## Quantum Fractals - The Algorithm

In applications the vectors n[i], i=1,2,...,N can be arbitrary except that they are of unit length, that is they all lie on the surface of the sphere of unit radius, and they add to zero vector: n[1] + n[2] +...+ n[N] = 0 - so they are balance each other in such a way that their common center of gravity is at the center of the sphere.

The algorithm describes a sequence of maps of the sphere. These maps are applied, sequentially, to a randomly chosen starting point. Thus what we get is a point on the sphere jumping in a somewhat chaotic way on the surface of the sphere. After 100 000 jumps or so a pattern appear. This pattern is independent of the starting point, as it is usual in chaotic dynamics with an attractor. What we see is an approximation of the attractor set. For instance, for a particular value of the fuzziness parameter alpha = 0.65, and the dodecahedron, we get the following images for 10 000,100 000, 1000 000 and 10 000 000 respectively:

The more jumps we generate, the more details of the attractor set shows up. The 100 000 000 dots image shows even more details:

There are two separate problems when creating these fractal images, namely that of generating data, and of rendering the generated date into an image.
Let us first discuss the data generating algorithm. This problem itself can be separated into two: a) generating the sequence of points on the sphere, and b) storing these data into an array. Storing the coordinates of all generated points is not a good idea, as it takes lot of space. For instance, if we use 2 bytes for a floating point number, then the picture on the left would require storing space of 600 MB of RAM (storing the data on the disk would essentially slow down the generation process). Therefore we create a double array d[i,j] of integers by dividing the unit square into JxJ "pixels", initialize it to 0 (or to 1), and with each generated point r = (x,y,z) on the unit sphere we increase d[i,j] by 1 if the (x,y) coordinates of the point are within (i,j) pixel. So, for instance, the image on the left was created using 400x400 array, and the highest occupation "pixel" got 8029 "hits". (In fact, we are using two arrays: one for z negative, and one for z positive. So the pictures here are obtained by "looking at the sphere from above" - thus using only z-positive data.).

Given a point r the algorithm first computes probabilities p(i,r) of sum 1: p(1,r)+p(2,r)+...+p(N,r) = 1 with which we select the map that is to be applied to r. The formula, explained in more details in the paper, reads:

p(i,r) = (1 + alpha2 + 2 alpha n[i].r )/( N( 1 + alpha2 ))

where n[i].r stands fro the scalar product of n[i] and r, that is, in our case, for the cosine of the angle between the vectors n[i] and r, and alpha is the "fuzzy" parameter 0 < a < 1 . Once, for a given current point r, the probability distribution p(i) = p(i,r) is computed, a vertex n[i] is selected accordingly. Here we follow the standard procedure of selecting a case according to the given probability distribution. In Java code:

// Get random number
double r=Math.random();

// Check which detector that clicked
double ptmp = 0;
double[] detector = null;

for (int i = 0; i < p.length; i++)
{

ptmp+=p[i];
if (r <= ptmp)

{

// We found which detector clicked
detector = n[i];
break;

}

}

Then the new point r' on the sphere is generated using the following transformation:

r ' = ( (1 - alpha2)r + 2 alpha( 1+ alpha n[i].r )n[i] )/( (1 + alpha2 + 2 alpha n[i].r ) )

The denominator here is nothing but the length of the vector in the numerator, so as to ensure that the resulting vector is of unit length again. This normalization is the only nonlinearity in the resulting "iterated function system" (IFS).

Rendering of the image

Once the data are collected in a pixel array d[i,j], they need to be converted into a picture, either using grayscale or some coloring scheme. For instance one can find the maximum m = max (d[i,j]) divide it into 256 equal intervals, and aasign a grayscale value between 0 and 1 to an (i,j) pixel according to the value of d[i,j]/m. In fractal images it is neccessary to make some transformation of the array values before, otherwise details of the pattern are lost. For rendering of the greyscale pictures above we have used log10(log10(d[i,j]+1)+1) transformation.

### Prticle tracks, Tunneling times ...

Quantum Fractals Java Applet

EEQT Lab Java Applet

Beta wersion of Win9x application

Note: These two applets crash on certain browsers. Internet Explorer seems to be OK and Mozilla seems to be OK. The applets crash with Opera (for reasons that are not understood) and with older Netscape. Therefore the best thing is to download the .jar files from SourceForge, unpack them, download Java SDK appropriate to the operating system, install it, and then run the .jar files with "java -jar qf.jar" or "java -jar wave.jar".

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