Fractal Sequences , Part 1 : Overview

Infinity is where things happen that don't. — S. Knight Ever since Benoit Mandelbrot published his book on fractals [1], we've become accustomed to seeing fantastic and beautiful fractal images such as the ones at the bottom of the pages of this article. A fractal is a mathematical object that exhibits self similarity — it looks the same at any scale. If you zoom in on an image of a fractal, you see the same structure no matter how far you go, at least to the resolution of the image. In an actual fractal, there is no limit. In the case of sequences, a fractal sequence contains an infinite number of copies of itself, embedded within itself, as strange as this may seem. which is the same as the original sequence, as far as it goes. Of course, we can't show the complete sequence — it is infinite — for this you have to have faith. Rules that remove the values in fractal sequences to show their self similarity are called fractal decimation rules. The Morse-Thue sequence [2] is a binary fractal sequence, consisting only of 0s and 1s: The Morse-Thue sequence can be generalized to include more different values. The three-valued Morse-Thue sequence is Generalized Morse-Thue sequences also are fractal sequences. Can you find decimation rules that show their self similarity? Another binary fractal sequence is the rabbit sequence [3]: Many fractal sequences increase without bounds. Examples are sig