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Day #21: Mandelbrot Set and the Fractal Culture

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Do you actually know what is the dimension of the boundary of the Mandelbrot set??…

A: The boundary of the Mandelbrot set has Hausdorf dimension… 2. Source: List of Fractals by Hausdorf Dimension and

Mandelbrot Set

Initial image of a Mandelbrot set zoom sequence with a continuously colored environmen
Progressive infinite iterations of the “Nautilus” section of the Mandelbrot Set rendered using webGL


Mandelbrot animation based on a static number of iterations per pixel


Mandelbrot set detail

The Mandelbrot set is the set of complex numbersc for which the function f c ( z ) = z 2 + c {\displaystyle f_{c}(z)=z^{2}+c} {\displaystyle f_{c}(z)=z^{2}+c} does not diverge when iterated from z = 0 {\displaystyle z=0} z=0, i.e., for which the sequence f c ( 0 ) {\displaystyle f_{c}(0)} {\displaystyle f_{c}(0)}, f c ( f c ( 0 ) ) {\displaystyle f_{c}(f_{c}(0))} {\displaystyle f_{c}(f_{c}(0))}, etc., remains bounded in absolute value.

The set is closely related to the idea of Julia sets, which produce similarly complex shapes. Its definition and name are due to Adrien Douady, in tribute to the mathematicianBenoit Mandelbrot.[1]

Mandelbrot set images may be created by sampling the complex numbers and determining, for each sample point c, whether the result of iterating the above function goes to infinity. Treating the real and imaginary parts of each number c as image coordinates, pixels may then be colored according to how rapidly the sequence diverges, with the color 0 (black) usually used for points where the sequence does not diverge.

Images of the Mandelbrot set exhibit an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The “style” of this repeating detail depends on the region of the set being examined. The set’s boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization.


The first published picture of the Mandelbrot set, by Robert W. Brooks and Peter Matelski in 1978
The Mandelbrot set has its place in complex dynamics, a field first investigated by the French mathematiciansPierre Fatou and Gaston Julia at the beginning of the 20th century. This fractal was first defined and drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of Kleinian groups.[2] On 1 March 1980, at IBM‘s Thomas J. Watson Research Center in Yorktown Heights, New York, Benoit Mandelbrot first saw a visualization of the set.[3]
Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980.[4] The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard,[1] who established many of its fundamental properties and named the set in honor of Mandelbrot.
The mathematicians Heinz-Otto Peitgen and Peter Richter became well known for promoting the set with photographs, books,[5] and an internationally touring exhibit of the German Goethe-Institut.[6][7]
The cover article of the August 1985 Scientific American introduced the algorithm for computing the Mandelbrot set to a wide audience. The cover featured an image created by Peitgen, et al.[8][9] The Mandelbrot set became prominent in the mid-1980s as a computer graphics demo, when personal computers became powerful enough to plot and display the set in high resolution.[10]
The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics and abstract mathematics, and the study of the Mandelbrot set has been a centerpiece of this field ever since. An exhaustive list of all the mathematicians who have contributed to the understanding of this set since then is beyond the scope of this article, but such a list would notably include Mikhail Lyubich,[11][12]Curt McMullen, John Milnor, Mitsuhiro Shishikura, and Jean-Christophe Yoccoz.


Benoit Mandelbrot


The Hausdorff Dimension

If we take an object residing in Euclidean dimension D and reduce its linear size by 1/r in each spatial direction, its measure (length, area, or volume) would increase to N=rD times the original. This is pictured in the next figure.
We consider N=rD, take the log of both sides, and get log(N) = D log(r). If we solve for D. D = log(N)/log(r) The point: examined this way, D need not be an integer, as it is in Euclidean geometry. It could be a fraction, as it is in fractal geometry. This generalized treatment of dimension is named after the German mathematician, Felix Hausdorff. It has proved useful for describing natural objects and for evaluating trajectories of dynamic systems.
The Length of a Coastline
Mandelbrot began his treatise on fractal geometry by considering the question: “How long is the coast of Britain?” The coastline is irregular, so a measure with a straight ruler, as in the next figure, provides an estimate. The estimated length, L, equals the length of the ruler, s, multiplied by the N, the number of such rulers needed to cover the measured object. In the next figure we measure a part of the coastline twice, the ruler on the right is half that used on the left.
Measuring the length of a coastline using rulers of varying lengths.
But the estimate on the right is longer. If the the scale on the left is one, we have six units, but halving the unit gives us 15 rulers (L=7.5), not 12 (L=6). If we halved the scale again, we would get a similar result, a longer estimate of L. In general, as the ruler gets diminishingly small, the length gets infinitely large. The concept of length, begins to make little sense.



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Updating Math In Our Mind & Heart!!...

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