In terms of visual impact, none has a greater effect than Fractals. They are the most recognizable math images. As static images, they are a form of algorithmic art: the pixels are colored according to a mechanical computation. However, they are also dynamic, as the math behind reveals this remarkable property of fractals: self-similarity: if you zoom in at a spot, you can see finer and finer details — you’ll discover copies of the original, identical or distorted, somewhere in those finer details. This is marvelously printed in The Beauty of Fractals by Heinz-Otto Peitgen and Peter Richter (1986).
Instead of zooming in, you can imagine zooming out, getting less and less details of the fractal. For the computer screen, this means getting bigger and bigger “pixels” — to be simulated by software. This suggests building fractals in stages: the first stage being a very coarse-grain pattern, then repeat the pattern in its sub-parts in the next stage for a less coarse-grain pattern. Keep repeating this process to get the ultimate fine-grain fractal.
This method of generating fractals is called IFS: Iterated Function Systems. You can get some hands-on experience with IFS by folding paper:
- Fold a sheet of paper once, then unfold it, the crease pattern (from side) is: |_.
- Fold the sheet twice, then unfold twice, now the crease pattern is: |_|▔.
- Fold the sheet three times, …. and so on.
If you’re smart enough, you can predict the successive crease patterns. Anyway, you’ll be fascinated by these fold-crease patterns as they develop. These fractals are called Dragon Curves, by iterating (i.e. repeating) a function (e.g. the method of folding).
In 1992 Scott Draves took a variation on this theme, using different functions in iteration and applying novel coloring schemes, leading to Fractal Flames.
Modern cellphones don’t have outside antenna. Instead they have fractal antenna inside, etched on the circuit board, to get better reception and pick up more frequencies such as bluetooth, cellular, and Wi-Fi all from one antenna at the same time!
The IFS technique had been applied to image compression algorithms, since natural objects like trees, mountains, clouds, etc. have an approximate fractal structure. Will verifying such fractal algorithms by a theorem prover be a worthwhile research project?