Michael Bradford Williams

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A Lindenmayer system, or L-System is a collection of symbols and set of rules for replacing any symbol with one or more other symbols. That may not sound immediately interesting, but when combined with a means to generate a graphical representation of a sequence of symbols, L-Systems are capable of producing a huge array of complex images.

More precisely, an L-System consists of three items:

1. A set of symbols, called the alphabet, which is composed of two disjoint subsets:
   1. variables
   2. constants
2. A set of replacement rules, which transform a symbol from the alphabet into a sequence of symbols from the alphabet, such that
   1. a variable can be replaced a sequence of any symbols in the alphabet
   2. a constant can only be replaced by itself
3. An initial sequence of symbols, called the axiom

Once an L-System is defined, the rules can be applied simulaneously to each symbol in the axiom to generate a new sequence of symbols. This process can be repeated for the new sequence, and so on.

A simple example of an L-System has alphabet \(\{A, B\}\) (both variables), rules \(\{A \rightarrow AB, B \rightarrow BA\}\), and axiom \(A\). Then repeated application of the rules, starting from the axiom, yields

1. \(A\)
2. \(AB\)
3. \(ABBA\)
4. \(ABBABAAB\)
5. etc.

To visualize an L-System, geometric notions are typically associated with the symbols. For example, a symbol might mean "draw a line segment of unit length" or "rotate by angle \(\pi/2\)". The following app has a collection of pre-defined L-Systems that can be loaded, and the rules can be applied iteratively and visualized. The details of the L-System will be printed below the image.

A huge gallery of interesting examples of L-Systems can be found here.

Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Example 7 Example 8 Example 9 Example 10 Example 11 Example 12 Example 13 Example 14 Example 15 Example 16 Example 17 Example 18 Example 19 Example 20 Example 21