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The terms fractal dimension and fractal were coined by Mandelbrot in 1975, [16] about a decade after he published his paper on self-similarity in the coastline of Britain. . Various historical authorities credit him with also synthesizing centuries of complicated theoretical mathematics and engineering work and applying them in a new way to study complex geometries that defied description in ...
To calculate this dimension for a fractal , imagine this fractal lying on an evenly spaced grid and count how many boxes are required to cover the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying a box-counting algorithm.
This results from the fractal curve-like properties of coastlines; i.e., the fact that a coastline typically has a fractal dimension. Although the "paradox of length" was previously noted by Hugo Steinhaus, [1] the first systematic study of this phenomenon was by Lewis Fry Richardson, [2] [3] and it was expanded upon by Benoit Mandelbrot. [4] [5]
This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects. The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.
Mobile phone and Wi-Fi fractal antennas have been produced in the form of few iterations of the Sierpiński carpet. Due to their self-similarity and scale invariance, they easily accommodate multiple frequencies. They are also easy to fabricate and smaller than conventional antennas of similar performance, thus being optimal for pocket-sized ...
In chaos theory, the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. [ 1 ] [ 2 ] [ 3 ]
In fractal geometry, the generalized Hurst exponent has been denoted by H or H q in honor of both Harold Edwin Hurst and Ludwig Otto Hölder (1859–1937) by Benoît Mandelbrot (1924–2010). [3] H is directly related to fractal dimension, D, and is a measure of a data series' "mild" or "wild" randomness. [4]
The Lyapunov spectrum can be used to give an estimate of the rate of entropy production, of the fractal dimension, and of the Hausdorff dimension of the considered dynamical system. [5]