Streams
When Thue meets Koch
This animation is realized by Joerg Endrullis, 2010, in the framework of our study of streams. In a remarkable paper by Holdener and Ma in 2005, called When Thue meets Koch, it is explained how the famous Thue-Morse sequence 1001 0110 0110 1001 …, a morphic sequence with start word 10 and morphisms 1 \to 10, 0 \to 01, relates to the equally famous Koch snowflake that we considered also in this site as one of the Aha insights. The sequence can be considered as a drawing program for a ‘smart turtle’, in turtle graphics. It draws successively better approximations of the snowflake fractal, as the animation shows. Btw: it was pointed out by Jean-Paul Allouche that this observation of the snowflake as rendered by a notation with complex numbers tantamount to this instance of turtle graphics, essentially was made twenty years earlier, early 1980’s, by Michiel Dekking, Delft University, the Netherlands. Dekker did not used a turtle for his fractal drawings, but sums of exponential complex numbers $\Sigma e^{\pi .i . s_n}$.
Actually, turtle graphics is more expressive or versatile in its use than summing complex numbers e^{\pi .i . s_n, because turtles can be easily equipped with an auxiliary device, namely a Finite State Transducer (FST), transforming the stream entries 0,1 into graphical output instructions, such as move a unit in the direction of the write head, turn write head over 60 degrees, move a unit without drawing ink on the paper etc. Such a turtle we call a ‘smart turtle, and it can draw various fractals using intermediate scaling-down to keep the drawing’ on the screen, just as in this animation. The successive approximations are converging, in the sense of the Hausdorff metric. A smart turtle thus can also draw the fractal known as the Cantor middle third set, or Cantor dust. As is well-known, smart turtles were proposed and used in the didactical popular geometric drawing tools introduced in Logo, developed by Seymour Papert et al. around 1960.