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The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński. The carpet is one generalization of the Cantor set to two dimensions (the other is Cantor dust). Higher-dimensional generalizations such as the 3-dimensional Menger sponge are also possible. The Mandelbrot set, named after its discoverer, is a famous example of a fractal. ...
Wacław Franciszek Sierpiński (March 14, 1882 — October 21, 1969), a Polish mathematician, was born and died in Warsaw. ...
The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...
Cantor dust, named after the mathematician Georg Cantor, is the two-dimensional version of the Cantor set. ...
The Menger sponge is a fractal solid. ...
Download high resolution version (728x729, 1 KB)moving over from meta File links The following pages link to this file: Sierpinski carpet Categories: GFDL images ...
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 illustration below shows the first few iterations in the construction process. In plane geometry, a square is a polygon with four equal sides and equal angles. ...
In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ...
In mathematics and computer science, recursion specifies (or constructs) a class of objects (or an object from a certain class) by defining a few very simple base cases (often just one), and then defining rules to break down complex cases into simpler cases. ...
Iteration is the repetition of a process, typically within a computer program. ...
| | | | | | | Order 0 | Order 1 | Order 2 | Order 3 | Order 4 | The Hausdorff dimension of the carpet is log 8/log 3 ≈ 1.8928. Image File history File links SierpinskiCarpet0. ...
Image File history File links SierpinskiCarpet1. ...
Image File history File links SierpinskiCarpet2. ...
Image File history File links SierpinskiCarpet3. ...
Image File history File links SierpinskiCarpet4. ...
In mathematics, the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infinite interval [0, ∞]) associated to any metric space . ...
Brownian motion on the Sierpinski carpet The topic of Brownian motion on the Sierpinski carpet has attracted scientific 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. An example of 1000 simulated steps of Brownian motion in two dimensions. ...
In mathematics and physics, a random walk is a formalization of the intuitive idea of taking successive steps, each in a random direction. ...
See also Image File history File links Commons-logo. ...
The Wikimedia Commons (also called Commons or Wikicommons) is a repository of free content images, sound and other multimedia files. ...
The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Wacław Sierpiński. ...
The T-Square is a fractal curve of infinite length inside finite area. ...
The Menger sponge is a fractal solid. ...
External link - Variations on the Theme of Tremas II
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