Pythagoras tree

We illustrate self-similarity by way of making a Pythagoras tree with construction paper. Let us begin with two squares of side b, one on top of the other, as shown in figure 6. After cutting along the two dotted diagonal lines of the top square, what we have is an outline of house with a symmetric roof. Because of diagonal cuts, the roof is an isosceles triangle of   45 - 90 - 45. Hence, the run and rise of roof are b/2, and the length of roof (hypotenuse) is by the Pythagorean theorem. (Can you show this?) The house outline is called motif which appears again and again as we build a Pythagoras tree as follows. First, we cut out a piece of construction paper for house motif 1 with b = 10cm, and for house motif 2, 3, and 4 with the decreasing bases by factor 0.707, as shown in figure 7. Table 3 summarizes the base length of house motifs and the number of them needed for the construction of a Pythagoras tree.

Figure 6. The motif for Pythagoras tree

Figure 7.  Self-similar house motifs

 

Table 3.  Self-similar house motifs for b = 10cm

  Label

  Square side (cm)

  No. of houses

1

10

1

2

10/ 7.07

2

3

10/ / = 10/2 = 5

4

4

10 / / / = 10 / ()3 3.54

8

 

In figure 8, we first put the house motif 2 on each roof of house motif 1 for the 1st generation branching. Then, we put two of the house motif 3 on the roof of house motif 2 for the 2nd generation branch, and so on. As shown in the third column of table 3, the number of house motifs doubles at each generation of branching, following the doubling sequence of numbers 2, 4, 8, 16, …, as we have already encountered in Lesson 1. After the paper construction of Pythagoras tree, you can step through successive branching by Prog #2 up to the 9th generation with 512 branches.

Note:
Prog #2 is available for download on the index page.
If you use a Macintosh, view the Flash version.

Figure 8. Pythagoras tree

 

To see how self-similarity persists in figure 8, let us chop off one branch of the Pythagoras tree at the base of house motif 2. In Figure 9 the chopped-off branch is again a Pythagoras tree, the only difference being it is smaller than the original one of figure 8. If we again chop off a Pythagoras tree at the base of house motif 3, what we get is again a Pythagoras tree, but only smaller. In other words, any part of the Pythagoras tree is a Pythagoras tree, no matter how many times you chopped it off. We point out that the idea of recurring theme is also common in literature and music. For instance, a novel of mystery story has in it several of smaller mysteries of varying degree to pique the curiosity of readers.

Figure 9. Small Pythagoras tree

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