Lindenmayer systems (L-systems, turtle graphics) are used to generate Fractal curves (Koch, Sierpinski, Levy, Dragon), Space filling curves (Hilbert, Peano-Gosper), Growth patterns in plants, etc. It is an iterative string rewriting method. Starting from the Axioma a new string is formed by replacing the string symbols according to a few given Rules. The final string contains instructions telling the turtle where to go, or how to draw the path. The list contains a number of well known examples. Choose one of them and push the Draw button to draw it in the graphical window. You may construct your own examples and add them to the list.
More information on Lindenmayer systems: Some history | Mathforum | Ridldle | Wikipedia | Mathworld | Algorithmic botany
Lindenmayer systems – Space filling curves
Hilbert’s curve: Axioma X, Rules: X => -YF+XFX+FY- and Y=> +XF-YFY-FX+ . This type of curves have the remarkable property that they can fill a 2 (or more) dimensional space. The curves below were drawn by increasing the order (1,2,3,4,5 and 6) and decreasing the line length (32,16,8,4,2,1). The number of points inside the square that are covered by the curve increases with each step. In fact there is no limit to the number of points that can be covered by the curve as you proceed in this way. This means that the curve is “space filling”.