Sierpinski Triangle Generator

The Sierpinski Triangle Generator draws the most famous fractal in computer science and recreational mathematics — at any depth, from any outer triangle, using either the deterministic recursive subdivision algorithm or the surprising chaos game random walk. Side-by-side mode draws both at once so you can see that randomness and recursion converge to exactly the same shape. The tool reports the leaf count, the precise remaining area, and the Hausdorff dimension (log 3 / log 2 ≈ 1.5849625), and exports a clean SVG suitable for slides, worksheets, or laser-cutting.

How the Sierpinski Triangle Is Built — Step by Step

What Makes This Sierpinski Generator Different

What Is the Sierpinski Triangle?

The Sierpinski triangle (also called the Sierpinski gasket or sieve) is a self-similar fractal first formally described by Polish mathematician Wacław Sierpiński in 1915. It is constructed by recursively removing the central inverted sub-triangle from every triangle that remains, leaving three smaller copies of the original at the corners. The process is repeated indefinitely; the limiting set has measure zero (no area at all) but contains uncountably many points and has a non-integer fractal dimension of log 3 / log 2 ≈ 1.5849625 — meaning it is "fatter" than a 1-dimensional curve but "thinner" than a 2-dimensional region.

The Chaos Game: Order from Randomness

The chaos game, popularized by Michael Barnsley in his 1988 book Fractals Everywhere, is one of the most striking results in dynamical systems. Pick any starting point inside the triangle and follow this rule: choose one of the three vertices uniformly at random, jump exactly halfway from your current point toward that vertex, and drop a dot. Repeat thousands of times. After a short burn-in period, every subsequent point lies on the Sierpinski triangle with probability 1 — the fractal is the unique attractor of this random walk. The deterministic recursive subdivision and the random chaos game are both instances of an Iterated Function System (IFS) with the same three midpoint maps; by the contraction mapping theorem, every IFS with strict contractions has a unique non-empty compact attractor that any randomized trajectory converges to.

Recursion Depth Reference

Where the Sierpinski Triangle Shows Up

• Pascal's triangle modulo 2: color each cell of Pascal's triangle black if it's odd and white if it's even. The black cells form the Sierpinski triangle exactly — a stunning bridge between combinatorics and fractal geometry.
• Cellular automaton Rule 90: the one-dimensional cellular automaton "Rule 90" of Stephen Wolfram, started from a single black cell, generates the Sierpinski triangle row by row.
• Fractal antennas: Sierpinski monopole and dipole antennas exploit self-similarity to achieve multiband resonance — a single antenna can cover many frequency ranges. They are used in modern mobile phones and Wi-Fi devices.
• Computer-science teaching: a canonical example for recursion, divide-and-conquer, IFS, and dimension theory. It also makes a great unit-test target for graphics libraries.
• Generative art and design: textiles, logos, laser-engraved coasters, music-festival posters — the fractal's combination of mathematical depth and visual simplicity makes it endlessly remixable.
• Tower of Hanoi state graph: the state graph of the Tower of Hanoi puzzle with N disks is exactly the depth-N Sierpinski graph — the same structure under a different cover.

Sierpinski Triangle vs Pascal's Triangle: A Surprising Identity

Write out Pascal's triangle for many rows, then color cells with odd binomial coefficients dark and cells with even coefficients light. The picture is a perfect Sierpinski triangle. The reason is Kummer's theorem on binomial coefficients modulo a prime: C(n, k) mod 2 equals 1 if and only if the binary representation of k is bit-wise less than or equal to that of n. Recursively this exactly produces the Sierpinski rule — three copies above, central one missing — and the limiting picture is the fractal. Switch this generator to "Pascal triangle layout" to see the connection in matching orientation.

Common Misconceptions

• "The Sierpinski triangle has zero area." True — but only in the infinite limit. At any finite depth N, the leaves still fill (3/4)N of the outer area. At depth 9 that's still about 7.5%, plenty visible.
• "You need an equilateral triangle to start." False. The recursion works on any triangle (right, obtuse, degenerate-as-long-as-not-collinear). The fractal shape is preserved by every affine transformation. Switch outer shapes in this tool to see for yourself.
• "The chaos game requires special random numbers." No — uniform integer-3 randomness is enough. Any starting point also works (after a short burn-in to forget the start).
• "Fractal dimension is just a fancy name for an integer." No — the Sierpinski triangle's dimension is genuinely between 1 and 2. There is no integer dimension that captures how it scales.

Frequently Asked Questions