L Systems

There was a time when people believed that a lot of different complex systems could be reduced to very simple descriptions, if only you could come up with the right mathematical key. Through various eras there have been tantalizing suggestions that this might work. In the 1960s and 1970s this concept was epitomized by catastrophe theory, originated by the french mathematician René Thom, which people hoped would be able to predict everything from wars to stock market crashes, by modeling them as simple shapes in higher dimensions.

In the 1980s computer graphics got its own version of this phenomenon in the form of L Systems, short for “Lindenmayer Systems”, named for the person who first came up with them. The basic idea is that you keep making grammatical substitutions to simple strings of symbols, generally replacing short sequences with longer sequences. If you think of each little sequence as a physical shape, like a tree branch, it becomes easy to build really cool fractal shapes that have some of the quality of real plants:

Using this general technique you can build up some really lovely computer graphic forms:

This is very exciting, until you realize that you can only use such a technique to produce variants on a narrow range of forms. I can use L-Systems to produce trees, but not elephants or rocks or mountains. If I want to get mountains, I might use a different set of techniques – generally known as fractal subdivision – originally developed by Benoit Mandelbrot and implemented in various ways by Richard Voss and others (including me, as it happens):

L-Systems can produce many different kinds of plants, but they more or less only produce different kinds of plants, whereas the fractal subdivision techniques only produce all different kinds of mountain terrains. None of these things really do what they at first seem – to create a robust recipe for dehydrated diversity – a few simple equations that can generate the entire Universe.

In a sense, the seeming ability to get something from nothing – to produce the endless variety of forms found in nature using only a few simple formulas – is an illusion, a kind of intellectual Ponzi scheme. For a while it seems as though you can generate anything using only some simple set of rules, but soon you hit a wall, and then you realize you need more and more rules, and different rules.

Natural complexity seems to defeat all human attempts to oversimplify and tame it. And maybe that’s a good thing.