Sierpinski carpet
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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.
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[edit] 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. The Hausdorff dimension of the carpet is log 8/log 3 ≈ 1.8928.
The area of the carpet is zero (in standard Lebesgue measure).
[edit] 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 n1/2 after n steps, but the random walk on the discrete Sierpinski carpet reaches only a mean distance proportional to n1/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.
[edit] References
Sierpiński, W.: Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée. C. r. hebd. Śeanc. Acad. Sci., Paris 162, 629--632 (1916)
[edit] See also
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