Weird Sets

Imagine a long (perhaps infinite) white string laid out across a table with three markings – one labelled F and the two others at equal distances on either side of it labelled S and S’, with S on the right. Now fold the string at F from left to right (so that the point which was as S’ ends up at S). Next, stretch the string away from the point S by a factor of three (so that the point originally at F ends up on the other side of S’). Note that in this process all points to the right of S stay there. We will not be interested in those points any more, so we could just cut them off, but instead let’s just colour them in a rinse of black dye.

Now repeat, rinse, and repeat.

At each stage, the end part of the last fold gets changed from white to black. And the pattern of white and black on the string gets more and more complicated. Try to imagine what happens if we continue repeating the process for ever. (It has a famous name, but “wierd” will do for now.)

Now imagine doing something similar in two dimensions with a fixed point F but with the relative positions of S and S’, and the stretch factor, depending on direction. I am sure you can see that this could get pretty messy real fast.

The Mandelbrot set is rather similar.

Imagine being a Filo (or flaky pastry) chef working in a pizza parlour. It would get boring pretty fast, so you might want to try something like this:

Roll out a bunch of dough onto a big round table. Now choose a point near the middle and make a straight cut through the dough out to the edge of the table. Next (and this, like porn sex, is easier to imagine than to do) take one side of the cut and swing it all the way around the centre point so that it comes back to where it started. Oh, and I’m so sorry I forgot to mention this, but before that last move you should have marked the dough with a bunch of radial lines and concentric circles in such a way that at each radius R the spacing of the radial lines exactly matches that of the concentric circles to create a pattern of little approximate squares. (Another advantage of imagination – either in place of, or in conjunction with reality – is that you can dress up your partner however you want – and even change the costume in mid-action.)

Now with the circles and rays in place (or imagined), choose some particular radius and push the dough away from that circle in order to re-square all the now sideways-stretched little former squares. This will push in everything inside the chosen radius and push out the rest.

You can see that if the cut and stretch are always based radially from exactly the same point, then the process, if repeated indefinitely, will move all points outside the special circle off the table – and those inside it will squeeze ever closer to the centre. But if some other point is moved to the centre before each iteration, then the pattern of what gets left on the table could be quite complicated – and that’s where the Mandelbrot set comes from!

(It’s basically the set of points on the table that can be used as the one that always gets moved to the centre without the bit of pastry that started there ever falling off the table.)