Draw Pythagoras Fractal

The Pythagoras tree is a plane fractal first described by the Dutch mathematics teacher Albert E. Bosman in his 1942 book “Het wondere onderzoekingsveld der vlakke meetkunde” (The Wondrous Field of Investigation of Plane Geometry). He drew it by hand. The construction: start with a square (the trunk). On top of the square, erect a right triangle. The two non-hypotenuse sides of the triangle become the sides of two new squares – and the same procedure is applied to each new square recursively.

If the branching angle is θ, the two child squares have sides cos(θ) and sin(θ) times the parent side. So the sum of their areas is cos²(θ) + sin²(θ) = 1 times the parent area – exactly the parent’s area. The total covered area is therefore conserved at every level: the tree always has the same total area as a single unit square, no matter how deep you go.

2 to the (N+1) minus 1. The tree is a binary recursion: 1 trunk at level 0, 2 children at level 1, 4 at level 2, …, 2 to the N at level N. The geometric series sums to 2 to the (N+1) minus 1. Iter 10 = 2,047. Iter 12 = 8,191. An older version of this tool’s code accidentally skipped the leaf generation (returned before pushing the last layer) so it produced 2 to the N minus 1 instead – half the correct count. That bug is fixed; the stats line shows the real number.

By the Pythagorean identity, the IFS (similarity) dimension is exactly 2.0 for every branching angle, not just 45°. Solve the Moran equation cos(θ) to the d plus sin(θ) to the d equals 1: at d = 2 we get cos²+sin² = 1, which holds for any θ. However, at angles far from 45° the tree self-overlaps, so the box-counting dimension of the actually visible figure dips below 2. The classical “the Pythagoras tree fills 2D space” statement strictly applies in the IFS sense; in the rendered image, only 45° has a non-overlapping tree that genuinely covers a finite area exactly.

Each square’s recursion depth – from 0 at the trunk to N at the leaves – is mapped to a color interpolated linearly between the trunk color and the leaf color. So the trunk takes the trunk color, the topmost leaves take the leaf color, and intermediate generations sit on a smooth gradient between them. It is the easiest way to see how many recursion levels are actually rendered.

At θ small, cos(θ) is close to 1 and sin(θ) is close to 0 – one child square is almost as big as the parent and the other is tiny. The tree grows mostly in one direction, and the small branches are barely visible. At θ close to 90° the same thing happens in mirror. At 45° both children are size √2/2 ≈ 0.707 of the parent and the tree is symmetric.

Real plants use very similar recursive branching – each node splits into two (or more) sub-branches at a roughly constant angle. The Pythagoras tree at 45° doesn’t look exactly like a maple, but the structural similarity is strong, which is why it appears in biology textbooks alongside L-system plant models. It is one of the canonical examples of “fractal in nature”.

No. The tool loads three static files (HTML, CSS, JS) and then runs entirely in your browser – recursion, drawing, PNG export, clipboard copy. You can disconnect from the internet after the page loads. No analytics, no tracking, no cookies.

Yes – free, unlimited, no signup, no watermark. Use the PNGs in lectures, papers, blog posts, or art freely. Attribution to is appreciated but not required.