Sierpinski carpet

The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions (another is Cantor dust). Sierpiński demonstrated that this fractal is a universal curve, in that it has topological dimension one, and every other compact metric space of topological dimension 1 is homeomorphic to some subset of it.

The technique of subdividing a shape into smaller copies of itself, removing one or more copies, and continuing recursively can be extended to other shapes. For instance, subdividing an equilateral triangle into four equilateral triangles, removing the middle triangle, and recursing leads to the Sierpinski triangle. In three dimensions, a similar construction based on cubes produces the Menger sponge, which is also a universal curve.

Construction

The construction of the Sierpinski carpet begins with a square. The square is cut into 9 congruent subsquares in a 3-by-3 grid, and the central subsquare is removed. The same procedure is then applied recursively to the remaining 8 subsquares, ad infinitum. It can be realised as the set of points in the unit square whose coordinates written in base three do not both have a digit '1' in the same position.^[1]

The process of recursively removing squares is an example of a finite subdivision rule.

The area of the carpet is zero (in standard Lebesgue measure). The Hausdorff dimension of the carpet is log 8/log 3 ≈ 1.8928.^[2]

The Sierpinski carpet can also be created by iterating every pixel in a square and using the following algorithm to decide if the pixel is filled. The following implementation is valid C, C++, and Java.

/**
* Decides if a point at a specific location is filled or not.  This works by iteration first checking if
* the pixel is unfilled in successively larger squares or cannot be in the center of any larger square.
* @param x is the x coordinate of the point being checked with zero being the first pixel
* @param y is the y coordinate of the point being checked with zero being the first pixel
* @return 1 if it is to be filled or 0 if it is open
*/
int isSierpinskiCarpetPixelFilled(int x, int y)
{
    while(x>0 || y>0) // when either of these reaches zero the pixel is determined to be on the edge 
                               // at that square level and must be filled
    {
        if(x%3==1 && y%3==1) //checks if the pixel is in the center for the current square level
            return 0;
        x /= 3; //x and y are decremented to check the next larger square level
        y /= 3;
    }
    return 1; // if all possible square levels are checked and the pixel is not determined 
                   // to be open it must be filled
}

Process

Brownian motion on the Sierpinski carpet

The topic of Brownian motion on the Sierpinski carpet has attracted interest in recent years.^[3] Martin Barlow and Richard Bass have shown that a random walk on the Sierpinski carpet diffuses at a slower rate than an unrestricted random walk in the plane. The latter reaches a mean distance proportional to n^1/2 after n steps, but the random walk on the discrete Sierpinski carpet reaches only a mean distance proportional to n^1/β for some β > 2. They also showed that this random walk satisfies stronger large deviation inequalities (so called "sub-gaussian inequalities") and that it satisfies the elliptic Harnack inequality without satisfying the parabolic one. The existence of such an example was an open problem for many years.

Wallis sieve

A variation of the Sierpinski carpet, called the Wallis sieve, starts in the same way, by subdividing the unit square into nine smaller squares and removing the middle of them. At the next level of subdivision, it subdivides each of the squares into 25 smaller squares and removes the middle one, and it continues at the ith step by subdividing each square into (2i + 1)² smaller squares and removing the middle one. By the Wallis product, the area of the resulting set is π/4,^[4][5] unlike the standard Sierpinski carpet which has zero limiting area.

References

• Sierpiński, Wacław (1916). "Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée". C. r. hebd. Seanc. Acad. Sci., Paris (in French) 162: 629–632. ISSN 0001-4036. JFM 46.0295.02. 

See also

Wikimedia Commons has media related to Sierpinski carpet.

• List of fractals by Hausdorff dimension
• Hawaiian earring

External links

• Variations on the Theme of Tremas II
• Sierpiński Cookies

v t e Fractals Characteristics Fractal dimension Hausdorff dimension Topological dimension Recursion Self-similarity Iterated function system Barnsley fern Cantor set Dragon curve Koch snowflake Menger sponge Sierpinski carpet Sierpinski triangle Space-filling curve T-square Strange attractor Multifractal system L-system Space-filling curve Escape-time fractals Burning Ship fractal Julia set Lyapunov fractal Mandelbrot set Nova fractal Random fractals Brownian motion Brownian tree Diffusion-limited aggregation Fractal landscape Lévy flight Percolation theory Self-avoiding walk People Georg Cantor Felix Hausdorff Gaston Julia Helge von Koch Paul Lévy Aleksandr Lyapunov Benoit Mandelbrot Lewis Fry Richardson Wacław Sierpiński Other "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" List of fractals by Hausdorff dimension