Lesson

In Lesson 2, the Pythagoras tree was constructed by replicating house motifs in a self-similar fashion. A square building with the roof of 90 ° - 45 ° - 45 ° isosceles triangle is house motif, and the self-similar house motifs are scaled by factor . A rule for construction is to place a smaller house motif on each slanted roof of the larger one. Although the rule is simple, a surprisingly complex pattern emerges after many iterations, while the size of branches shrinks by factor at each stage of branch doubling. In any event, we find Pythagoras trees of endlessly smaller size within a Pythagoras tree. Although this is somewhat akin to seeing a finer detail of cells with a more powerful microscope, what we see here is always a Pythagoras tree due to the self-similarity. In this lesson, complex patterns are generated by the motifs of the Koch curve, dragon curve, and modified Pythagoras trees.

Koch curve

Unlike the Cantor function which cuts up a line segment and throws away the middle one-third piece, the Koch function actually increases the input line segment by 1/3, as shown in figure 1.

Figure 1. The Koch function of type 1

In essence, it divides the input line segment of, say, 1 meter long into three equal lengths, replaces the middle one-third section with an equilateral triangle of side 1/3 meter, and then removes the base of the equilateral triangle;

We therefore end up with a 1/3 meter longer line segment than the input, that is, meter long. This is the Koch curve proposed by the Swedish mathematician Helge von Koch in 1904. We now repeat the Koch function. Figure 2 shows that each of the 1/3 line segments picks up an equilateral triangular bump, so that the total length is now meter (Can you explain this?). Prog#3a will let you step through up to the 4th generation Koch curve. How many line segments are in the 3rd and 4th generation Koch curves and what are the total lengths? You may fill in the second and third column entries of table 1. Also, had you iterated 10 times, how long is Koch curve of the 10th generation? As you iterate the Koch function (infinitely) many times, not only the number of line segments increases very rapidly but the total length also becomes (infinitely) long. You may compare the Koch curve (table 1) with Cantor rod summarized in table 4 of Lesson 1. Recall that the Cantor function has thrown away almost the entire input rod after, say, 10 iterations.

Figure 2. First two generations of Koch curve

Table 1. Koch curve

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