Let’s say I have a function that takes in a real number, adds the negative of one half that real number to itself and returns this new number. Now, take that new number and feed it in to the function. Repeat this as many times as you please and you can see that the output is tending towards zero. Here, zero is an attractor, and zero’s basin of attraction is the entire real number line, not including zero itself (for zero the value is fixed and that’s just boring). An attractor doesn’t have to be a single point, it can be a curve, or a set of a few discrete points, or whatever. A strange attractor is an attractor that is a fractal.
Well, that is at least the cartoon version, as we say.
So, on to that neat result. Let’s say that on the real plane you pick 3 distinct points. Here we will pick them such that they mark the corners of an equilateral triangle, but this is not necessary. Now, place a point at any finite value at all on the real plane. It doesn’t matter where, inside the three points, outside the three points, directly between two points, it doesn’t matter. Now, choose any one of those three points at random (we will call those special points corners). Draw a new point one half of the distance between that first random point and the chosen corner. Now, pick a corner at random, draw a new point half way between that last point and the new chosen corner. Repeat this as many times as you please. We are picking corners at random but a very regular shape starts to emerge.
The shape is this function’s strange attractor, which just happens to be Sierpinski’s Triangle. Sierpinski’s Triangle is classically made by taking an equilateral triangle, finding the midpoint of each side and removing the triangle described from those points from the triangle’s centre. Then do the same with all three new triangles you have. Then with all nine new triangles etc, an infinite number of times. Yet we can approach that same strictly defined shape using a process that is in fact dependent on randomness.
Below are two images of this from some software I whipped up (still in progress as it’s a bit clunky). The larger of the two is what this process looks like after 40,000 iterations (sort of, I’m rounding off to the nearest pixel right now with it). The smaller one is a blow up of one region of the image.
You can see the initial point which was clearly outside the attractor altogether. It starts nearish one of the corners, and it seems the same corner was picked a few times in a row so there is another point halfway between it and the same corner, and another point halfway between that one and the same corner again. A different corner was chosen after that and the points start to be drawn exclusively inside the outer boundaries of the triangle, and once in, it never escapes again. The new points just wanders around inside drawing out this fractal.
If you pick four corners in a square and preform similar transforms you can get a different fractal, in this case a 2 dimensional version of the Cantor Set. I hesitate to call the result Cantor Dust or Sierpinski’s Carpet as the method to produce these points is so different, but like the above, either do describe the shape of the strange attractor.