Sierpiński Carpet

Fractal dimension: ≈ 1.893

Visualization

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AIFS program

@
$dim=2
g=3
A=g^-1*([0,0]|[1,0]|[2,0]|[0,1]|[2,1]|[0,2]|[1,2]|[2,2])*A

Overview

The Sierpiński Carpet is a self-similar plane fractal first described by Polish mathematician Wacław Sierpiński in 1916. It belongs to the same family as the Cantor set (1D) and the Menger sponge (3D): all three are defined by the same recursive removal principle applied to progressively higher-dimensional cubes.

The carpet is constructed by repeatedly subdividing a square into a 3×3 grid of nine equal sub-squares and removing the central one, then applying the same rule to each of the eight remaining sub-squares. Despite having zero area, it is a surprisingly rich object: it is simultaneously a fractal, a topological curve, and a universal compact metric space.

History

Sierpiński introduced the carpet in the 1916 note Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée (“On a Cantorian curve that contains a bijective and continuous image of every given curve”). The title itself states the carpet’s key property: it is a universal plane curve. This means every compact, locally connected, one-dimensional metric space embeds homeomorphically into the Sierpiński Carpet — a remarkable topological result proved by Sierpiński in the same paper.

The carpet is also the defining example used by Karl Menger in his 1926 work on topological dimension theory.

Construction

At each iteration, divide the current square into a 3×3 grid and delete the centre cell. After $n$ steps, $8^n$ sub-squares of side length $3^{-n}$ remain.

The attractor is the limit $S = \lim_{n\to\infty} S_n$ and is defined exactly by the eight IFS maps.

Definition

In the algebraic form, the expansion $g = 3$ acts on integer vectors in $\mathbb{Z}^2$ . The eight translations are all integer points of the $3\times 3$ grid except the centre $[1,1]$ :

$A = g^{-1}\bigl([0,0]\mid[1,0]\mid[2,0]\mid[0,1]\mid[2,1]\mid[0,2]\mid[1,2]\mid[2,2]\bigr)(A)$

Each term $g^{-1}(\mathbf{x}+\mathbf{t})$ maps the attractor to a $\frac{1}{3}$ -scaled copy translated to position $\mathbf{t}/3$ — exactly the eight non-central cells of the $3\times 3$ grid. All coordinates are exact integers; no floating-point approximation is needed.

Properties

Hausdorff dimension: $\dfrac{\log 8}{\log 3} \approx 1.893$ , derived from the Moran equation $8 \cdot (1/3)^d = 1$
Lebesgue measure: exactly $0$ — at step $n$ the remaining area is $(8/9)^n \to 0$
Topological dimension: $1$ — despite living in the plane, the carpet contains no open 2D regions; it is a fractal “curve”
Universal curve: every compact, locally connected, one-dimensional metric space is homeomorphic to a subset of the Sierpiński Carpet (Sierpiński 1916)
Connectivity: the carpet is connected and locally connected; the complement in the unit square consists of countably many open squares
Number of transforms: 8, all with contraction ratio $\frac{1}{3}$ , all axis-aligned (no rotation or reflection)

Dimension	Fractal	Construction	Hausdorff dim
1D	Cantor set	Remove middle third of intervals	$\log 2 / \log 3 \approx 0.631$
2D	Sierpiński Carpet	Remove centre of 3×3 grid	$\log 8 / \log 3 \approx 1.893$
3D	Menger Sponge	Remove centre of 3×3×3 grid	$\log 20 / \log 3 \approx 2.727$

All three are universal in their respective dimensions: the Cantor set is universal among compact metric spaces, the Carpet among compact curves, and the Menger Sponge among compact 1-dimensional spaces embedded in 3D.