Skip to main content

sierpinski-triangle

The Sierpinski Triangle is a classic fractal: take an equilateral triangle, remove the central upside-down triangle, repeat on each remaining sub-triangle, repeat again. After infinite iterations, the result has zero area but infinite perimeter — a fractal of dimension log(3)/log(2) ≈ 1.585. The ZTools Sierpinski Triangle Generator draws the construction live in SVG up to depth 9 (19,683 triangles), with custom colours, sizes, and optional chaos-game generation as an alternative to recursion.

Use cases​

Math classroom — illustrate fractals​

Increment depth from 0 to 7 to show how iteration produces the fractal. Visually unforgettable.

Generative wall art​

High-depth Sierpinski with custom colours produces striking minimalist art.

Algorithm viz​

Demonstrate divide-and-conquer / recursion visually. Each call splits the problem into 3 sub-problems.

Teach Pascal's Triangle connection​

Color Pascal's Triangle entries by parity (odd/even) — the pattern that emerges IS the Sierpinski Triangle.

How it works​

  1. Pick depth — 0 = single triangle. 1 = 3 sub-triangles. N = 3^N triangles.
  2. Choose colours — Single fill or gradient by depth (top triangle one colour, deeper sub-triangles another).
  3. Pick generation method — Recursive (exact, deterministic) or chaos game (random — converges to the same shape after ~10,000 points).
  4. Export — SVG (clean vector), PNG (rasterised at chosen size).

Examples​

Input: depth 5, recursive

Output: 243 small triangles arranged in the famous self-similar pattern.

Input: depth 7, recursive

Output: 2,187 triangles — fine detail visible at high res.

Input: chaos game, 10,000 points

Output: Each point random + halfway-to-vertex; result converges to the Sierpinski shape after ~5,000 iterations.

Frequently asked questions​

What's the chaos game?

Pick 3 vertices (corners of an equilateral triangle). Start at any point. Repeatedly: pick a random vertex, move halfway towards it, plot. After thousands of iterations, the plotted points form the Sierpinski Triangle. Counter-intuitive but provably correct.

Pascal's Triangle connection?

Color row n column k of Pascal's Triangle black if C(n, k) is odd, white if even. The resulting pattern is a Sierpinski Triangle (visible from row 16 or so).

How big can depth be?

9 in this tool (19,683 triangles). Beyond, SVG path count causes lag. For higher depth, use the chaos game with 100,000+ points.

Privacy?

All in browser.

Tips​

  • For teaching, animate from depth 0 to depth 5 — each step is dramatic.
  • Chaos game shows that randomness can produce structure — counter-intuitive demo for students.
  • Connect to Pascal's Triangle visually — same fractal, two derivations.
  • Export SVG for slides; PNG only when a raster system needs it.

Try it now​

The full sierpinski-triangle runs in your browser at https://ztools.zaions.com/sierpinski-triangle — no signup, no upload, no data leaves your device.

Open the tool ↗

Last updated: 2026-05-06 · Author: Ahsan Mahmood · Edit this page on GitHub