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Sierpiński Carpet — Definition, Formula & Examples

The Sierpiński Carpet is a fractal created by repeatedly dividing a square into 9 equal smaller squares and removing the center one, then repeating this process on every remaining square infinitely.

The Sierpiński Carpet is a self-similar plane fractal obtained by starting with a unit square, subdividing it into a 3×3 grid of congruent subsquares, removing the open central subsquare, and iterating this procedure on each of the 8 remaining subsquares ad infinitum. Its Hausdorff dimension is ln⁡8ln⁡3≈1.893\frac{\ln 8}{\ln 3} \approx 1.893.

Key Formula

d=ln⁡8ln⁡3≈1.893d = \frac{\ln 8}{\ln 3} \approx 1.893
Where:
  • dd = Fractal (Hausdorff) dimension of the Sierpiński Carpet
  • 88 = Number of self-similar pieces at each iteration
  • 33 = Scaling factor (each piece is 1/3 the side length of the previous)

How It Works

Start with a solid square (iteration 0). Divide it into a 3×3 grid of 9 equal squares and remove the center square, leaving 8 squares (iteration 1). For each of those 8 squares, repeat the same process: subdivide into 9, remove the center (iteration 2). Each iteration multiplies the number of filled squares by 8 while shrinking each square's side length by a factor of 3. After infinitely many iterations, you get a set with zero area but infinite perimeter — a hallmark of fractal geometry.

Worked Example

Problem: A Sierpiński Carpet starts as a square with side length 1. After iteration 2, how many small filled squares are there and what is the total remaining area?
Iteration 0: You begin with 1 filled square. Its area is 1.
Squares=1,A0=1\text{Squares} = 1, \quad A_0 = 1
Iteration 1: Divide into 9 subsquares and remove the center. You keep 8 squares, each with side length 1/3.
Squares=8,A1=8×(13)2=89\text{Squares} = 8, \quad A_1 = 8 \times \left(\tfrac{1}{3}\right)^2 = \frac{8}{9}
Iteration 2: Each of the 8 squares is subdivided into 9 and loses its center, producing 8 × 8 = 64 squares, each with side length 1/9.
Squares=64,A2=64×(19)2=6481=(89)2\text{Squares} = 64, \quad A_2 = 64 \times \left(\tfrac{1}{9}\right)^2 = \frac{64}{81} = \left(\frac{8}{9}\right)^2
Answer: After iteration 2, there are 64 filled squares with a combined area of 6481≈0.790\frac{64}{81} \approx 0.790. In general, after iteration nn, the area is (89)n\left(\frac{8}{9}\right)^n, which approaches 0 as n→∞n \to \infty.

Why It Matters

The Sierpiński Carpet appears in courses on fractal geometry, discrete math, and topology. It serves as a universal plane curve — every compact one-dimensional curve in the plane is homeomorphic to a subset of it. Understanding its construction also builds intuition for geometric series, self-similarity, and non-integer dimensions.

Common Mistakes

Mistake: Confusing the Sierpiński Carpet (2D, based on squares) with the Sierpiński Triangle (2D, based on triangles).
Correction: The Carpet starts with a square divided into 9 parts (removing 1 center), while the Triangle starts with a triangle divided into 4 parts (removing 1 center). Their fractal dimensions differ: approximately 1.893 vs. 1.585.