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 any possible one-dimensional graph, projected onto the two-dimensional plane, is homeomorphic to a subset of the Sierpinski carpet. For curves that cannot be drawn on a 2D surface without self-intersections, the corresponding universal curve is the Menger sponge, a higher-dimensional generalization.

The technique can be applied to repetitive tiling arrangement; triangle, square, hexagon being the simplest. It would seem impossible to apply it to other than rep-tile arrangements.

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.

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

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. 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.

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

• List of fractals by Hausdorff dimension
• Sierpinski triangle
• Hawaiian earring
• Menger sponge

External links

• Variations on the Theme of Tremas II
• Sierpiński Cookies