The Sierpiński Carpet: A Classic Fractal of Infinite Complexity

The Sierpiński carpet is one of the most famous examples of a self-similar fractal, first described by Wacław Sierpiński in 1916. It is a two-dimensional generalization of the Sierpiński triangle, constructed by recursively removing central squares from a grid. Despite being infinitely complex, it has elegant mathematical properties involving dimension, measure, and topology.


1. Construction of the Sierpiński Carpet

The Sierpiński carpet is built using the following recursive algorithm:

• Start with a unit square (generation 0).
• Divide it into 9 equal smaller squares (like a tic-tac-toe grid).
• Remove the open middle square.
• Apply the same process recursively to each of the 8 remaining squares.
• Repeat this process infinitely.

Let denote the th iteration of the Sierpiński carpet.

2. Area and Perimeter

Let’s calculate the total area after iterations:

At each iteration:

• The total number of squares increases by a factor of 8.
• Each square is scaled by a factor of in both dimensions.

So:

• Number of squares at iteration :
• Area of each square:
• Total area:

Hence:

Thus, the area of the Sierpiński carpet is zero, even though it occupies a full square in the limit.

Perimeter becomes infinite as the number of boundaries grows without bound.

3. Hausdorff Dimension

To find the Hausdorff dimension, we use the formula for self-similar fractals:

If the carpet is made of self-similar copies of scale , then:


So, the dimension lies between 1 and 2 — it is more than a line, but less than a surface.

4. Topological and Geometric Properties

• The Sierpiński carpet is nowhere dense and totally disconnected in measure.
• It is self-similar — it looks the same at any magnification.
• It is universal for 1-dimensional planar curves: any such curve can be embedded in it.
• It has zero Lebesgue measure but is uncountable.

5. Generalizations

• The Menger sponge is the 3D analog of the Sierpiński carpet.
• The Sierpiński–Menger cube is another hybrid.
• Higher-dimensional analogs can be defined using similar removal patterns.

6. Applications

The Sierpiński carpet appears in:

• Modeling porous materials.
• Antenna design (e.g. fractal antennas).
• Image compression and recursive rendering.
• Topological counterexamples in mathematical logic.

7. References

• Wikipedia: Sierpiński Carpet
• Falconer, K. Fractal Geometry: Mathematical Foundations and Applications
• Mandelbrot, B. The Fractal Geometry of Nature
• AoPS Wiki: Sierpiński Carpet