In colloquial usage, a fractal is a shape that is recursively constructed or self-similar, that is, a shape that appears similar a... first described by Waclaw SierpinskiWaclaw Sierpinski
Waclaw Franciszek Sierpinski , a Polish mathematician, was born and died in Warsaw.... in 1916. The carpet is a generalization of the Cantor setCantor set Overview
The Cantor set, introduced by German mathematician Georg Cantor, is a construction of a set of points lying on a single lin... to two dimensions (another is Cantor dustCantor dust
Cantor dust is a multi-dimensional version of the Cantor set.... ). Sierpinski 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 spongeMenger sponge
In mathematics, the Menger sponge is a fractal curve.... , 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 squareSquare (geometry)
In plane geometry, a square is a polygon with four equal sides, four right angles, and parallel opposite sides.... . The square is cut into 9 congruentCongruence (geometry)
In geometry, two sets are called congruent if one can be transformed into the other by an isometry, i.e., a combination of ... subsquares in a 3-by-3 grid, and the central subsquare is removed. The same procedure is then applied recursivelyRecursion
In mathematics and computer science, recursion specifies a class of objects or methods by defining a few very simple base ... to the remaining 8 subsquares, ad infinitum. The Hausdorff dimensionFacts About Hausdorff dimension
In mathematics, the Hausdorff dimension is an extended non-negative real number associated to any metric space.... of the carpet is log 8/log 3 ˜ 1.8928.
The area of the carpet is zero (in standard Lebesgue measureLebesgue measure
In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space... ).
The term Brownian motion refers to either... on the Sierpinski carpet has attracted interest in recent years. Martin Barlow and Richard Bass have shown that a random walkRandom walk
In mathematics and physics, a random walk, sometimes called a "drunkard's walk," is a formalisation of the intuitive idea of... 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/ß for some ß > 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.
A Java applet is an applet delivered in the form of Java bytecode.... draws a Sierpinski carpet by means of a method that recursivelyRecursion
In mathematics and computer science, recursion specifies a class of objects or methods by defining a few very simple base ... calls itself: import java.awt.*; import java.applet.*;
