Iterated function systems

Pythagoras tree

Dutch mathematics teacher Albert Bosman invented the Pythagoras tree in 1942. Each square spawns two smaller squares atop a right triangle whose hypotenuse is the parent’s top edge, recreating the Pythagorean theorem at every step.

At a glance

Construction

1. Start with a unit square.
2. On its top edge place a right triangle with legs a, b.
3. Build a square on each leg (sides a, b).
4. Recurse on each new square.

Pythagorean theorem at every step

The hypotenuse of the right triangle has length 1, and the squares on the legs have areas a² and b², satisfying a2 + b2 = 1. Each iteration of the tree is a visual proof of a² + b² = c².

Acute, right, obtuse variants

For acute triangles the children do not overlap and the tree remains visually a tree; at 45° (isoceles right) the tree eventually overlaps itself, producing a space-filling object; for obtuse triangles the tree spreads out and may self-overlap early.

References

• Bosman, A. E., Het Wondere Onderzoekingsveld der Vlakke Meetkunde, 1942.
• Mandelbrot, B., The Fractal Geometry of Nature, W. H. Freeman, 1982.
• IFS Playground

Try it

Run an interactive playground at /tools/pythagoras-tree.