Lindenmayer Systems and the Logo Turtle

Lindenmayer Systems, or L-Systems:

Take a string as a set of instructions in turtle graphics. “L” means “turn left”, “R” means “turn right”, and “F” means “go forward”.

Now imagine starting with a string “A” and a set of rules. You replace the characters in this string according to the rule a fixed number of times. Then for every type of character, you can either replace it with one of “L“, “R“ or “F“, or ignore/remove it.

For example, suppose you have one rule,

and a final rule

In this example, it takes 12 or 12 turns to turn completely around.

We start with and apply the rule to get . We apply the rule again to get , and so forth. In my example, I stopped when I got a string that would have 5000 forward moves in it.

Here's what it looks like:

That’s one continuously traced path, all thanks to string rewriting. I’ve also created a page with a large number of random rules. It takes a lot of memory because of all the thumbnails, but if you’re curious about the rules used to generate a shape, simply follow the link to the SVG and “view source” to see what generated that shape.