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Fractal

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   The boundary of the Mandelbrot set is a famous example of a fractal.
   Enlarge
   The boundary of the Mandelbrot set is a famous example of a fractal.
   Another view of the Mandelbrot set.
   Enlarge
   Another view of the Mandelbrot set.

   In colloquial usage, a fractal is "a rough or fragmented geometric
   shape that can be subdivided in parts, each of which is (at least
   approximately) a reduced/size copy of the whole". The term was coined
   by Benoît Mandelbrot in 1975 and was derived from the Latin fractus
   meaning broken or fractured.

   A fractal as a geometric object generally has the following features:
     * fine structure at arbitrarily small scales
     * is too irregular to be easily described in traditional Euclidean
       geometric language.
     * is self-similar (at least approximatively or stochastically)
     * has a Hausdorff dimension that is greater than its topological
       dimension (although this requirement is not met by space-filling
       curves such as the Hilbert curve)
     * has a simple and recursive definition.

   Due to them appearing similar at all levels of magnification, fractals
   are often considered to be 'infinitely complex'. Obvious examples
   include clouds, mountain ranges and lightning bolts. However, not all
   self-similar objects are fractals — for example, the real line (a
   straight Euclidean line) is formally self-similar but fails to have
   other fractal characteristics.

History

   A Koch snowflake is the limit of an infinite construction that starts
   with a triangle and recursively replaces each line segment with a
   series of four line segments that form a triangular "bump". Each time
   new triangles are added (an iteration), the perimeter of this shape
   grows by a factor of 4/3 and thus diverges to infinity with the number
   of iterations. The length of the Koch snowflake's boundary is therefore
   infinite, while its area remains finite. For this reason, the Koch
   snowflake and similar constructions were sometimes called "monster
   curves."
   Enlarge
   A Koch snowflake is the limit of an infinite construction that starts
   with a triangle and recursively replaces each line segment with a
   series of four line segments that form a triangular "bump". Each time
   new triangles are added (an iteration), the perimeter of this shape
   grows by a factor of 4/3 and thus diverges to infinity with the number
   of iterations. The length of the Koch snowflake's boundary is therefore
   infinite, while its area remains finite. For this reason, the Koch
   snowflake and similar constructions were sometimes called "monster
   curves."

   Objects that are now described as fractals were discovered and
   described centuries ago. Ethnomathematics like Ron Eglash's African
   Fractals ( ISBN 0-8135-2613-2) describes pervasive fractal geometry in
   indigenous African craft work. In 1525, the German Artist Albrecht
   Dürer published The Painter's Manual, in which one section is on "Tile
   Patterns formed by Pentagons". The Dürer's Pentagon largely resembled
   the Sierpinski carpet, but based on pentagons instead of squares..

   The idea of "recursive self-similarity" was originally developed by the
   philosopher Leibniz and he even worked out many of the details. In
   1872, Karl Weierstrass found an example of a function with the
   nonintuitive property that it is everywhere continuous but nowhere
   differentiable — the graph of this function would now be called a
   fractal. In 1904, Helge von Koch, dissatisfied with Weierstrass's very
   abstract and analytic definition, gave a more geometric definition of a
   similar function, which is now called the Koch snowflake. In 1915,
   Waclaw Sierpinski constructed his triangle and one year later his
   carpet. Actually, these fractals were described as curves, which is
   hard to realize with the well-known modern constructions. The idea of
   self-similar curves was taken further by Paul Pierre Lévy who, in his
   1938 paper Plane or Space Curves and Surfaces Consisting of Parts
   Similar to the Whole, described a new fractal curve, the Lévy C curve.

   Georg Cantor gave examples of subsets of the real line with unusual
   properties — these Cantor sets are also now recognized as fractals.
   Iterated functions in the complex plane had been investigated in the
   late 19th and early 20th centuries by Henri Poincaré, Felix Klein,
   Pierre Fatou and Gaston Julia. However, without the aid of modern
   computer graphics they lacked the means to visualize the beauty of many
   of the objects that they had discovered.

   In the 1960s Benoît Mandelbrot started investigating self-similarity in
   papers such as How Long Is the Coast of Britain? Statistical
   Self-Similarity and Fractional Dimension. This built on earlier work by
   Lewis Fry Richardson. In 1975, Mandelbrot coined the word fractal to
   denote an object whose Hausdorff-Besicovitch dimension is greater than
   its topological dimension. He illustrated this mathematical definition
   with striking computer-constructed visualizations. These images
   captured the popular imagination; many of them were based on recursion,
   leading to the popular meaning of the term "fractal".

Examples

   A Julia set, a fractal related to the Mandelbrot set
   Enlarge
   A Julia set, a fractal related to the Mandelbrot set

   A relatively simple class of examples is given by the Cantor sets,
   Sierpinski triangle and carpet, Menger sponge, dragon curve,
   space-filling curve, Koch curve. Additional examples of fractals
   include the Lyapunov fractal and the limit sets of Kleinian groups.
   Fractals can be deterministic (all the above) or stochastic (that is,
   non-deterministic). For example the trajectories of the Brownian motion
   in the plane have Hausdorff dimension 2.

   Chaotic dynamical systems are sometimes associated with fractals.
   Objects in the phase space of a dynamical system can be fractals (see
   attractor). Objects in the parameter space for a family of systems may
   be fractal as well. An interesting example is the Mandelbrot set. This
   set contains whole discs, so it has the Hausdorff dimension equal to
   its topological dimension of 2 —but what is truly surprising is that
   the boundary of the Mandelbrot set also has the Hausdorff dimension of
   2 (while the topological dimension of 1), a result proved by M.
   Shishikura in 1991. A closely related fractal is the Julia set.

Self-similarity dimension

   The self-similarity dimension is a simplification of the Hausdorff
   dimension which can be applied to exactly self-similar objects.

   The following analysis of the Koch Snowflake suggests how
   self-similarity can be used to analyze fractal properties.

   The total length of a number, N, of small steps, L, is the product NL.
   Applied to the boundary of the Koch snowflake this gives a boundless
   length as L approaches zero. But this distinction is not satisfactory,
   as different Koch snowflakes do have different sizes. A solution is to
   measure, not in meter, m, nor in square meter, m², but in some other
   power of a meter, m^x. Now 4N(L/3)^x = NL^x, because a three times
   shorter steplength requires four times as many steps, as is seen from
   the figure. Solving that equation gives x = (log 4)/(log 3) ≈ 1.26186.
   So the unit of measurement of the boundary of the Koch snowflake is
   approximately m^1.26186.

   More generally, suppose that a fractal consists of N identical parts
   that are similar to the entire fractal with the scale factor of L and
   that the intersection between part is of the Lebesgue measure 0. Then
   the Hausdorff dimension of the fractal is \frac{\log N}{\log L}\,\! .
   For example, the Hausdorff dimension of
     * the Cantor set is \frac{\log 2}{\log 3}\approx 0.63\,\! ,
     * the Sierpinski gasket is \frac{\log 3}{\log 2}\approx 1.58\,\! ,
     * the Sierpinski carpet is \frac{\log 8}{\log 3}\approx 1.89\,\! ,

   and so on. Even more generally one may assume that each of N parts is
   similar to the fractal with a different scale factor L[i], i = 1...N.
   Then the Hausdorff dimension can be calculated by solving the following
   equation in the variable s:

          \sum_{i=1}^N L_i^s = 1.

Generating fractals

   The whole Mandelbrot set
   Mandelbrot zoomed 6x
   Mandelbrot Zoomed 100x
   Mandelbrot Zoomed 2000x Even 2000 times magnification of the Mandelbrot
   set uncovers fine detail resembling the full set.

   Three common techniques for generating fractals are:

          + Escape-time fractals — These are defined by a recurrence
            relation at each point in a space (such as the complex plane).
            Examples of this type are the Mandelbrot set, Julia set, the
            Burning Ship fractal and the Lyapunov fractal.
          + Iterated function systems — These have a fixed geometric
            replacement rule. Cantor set, Sierpinski carpet, Sierpinski
            gasket, Peano curve, Koch snowflake, Harter-Heighway dragon
            curve, T-Square, Menger sponge, are some examples of such
            fractals.
          + Random fractals — Generated by stochastic rather than
            deterministic processes, for example, trajectories of the
            Brownian motion, Lévy flight, fractal landscapes and the
            Brownian tree. The latter yields so-called mass- or dendritic
            fractals, for example, diffusion-limited aggregation or
            reaction-limited aggregation clusters.

Classification of fractals

   Fractals can also be classified according to their self-similarity.
   There are three types of self-similarity found in fractals:

          + Exact self-similarity — This is the strongest type of
            self-similarity; the fractal appears identical at different
            scales. Fractals defined by iterated function systems often
            display exact self-similarity.
          + Quasi-self-similarity — This is a loose form of
            self-similarity; the fractal appears approximately (but not
            exactly) identical at different scales. Quasi-self-similar
            fractals contain small copies of the entire fractal in
            distorted and degenerate forms. Fractals defined by recurrence
            relations are usually quasi-self-similar but not exactly
            self-similar.
          + Statistical self-similarity — This is the weakest type of
            self-similarity; the fractal has numerical or statistical
            measures which are preserved across scales. Most reasonable
            definitions of "fractal" trivially imply some form of
            statistical self-similarity. (Fractal dimension itself is a
            numerical measure which is preserved across scales.) Random
            fractals are examples of fractals which are statistically
            self-similar, but neither exactly nor quasi-self-similar.

Fractals in nature

   A fractal fern computed using an Iterated function system
   Enlarge
   A fractal fern computed using an Iterated function system

   Approximate fractals are easily found in nature. These objects display
   self-similar structure over an extended, but finite, scale range.
   Examples include clouds, snow flakes, mountains, river networks,
   cauliflower or broccoli, and systems of blood vessels.

   Trees and ferns are fractal in nature and can be modeled on a computer
   by using a recursive algorithm. This recursive nature is obvious in
   these examples — a branch from a tree or a frond from a fern is a
   miniature replica of the whole: not identical, but similar in nature.

   The surface of a mountain can be modeled on a computer by using a
   fractal: Start with a triangle in 3D space and connect the central
   points of each side by line segments, resulting in 4 triangles. The
   central points are then randomly moved up or down, within a defined
   range. The procedure is repeated, decreasing at each iteration the
   range by half. The recursive nature of the algorithm guarantees that
   the whole is statistically similar to each detail.

   Fractal patterns have been found in the paintings of American artist
   Jackson Pollock. While Pollock's paintings appear to be composed of
   chaotic dripping and splattering, computer analysis has found fractal
   patterns in his work.

   A fractal is formed when pulling apart two glue-covered acrylic sheets.

   High voltage breakdown within a 4″ block of acrylic creates a fractal
   Lichtenberg figure.

   Fractal branching occurs on a microwave-irradiated DVD

   Romanesco broccoli showing very fine natural fractals

   A DLA cluster grown from a copper sulfate solution in an
   electrodeposition cell

Applications

   As described above, random fractals can be used to describe many highly
   irregular real-world objects. Other applications of fractals include:
     * Classification of histopathology slides in medicine
     * Generation of new music
     * Generation of various art forms
     * Signal and image compression
     * Seismology
     * Computer and video game design, especially computer graphics for
       organic environments and as part of procedural generation
     * Fractography and fracture mechanics
     * Fractal antennas — Small size antennas using fractal shapes
     * Neo-hippies t-shirts and other fashion.
     * Generation of patterns for camouflage, such as MARPAT.
     * Digital sundial

   A fractal that models the surface of a mountain (animation)
   Enlarge
   A fractal that models the surface of a mountain (animation)

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