Modification #2: To make the tree trunk longer between branching, a new house motif is defined by the following rules;

1. Stack up two squares of side a for the building.
2. Divide the top side of the building into ar and a(1-r), where r is a number between (0, 0.5).
3. Place a right-angled triangle of (₂, 90°, 90°-₂) along the side a(1-r).
4. Place another right-angled triangle of (₁, 90°, 90°-₁) along the side ar (angle ₁ must be small than ₂).
5. Put a square of side a(1 - r)/cos₂ on the right-hand-side roof.
6. Put a square of side arcos₁ + a(1 - r)sin₁tan₂ on the left-hand-side roof and cut off the overhang.

It is perhaps easier to visualize the above rules as depicted in figure 7. In any event, for the convenience of readers, the construction of motif by Modification #2 has been coded into Prog#3f which first demonstrates the 10th generation branching with the pre-set parameters, r=0.4, ₁=30° and ₂=36°. Afterwards, you may experiment with Prog#3f by choosing r in the range (0.15, 0.5). We, however, suggest stepping through the following values of r

r = (0.15, 0.19, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5).

Figure 7. Motif for Pythagoras tree (modification #2)

We make the following observations. For small r, the main tree trunk wraps around completely so that it looks more like a wreath than a tree. In particular, at r = 0.19 it is a circular wreath. As r increases toward 0.5, the tree looks more natural. Since branching brings about turning, there appear many swirling branches of different radii in the Pythagoras tree. A big swirling branch has small swirling branches, which in turn have smaller swirling branches, and so on. This is a manifestation of self-similarity.

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