Generalized results on the convergence of Thue-Morse turtle curves

TL;DR

This paper establishes generalized conditions under which Thue-Morse turtle curves converge to fractal curves like the Koch curve, connecting turtle graphics visualizations with complex sum analysis and fractal convergence in the Hausdorff metric.

Contribution

It explicitly links Thue-Morse turtle curves to complex sums and proves their convergence to fractals, confirming a conjecture about Koch curve convergence.

Findings

Thue-Morse turtle curves can coincide with Dekking's complex sums

Scaled turtle curves can converge to fractals in the Hausdorff metric

Conditions are identified for convergence to the Koch curve

Abstract

Work by Ma and Holdener in 2005 revealed that using turtle graphics to visualize the Thue-Morse sequence can result in curves which approximate the Koch fractal curve. A 2007 paper by Allouche and Skordev pointed out that this phenomenon is connected to certain complex sums considered by F. M. Dekking in 1982. We make this connection explicit by showing that a broad class of Thue-Morse turtle curves will periodically coincide with sums of the form considered by Dekking, and we use this result to prove that scaled versions of these curves can converge to various fractal curves in the Hausdorff metric. In particular, we give a condition under which any Thue-Morse turtle curve will converge to the Koch curve, confirming a conjecture made by Hans Zantema in 2016.