Lindenmayer Systems

1890/1891 Peano and Hilbert detected two-dimensional curves that converge to a function that maps the interval [0,1] of the real numbers surjectively (but not bijectively) to the two-dimensional area [0,1] x [0,1]. This function is continuous but nowhere differentiable.

In 1968 a biologist, Aristid Lindenmayer, on looking for mathematical models of plant development, developed a string rewriting system that can be used to describe these curves:

Initial text: "L"

"L" -> "+RF-LFL-FR+"

"R" -> "-LF+RFR+FL-"

This assumes a turtle graphic:

"+" Turn left (by 90 degrees)

"-" Turn right (by 90 degrees)

"F" Move forward and draw a line (of one unit length)