First choose a right-angled triangle, indicating on the semicircle where the vertex of the right angle is (the hypothenuse will be the diameter of the semicircle).

Then use the watering can button to grow the Pythagorean tree…

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The Pythagoras tree is a plane fractal constructed from squares. Invented by the Dutch mathematics teacher Albert E. Bosman in 1942, it is named after the ancient Greek mathematician Pythagoras because each triple of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem. If the largest square has a size of L × L, the entire Pythagoras tree fits snugly inside a box of size 6L × 4L. The finer details of the tree resemble the Lévy C curve.

https://en.m.wikipedia.org/wiki/Pythagoras_tree_(fractal)

Begin with a square. Construct a right isosceles triangle whose hypotenuse is the top edge of the square. Construct squares along each of the other two sides of this isosceles triangle.

Iterated Function System

Take the initial square to be a unit square with lower left corner at the origin. Let α be the left angle shown in the figure below.

For the first function, corresponding to the upper left square, we must scale by cos(α) and rotate counterclockwise by α, then translate straight up. For the second function, corresponding to the upper right square, we must scale by sin(α) and rotate clockwise by 90°−α, then translate the point at the origin to the point at the right angle in the triangle. Finally, the third function is just the identity. This keeps the squares already drawn in their current locations while the first two functions add additional squares.

https://larryriddle.agnesscott.org/ifs/pythagorean/pythTree.htm

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Fractals

From the book MATH TALK: Mathematical Ideas in Poems for Two Voices by Theoni Pappas.

https://www.larryriddle.agnesscott.org/ifs/poem.htm

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Jos de Mey

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Pythagoras tree (for 3D print and laser cut)

https://www.printables.com/model/986086-pythagoras-tree-for-3d-print-and-laser-cut