Archive for the ‘Fractals’ Category

It’s Compilin’ Time!

January 10, 2010 Leave a comment

Further to my last post, I actually have coded a very simply Mandelbrot set renderer in C++.

I’m quite familiar with the basic syntax of C++, so at the outset I though the main challenges would be:

  1. dealing with complex numbers
  2. saving the results as an image file

After some mild Googling both of these challenges were surmountable.

C++ has a standard library header “complex,” which implements complex as a template class. Whatever that means. Anyway, import “complex” and you now have complex number support. See the code below for a demonstration.

For PNG support I used PNGwriter (also available in the Ubuntu repos). PNGwriter’s project site has plenty of example code, and the usage seems very straight forward. The only stumbling block (for me at least) was getting the compiling options right to “link” the PNGwriter library. As it turns out, some or all of the following command is required to compile this code:

g++ main.cpp -o my_program `freetype-config –cflags` -I/usr/local/include -L/usr/local/lib -lpng -lpngwriter -lfreetype

Aside from that bit of unpleasantness, it was really no harder to write this in C++ than in Python. Here it the finished code:

#include <iostream>
#include <complex>
#include <pngwriter.h>

using namespace std;

int main()
    int bailout = 256;
    int size = 800;
    pngwriter png1(size,size,0,"mandel2.png");

    for(int x=0;x<size;x++)
        for(int y=0;y<size;y++)
            int i = 0; //iterations
            complex<double> c(x/(size/4.)-2, y/(size/4.)-2);
            complex<double> z(0, 0);

            while(abs(z)<2 and i<bailout)
                // iterate through mandelbrot dynamic
                z = pow(z, 2) + c;

                // c is outside the mandelbrot set
                float value = (float) i / bailout;
                png1.plot(x, y, value, value, value);

        } // for y
    } //for x


    cout << "done!" << endl;

    return 0;

Remember, if you want to try running this code you will need the PNGwriter library.

The output is of course no different from the Python version that I posted previously, but here it is for the sake of completeness:

The Mandelbrot Set

The Mandelbrot Set

See Plus Plus

January 7, 2010 Leave a comment

These Buddhabrot scripts can sure be processor intensive. I would be better off programming them in C or C++, much as I do love Python. They are rather trivial, so it shouldn’t be too difficult.

One purpose of this blog is to openly record my musings of the various things that I think I ought to do—in the hope that I may remember to actually do them. This is but one such open musing.

Deeper into the Buddhabrot

January 3, 2010 2 comments

My previous Buddhabrot images were built up from many thousands of ‘z‘ orbits. Over time the trails left by these orbits average out into nice soft symmetrical clouds. Most of what you can see are the trails left by the orbits that survive for relatively few iterations before escaping to infinity (there are so many more of this kind of orbit).

If you filter out these “shallow” iteration orbits and instead render only orbits with deep iterations something very interesting happens. The paths of the individual orbits become discernable. They almost look like temporary strange attractors, spinning and swirling before finally blowing away on the winds of chaos.

Take a look at these fantastic patterns (click to enlarge):

Deeply iterated Buddhabrot

This image consists only of orbits with between 20,000 and 30,000 iterations.

As with my strange attractor images, due to a few very bright pixels, it was necessary to render in 16-bit greyscale and then equalize the images to bring out the details which would otherwise be too dark to see. Normalize or auto-levels won’t do the trick. I used ImageJ for the equalize function.

If more orbits were calculated, over time a regular and symmetrical pattern should arise.

Here is the same image as above allowed to accumulate roughly 3x the orbits:

Deeply iterated Buddhabrot

Some interesting discussion of deeply iterated Buddhabrots can be found here.

Buddhabrot Chromatica

January 2, 2010 1 comment

In my previous post on Buddhabrots I demonstrated that different iteration or “bail-out” levels can produce different looking Buddhabrots. Buddhabrot discoverer Melinda Green also described the method of mapping differently-iterated Buddhabrots to the different channels of an RGB image. This can result in strikingly “Nebulous” images, reminiscent of those famous images from the Hubble space telescope.

Obviously this was the next thing I had to try.

3000r, 1500g, 500b. Channels normalized individually.

This image uses 3000 iterations on the red channel, 1500 on the green channel, and 500 on the blue channel. I have normalized the channels individually which seems to favor the channel with the least iterations.

3000r, 1500g, 500b. Channels normalized together.

This images uses the same configuration of iterations per channel as the last. This seems to favor the channel with the most iterations.

It strikes me that both of these approaches are rather arbitrary. The best solution would be to combine the channels in a decent image editing program and adjust the levels of each until the result is visually appealing (These images had their channels combined at creation time in Python). Unfortunately GIMP is not the decent image editing program I’m looking for. CinePaint, however, seems perfect for this job.

1000r, 200g, 100b. Adjusted in CinePaint

“To infinity… and beyond!”

1000 iterations red, 200 green, 100 blue. Adjusted to Hubbletastic perfection. Thanks CinePaint!

The next step is probably to render some super-big color images.

The Buddhabrot

January 2, 2010 2 comments

The Buddhabrot is closely related to the Mandelbrot set. For both, you would calculate the orbit of z under the dynamic z\to z^2 + c for various values of the constant c (where z and c are numbers on the complex plain).

When rendering the Mandelbrot set we are chiefly concerned with values of c corresponding to pixels in our final image. We check which values of c cause the path of z to enter an orbit around the origin, and which values of c cause z to fly off to infinity (and how quickly). Pixels are then coloured accordingly.

For a Buddhabrot, the value of c is less important. Instead, the frequency with which the orbiting point z visits various pixels is recorded — lighter pixels have received a higher frequency of visits from z. While values of c are chosen at random, the classic Buddhabrot uses only those values of c which will cause z to escape to infinity (non Mandelbrot-set points). I think of this process as being a cross between a Mandelbrot set and a strange attractor.

Here is my first attempt at rendering a Buddhabrot using a program I wrote in Python:

This is a 100 iteration image This means that, when checking to see if an orbiting z point is escaping to infinity, we bail-out after 100 iterations and assume all remaining points are within the Mandelbrot set (their orbits are not used in the Buddhabrot). More iterations means more of the kind of orbit that take longer to escape.

It turns out that adjusting this ‘bail-out’ level can have a dramatic impact on the resulting image. For example, here is a 1000 iteration image:

We can see that higher bail-out values result in finer details.

Finally, a 5000 iteration image (click for super-high-res):

Here is the Python code in full, with gratuitous comments and print statements included — remember that you will need PyPng and Numpy to run this code.

#! /usr/bin/python
import png
import numpy as np
from random import random

def c_set(num_samples, iterations):
    # return a sampling of complex points outside of the mset

    # Allocate an array to store our non-mset points as we find them.
    non_msets = np.zeros(num_samples, dtype=np.complex128)
    non_msets_found = 0

    # create an array of random complex numbers (our 'c' points)
    c = (np.random.random(num_samples)*4-2 + \

    # Optimizations: most of the mset points lie within the
    # within the cardioid or in the period-2 bulb. (The two most
    # prominant shapes in the mandelbrot set. We can eliminate these
    # from our search straight away and save alot of time.
    # see:

    print "%d random c points chosen" % len(c)
    # First elimnate points within the cardioid
    p = (((c.real-0.25)**2) + (c.imag**2))**.5
    c = c[c.real > p- (2*p**2) + 0.25]
    print "%d left after filtering the cardioid" % len(c)

    # Next eliminate points within the period-2 bulb
    c = c[((c.real+1)**2) + (c.imag**2) > 0.0625]
    print "%d left after filtering the period-2 bulb" % len(c)

    # optimizations done.. time to do the escape time algorithm.
    # Use these c-points as the initial 'z' points.
    # (saves one iteration over starting from origin)
    z = np.copy(c)

    for i in range(iterations):
        # apply mandelbrot dynamic
        z = z ** 2 + c

        # collect the c points that have escaped
        mask = abs(z) < 2
        new_non_msets = c[mask == False]
                  = new_non_msets
        non_msets_found += len(new_non_msets)

        # then shed those points from our test set before continuing.
        c = c[mask]
        z = z[mask]

        print "iteration %d: %d points have escaped!"\
        % (i + 1, len(new_non_msets))

    # return only the points that are not in the mset
    return non_msets[:non_msets_found]

def buddhabrot(c, size):
    # initialise an empty array to store the results
    img_array = np.zeros([size, size], int)

    # use these c-points as the initial 'z' points.
    z = np.copy(c)


        print "%d orbits in play" % len(z)
        # translate z points into image coordinates
        x = np.array((z.real + 2.) / 4 * size, int)
        y = np.array((z.imag + 2.) / 4 * size, int)

        # add value to all occupied pixels
        img_array[x, y] += 1

        # apply mandelbrot dynamic
        z = z ** 2 + c

        # shed the points that have escaped
        mask = abs(z) < 2
        c = c[mask]
        z = z[mask]

    return img_array

if __name__ == "__main__":

    size = 400 # size of final image
    iterations = 200 # bailout value -- higher means more details
    samples = 10000000 # number of random c points chosen

    img_array = np.zeros([size, size], int)

    i = 0

    while True:

        print "get c set..."
        c = c_set(samples, iterations)
        print "%d non-mset c points found." % len(c)

        print "render buddha..."
        img_array += buddhabrot(c, size)

        print "adjust levels..."
        e_img_array = np.array(img_array/float(img_array.max())*((2**16)-1), int)

        print "saving buddhabrot_n_%di_%03d.png" % (iterations,i)
        # save to final render to png file
        imgWriter = png.Writer(size, size, greyscale=True, alpha=False, bitdepth=16)
        f = open("buddhabrot_n_%di_%03d.png" % (iterations,i), "wb")
        imgWriter.write(f, e_img_array)

        print "Done."
        i += 1