WHEN THUE-MORSE MEETS KOCH
In this paper, we reveal a remarkable connection between the Thue-Morse sequence and the Koch snow°ake. Using turtle geometry and polygon maps, we realize the Thue-Morse sequence as the limit of polygonal curves in the plane. We then prove that a sequence of such curves converges to the Koch snow°ake in the Hausdor® metric. In the ¯nal section we consider generalized Thue-Morse sequences and provide a characterization of those that encode curves converging to the Koch snow°ake.
"When Thue-Morse meets Koch," Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society,13(2005) 191-206 (co-authored with Kenyon student Jun Ma, Class of 2005)