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The Cantor set is the prototype of a fractal. It is self-similar, because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated.
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." [ 1 ] Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension.
After black intervals have been removed, the white points which remain are a nowhere dense set of measure 1/2. In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, [1] or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure.
Different Cantor functions, or Devil's Staircases, can be obtained by considering different atom-less probability measures supported on the Cantor set or other fractals. While the Cantor function has derivative 0 almost everywhere, current research focuses on the question of the size of the set of points where the upper right derivative is ...
6 steps of a Sierpiński carpet. The Sierpiński 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 such generalization is the Cantor dust.
Indeed, the Mandelbrot set is defined as the set of all c such that () is connected. For parameters outside the Mandelbrot set, the Julia set is a Cantor space: in this case it is sometimes referred to as Fatou dust. In many cases, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c.
In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) [1] [2] [3] is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet.
For example, = {,,, …} is a fractal string corresponding to the Cantor set. A fractal string is the analogue of a one-dimensional "fractal drum," and typically the set has a boundary which corresponds to a fractal such as the Cantor set. The heuristic idea of a fractal string is to study a (one-dimensional) fractal using the "space around the ...