### 15.1 Formal definition

A L-System is a formal grammar with :

1. An alphabet V : The set of the variables of the L-System. V * stands for the set of the “words” we could generate with any symbols taken from alphabet V , and V + the set of “words” with at least one symbol.
2. A set of constant values S. Some of this symbol are common to all L-System. (in particular with the turtle!).
3. A start awiom ω taken from V + , it is the initial state.
4. A set of prodution rules P of the V symbols.

Such a L-System is defined as a tuple {V,S,ω,P}.

Let’s consider the following L-system:

• Alphabet : V = {A,B}
• Constants : S = {∅}
• Start Axiom: ω = A
• Rules :
 A → AB B → A

The two production rules are rewriting rules. On each step, the symbol A is replaced by the séequence AB, and the symbol B is replaced by A. Here are the first iterations of this Lindemayer system: • Itération 1: A
• Itération 2: AB
• Itération 3: ABA
• Itération 4: ABAAB

Ok, ok but concretely? Let’s read next section!