Recursive turtle programs and iterated affine transformations

摘要

We provide a formal proof of equivalence between the class of fractals created by recursive-turtle programs (RTP) and iterated affine transformations (IAT). We begin by reviewing RTP (a geometric interpretation of non-bracketed L-systems with a single production rule) and IAT (iterated function systems restricted to affine transformations). Next, we provide a simple extension to RTP that generalizes RTP from conformal transformations to arbitrary affine transformations. We then present constructive proofs of equivalence between the fractal geometry generated by RTP and IAT that yield conversion algorithms between these two methods. We conclude with possible extensions and a few open questions for future research.