Sierpinski Carpet: Removing the Middle, Forever

Remove the middle, forever

The Sierpinski carpet is built by one rule applied recursively. Split a square into a 3×3 grid of nine equal sub-squares, delete the centre one, and then apply the same rule to each of the eight remaining sub-squares. Repeat as many levels as you like. Waclaw Sierpinski described it in 1916, two years before the more famous triangle that shares his name, and it is the two-dimensional analogue of the Cantor set — the same removing-the-middle idea, one dimension up, in a grid instead of a line.

function carpet(x, y, size, level) {
  if (level === 0) { fillSquare(x, y, size); return; }
  const s = size / 3;
  for (let row = 0; row < 3; row++)
    for (let col = 0; col < 3; col++) {
      if (row === 1 && col === 1) continue;  // skip the centre cell
      carpet(x + col * s, y + row * s, s, level - 1);
    }
}

How much area survives

Every level keeps 8 of the 9 sub-squares, so after n levels the remaining area is (8/9)ⁿ of the original square. That ratio shrinks slowly at first — 88.9% after one level, 79% after two — but it is a genuine geometric decay: by level 10 barely under 28% of the square remains, and by level 20 less than 8%. In the limit as n → ∞, the area goes to exactly zero, even though the carpet still contains infinitely many points and, remarkably, is still path-connected — you can walk between any two surviving points without leaving the carpet, unlike the fully-disconnected Cantor set.

A dimension between a line and a plane

The carpet's fractal (Hausdorff) dimension follows directly from the construction: each level replaces 1 square with 8 copies at 1/3 scale, so the dimension solves 8 = 3^D, giving D = log 8 / log 3 ≈ 1.8928. That number sits between 1 (a plain line) and 2 (a filled plane) precisely because the carpet is almost a filled square — it retains most of its area at low iteration counts and only asymptotically thins to zero-area, dimension ≈1.89 dust.

This is the same log(count)/log(scale) formula that gives the Sierpinski triangle its dimension of log 3 / log 2 ≈ 1.585 (3 copies at half scale) and the Cantor set its dimension of log 2 / log 3 ≈ 0.631 (2 copies at one-third scale). Comparing the three side by side is the fastest way to build intuition for what "fractional dimension" is actually measuring: how densely a self-similar shape fills the space it's embedded in.

The 3D cousin: the Menger sponge

Extend the same subdivide-and-remove idea to a cube split into a 3×3×3 grid of 27 sub-cubes, removing the centre cube of each face plus the very centre cube (7 removed, 20 kept), and you get the Menger sponge — a solid with zero volume, infinite surface area, and a cross-section through the middle that is exactly a Sierpinski carpet. Its dimension is log 20 / log 3 ≈ 2.727, again strictly between the dimension of a surface (2) and a solid (3).

Where this pattern shows up outside pure math

The carpet's fractal antenna cousins are a real engineering application: antennas built with Sierpinski-carpet or Sierpinski-triangle geometry are naturally multi-band, because each iteration level resonates at a different self-similar scale, letting one physical antenna respond efficiently across several separated frequency bands instead of needing one antenna per band. The same recursive void-cutting idea also appears in truss and lattice engineering, where removing material at every scale trades a small loss of stiffness for a large reduction in weight.

Frequently asked questions

Try it live

Everything above runs in your browser — open Sierpinski Carpet and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.