite well: if we stop the iteration when |z|>R, the relative error will be of order 1/R. Now assume the picture is drawn for $|c|<10$ (anyway all points of M satisfy $|c|\leq 2$) and at some point in the iteration of $0$ we reach $|z|>4$. Then a value of $|z|>1000$ is reached pretty fast because $|P(z)|=|z^2+c|>|z|^2-5$: one sees that in 4 more iterations we have at least |z|>443556...

In the computation, when |z|=R after n iteration, it means $V(c)=\log(R)/2^n$. Hence taking R=1000 in the previous algorithm (see the picture above) yields color regions whose boundaries very closely match the equipotentials $V(c)=\log(1000)/2^n$.

But we can do better: as soon as $|z|>1000$, we have a good approximation $V(c)\approx\log|z|/2^n$, and we can thus choose a continuous coloring scheme depending on $V(c)$.

Here is a possible algorithm:

def f(V): # this function returns a color depending on a float V
  return ... # put here your custom code
for p in allpixels:
  c = p.affix
  z = 0j
  color = inside_color # color may be modified below and assigned to p
  pow = 1. # pow will hold 2^n
  for n in range(0,N):
    R2 = squared_modulus(z)
    if R2>1000000: # 1000 squared
      V = log(R2)/pow
      color = f(V)
      break: # this will break the innermost for loop and jump just after
    z = z*z+c
    pow = pow * 2.
  p.color = color

And here an example of result: computed with N=2000 and 4800x4000px and a color function that starts black away from M, then slighlty oscillate in shades of blue like a sine function of log(V), but when V gets small a uniform egg yellow components takes over, and even closer it is a black one. The set M is in dark bordeaux red.

Whole object, downsampled to 800x677 (factor 6)

downsampled by factor 4 and cropped

enlargment on a small portion of original image, no downsampling

Deep zooms and log-potential scale

The function $V$ is continuous and has value $0$ exactly on M.

The function $\log V$ is better suited than $V$ for coloring deep zooms. For instance in many points (especially tips and branch points which are asymptotically self-similar [Tan Lei]) a zoom will essentially shift the value of $\log V$ by a adding a constant that depends on the situation.

A simple choice is to have a color that cycles when some chosen constant is added to $\log V$. This choice is also a heritage from when images had a limited palette. So choose $K>0$ and set $$x=\log(V)/K$$ $$\mathrm{color}=g(x)$$ for some function g with g(x+1)=g(x). You can enforce periodicity by reducing x modulo 1 to [0,1[ but that may introduce discontinuities if g was not 1-periodic. For instance $$g(x)=(R,G,B)=(255,255,255)\times\frac{1+\cos(2\pi x)}{2}$$ will yield a smooth wave pattern, from white to black and back.

Taking $K= \log 2$ is a good starting choice, as it is "natural" for reasons we will explain later. You may already notice that $\log \log P(z) \approx (\log\log |z|) + \log 2$ when z is big. However, M gets quite furry in some deep zooms, it is then better to divide by a bigger constant in this case.

Below, some illustrations: Instead of a periodic function we chose a quasiperiodic one: $$g(x)=(R,G,B)=\left(255\times\frac{1-\cos(a x)}{2},255\times\frac{1-\cos(b x)}{2},255\times\frac{1-\cos(c x)}{2}\right)$$ with $(a,b,c)=\left(1,\frac{1}{3\sqrt2},\frac{1}{7\cdot3^{1/8}}\right)$ linearly independent over $\mathbb{Z}$. (It is not the prettiest choice; this is not the point, though.)

...

Boundary detection methods

Milnor's

Variation: built-in antialias

Henriksen's

Fancy coloring of the outside

Images

Bump map effect

Radial strands