L-Systems: Patterns with Grammar

Hilbert curve example

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+

The prodction sequence begins

x -> -yF+xFx+Fy- -> - +xF-yFy-Fx+ F+ -yF+xFx+Fy- F -yF+xFx+Fy- +F +xF-yFy-Fx+ -

= xF-yFy-Fx+FyF+xFx+Fy-F-yF+xFx+FyF+xF-yFy-Fx

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.

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