Fractals

In this video, we’re going to explore briefly the use of fractals which can be used in the production of artificial life.

The Boids video that you’ve seen demonstrates how apparently complex behaviour can emerge from very simple rules. And when we’ve watched our little robots in the past, we’ve seen similar emergence. So what we want to do in this short video is to explore briefly the concept of fractals, where interesting forms appear from very simple rules. And some of which can give a good appearance of artificial life. Some are just pretty patterns. There is a web page which I shall show which you can also explore.

So this is the web page and there are various options that you could run. I’ll explain what’s going on. For the basic, simple fractal you have what’s called an initiator, which in this case is just a straight line, and a generator which comprises another set of lines. And the idea is that you take every line in the initiator and replace it by the generator. And if you do that you see something like this. But of course, it’s produced some more lines. So we can then say, well, actually let’s replace all the lines that we’ve generated with the lines in the generator. And you get that. And then you do it again and again.

And you end up with this sort of pretty pattern which doesn’t look particularly realistic or otherwise. If you click the snowflake option it’s so called because it looks like a snowflake. And this is achieved by having an initiate which is a triangle and the same sort of replication. You can also do the so-called forest. Again, this is a straight line where the generator just consists of a line, a short peak, another line. And if you add more and more of replacements, you end up with something that looks a bit like a forest, or at least a forest perhaps that’s been hit by acid rain.

But that sort of idea you can use quite interesting shapes, like sort of something like a plant or a tree. All achieved just by taking a line and replacing it with a series of lines. You can, in this program, actually add some randomness to it. And then if you keep on redrawing it you end up with a tree that moves around. Quite fun. Another example of a tree that is quite famous is the Pythagoras tree, which looks like that. If I just go fewer and fewer replacements, you’ll see that initially it’s the trunk with two leaves on it. And then the leaves– you then become a branch with leaves on it. And then again, and again.

And you end up with this. I’ve got randomness on there. Let’s take the randomness off. And it’s a rather stylistic tree. You can have an uneven one where it’s a bit leaned on to the side. Where, with having the randomness, makes it more fun. But a instead of having lines which get replaced, you can also have– and I should put it down to fewer applications– a Pythagoras square tree. And this is called the Pythagoras tree because you’ve got a square and then you’ve got two other squares. And they meet, forming a triangle. And that reflects sort of Pythagoras’ theorem, that this squared plus this squared equals this squared.

And if you have an uneven Pythagoras tree, you have something like that. And then if you keep on doing replacements, you end up with this quite interesting looking tree. Very artificial, of course, but that’s fun. Another interesting shape is a so-called dragon. This goes back to the original one where you have a straight line which you replace by two lines which are effectively at right angles. So that as its drawn there doesn’t look like a right angle. Why is it called a dragon? Well, it looks a bit funny at the moment. But if I do more and more replications, you end up with something that looks like a sort of Chinese dragon.

That has emerged just out of taking one line and replacing it with two lines at right angles, many, many times. A completely different form of fractal is the fern, invented by Michael Barnsley. That’s achieved by drawing a series of dots, where the position of one dot is calculated from the previous position. And as you put more and more replacements on, you end up with something that’s almost a complete fern. This program you can run to explore. I will finish just by showing two other related things that they’re really fractals, they’re called space fitting curves. This case, the Sierpinski one– if I do fewer replacements, you can see it. So that’s the basic shape.

If you then start replacing things, it becomes that and that, and that. Which you can show separate, as well.

Have a play with the program to see what it does.

So in summary, we’ve done a short introduction to the idea of fractals, often achieved by replacing a line with multiple lines. And you get interesting effects that emerge, some of which are sometimes natural looking figures. Sometimes quite artificial. And these concepts can be developed further to produce more realistic images, for instance of landscapes, for clouds, and for other animals. A page also includes, as I say, the Sierpinski space filling curves. You can read up on those, if you’re interested.