Tool

Pythagoras tree playground

Each square spawns two smaller squares atop a right triangle whose legs form the children’s sides. Tune the angle of the triangle and the recursion depth.

Triangle angle α: 45° (symmetric / 45°)

Depth: 10 (squares: 2,047)

Construction rule

Triangle angle α = 45°  (so β = 45°, hypotenuse = parent edge)

Each square of side s, rotated by θ, spawns:
  · a right triangle on its top edge with legs
      L = s · cos α   (left leg)
      R = s · sin α   (right leg)
  · a left  child square of side L, rotated by  θ + α
  · a right child square of side R, rotated by  θ − (π/2 − α)

Pythagorean: L² + R² = s²  (visual proof at every step)

At α = 45°: L = R = s/√2, two copies at scale 1/√2 each →
  similarity dim = log 2 / log √2 = 2  (tree fills space).

Background at /fractals/pythagoras-tree.

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