L-Systems

L-System Geometry

Take the alphabet A = {F, +, -, x, y}, the axiom S = x, and the production rules

F -> F

+ -> +

- -> -

x -> -yF+xFx+Fy-

y -> +xF-yFy-Fx+

Here we give F, +, and - these interpretations

F means move one step forward while drawing a line

+ means turn counterclockwise by 90 degrees

- means turn clockwise by 90 degrees

Starting from a horizontal line segment, here are the geometric realizations of the first six steps.

With this rescaling, we get the sense that successive steps converge to a limiting shape, and that shape appears to fill a square. In fact, this is a way to construct Hilbert's space-filling curve.

Return to L-System Geometry.