Draw Sierpinski Fractal

The Sierpinski triangle (also called the Sierpinski gasket) is a fractal described by the Polish mathematician Wacław Sierpiński in his 1915 paper “Sur une courbe dont tout point est un point de ramification”. It is constructed by starting with an equilateral triangle and recursively removing the central triangle formed by joining the midpoints of the sides. The same construction had appeared in mosaic floors centuries earlier – in particular in 13th-century Cosmati pavements in Rome.

An IFS (Iterated Function System) construction: start with one triangle; each iteration replaces it with 3 smaller copies at half-scale at the corners. The bottom-most layer holds 3 to the N triangles. An older version of this tool labeled this “L-system” because logic.js contained a generateLSystem function that built the Sierpinski L-system string (axiom F-G-G, rules F → F-G+F+G-F and G → GG). That function was never actually called – rendering always used IFS subdivision. The dead code has been removed and the label corrected. Both formalisms yield identical figures, so visually nothing changes.

The chaos game is a randomised algorithm that converges onto the Sierpinski triangle. Place 3 vertices in a triangle. Start at any point. Pick one of the 3 vertices uniformly at random. Move halfway toward it. Plot the new point. Repeat. Despite the randomness, the points fall onto the Sierpinski attractor. It was popularised by Michael Barnsley in his 1988 book “Fractals Everywhere” as the canonical demonstration that an IFS attractor can be sampled by random iteration. This tool skips the first 100 iterations as “warm-up” so the starting position does not leave outlier points.

The similarity dimension is log(3) divided by log(2), about 1.5850. It is computed from 3 self-similar copies at scale 1/2: dim = log(N) / log(1/r) = log(3) / log(2). At iteration 0 the rendered figure is a single solid triangle (genuinely 2D); at higher iterations the figure approaches the attractor, whose dimension is the value displayed. The stats line marks this as “asymptotic” so the value is not mistaken for the dimension of the finite drawing.

At each iteration, 1/4 of the area of every triangle is removed (the central sub-triangle), leaving 3/4 of the parent area. After N iterations the remaining area is (3/4) to the N times the original. (3/4)^9 ≈ 0.075 – about 7.5% of the starting area remains after 9 iterations. The limit is 0 as N goes to infinity. The perimeter, meanwhile, scales by 3/2 per iteration and goes to infinity.

The starting point can be anywhere (random), including outside the triangle. The first few iterations bring it onto the attractor (the Sierpinski set), but those early points are not on the fractal yet. Skipping the first 100 prevents stray “outlier” points from being plotted. After 100 iterations the point is effectively on the attractor – every subsequent plot is genuine Sierpinski.

The Sierpinski triangle is the unique fixed point (attractor) of the IFS with three contractions, each scaling by 1/2 toward one of the triangle’s vertices. Recursive subdivision applies all three contractions at every step in lockstep (deterministic); the chaos game picks one of them randomly at every step (stochastic). Either way the trajectory converges to the same attractor – a result formalised in Barnsley’s IFS theorems.

No. The tool loads three static files (HTML, CSS, JS) and then runs entirely in your browser – recursion, chaos game iteration, 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.