Take the alphabet
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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