Fractals can have infinitely complex patters that are self replicating at multiple scales. These shapes can range in appearance from simple to complicated and can be usually generated from a very simple mathematical formula. For each pixel in an image, a point is associated with x and y coordinates. These starting coordinates are then fed into the formula resulting in new coordinates, which are plugged into the same formula for the next iteration. The pixel in the image is then assigned a colour depending on what happens to these coordinates and how many iterations they go through. For a 1000 by 1000 image, there are one million points. To decide what color to chose, we may need thousands of computations with this formula, and billions of computations for such an image.
Fractal images are beautiful. They become even more interesting when we zoom deeper and deeper into them. Imagine the whole eastern coast of Canada on a map. Now as you zoom in and get closer and closer, you can see the actual coast line. Zooming further you see details of a beach, and then you start to make out individual stones. You continue to zoom in and can now see individual pieces of sand, and then every molecule that makes up a grain of sand and so on. The further we zoom into a fractal, the more complicated the shapes can be. As the scales get ever smaller, we need more and more precision in the computations, which result in longer and longer computing times.
When the computation time grows so large that it may take months or even years on a regular computer, we need to employ more computing power: in these scenarios supercomputers take centre stage. Supercomputers currently have large number of processor cores usually ranging from tens of thousands to millions of cores. The number one supercomputer in the world, in the Top500 list, has 3,120,000 cores. The fastest computer in Canada is Blue Gene/Q of SciNet (Southern Ontario Smart Computing Innovation Consortium/University of Toronto). With its 32,768 cores and computing power, it is ranked 89 in the November 2013 Top500 ranking.
The students in the fourth-year Distributed Computer Systems course at the Department of Computing and Software utilized 1024 cores on Blue Gene/Q to compute deep fractals at high precision. Furthermore, they produced videos of zooming in into the so-called Mandelbrot sets, where thousands of frames (images) were computed and then stitched together. You can watch these videos at www.cas.mcmaster.ca/~nedialk/COURSES/4f03/2014-Project/