Math Is Fun Forum

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 46,424

Fractal

Fractal

Gist

a) Mathematics, Physics. an irregular geometric structure that cannot be described by classical geometry because magnification of the structure reveals repeated patterns of similarly irregular, but progressively smaller, dimensions: fractals are especially apparent in natural forms and phenomena because the geometric properties of the physical world are largely abstract, as with clouds, crystals, tree bark, or the path of lightning.
b) Architecture, Decorative Art. a design or construction that uses the concept and mechanics of fractal geometry:
Fractals distinguish the facade of the library, revealing recursive patterns, the smaller parts mirroring the larger parts.

Summary

Fractal, in mathematics, any of a class of complex geometric shapes that commonly have “fractional dimension,” a concept first introduced by the mathematician Felix Hausdorff in 1918. Fractals are distinct from the simple figures of classical, or Euclidean, geometry—the square, the circle, the sphere, and so forth. They are capable of describing many irregularly shaped objects or spatially nonuniform phenomena in nature such as coastlines and mountain ranges. The term fractal, derived from the Latin word fractus (“fragmented,” or “broken”), was coined by the Polish-born mathematician Benoit B. Mandelbrot.

Although the key concepts associated with fractals had been studied for years by mathematicians, and many examples, such as the Koch or “snowflake” curve were long known, Mandelbrot was the first to point out that fractals could be an ideal tool in applied mathematics for modeling a variety of phenomena from physical objects to the behavior of the stock market. Since its introduction in 1975, the concept of the fractal has given rise to a new system of geometry that has had a significant impact on such diverse fields as physical chemistry, physiology, and fluid mechanics.

Many fractals possess the property of self-similarity, at least approximately, if not exactly. A self-similar object is one whose component parts resemble the whole. This reiteration of details or patterns occurs at progressively smaller scales and can, in the case of purely abstract entities, continue indefinitely, so that each part of each part, when magnified, will look basically like a fixed part of the whole object. In effect, a self-similar object remains invariant under changes of scale—i.e., it has scaling symmetry. This fractal phenomenon can often be detected in such objects as snowflakes and tree barks. All natural fractals of this kind, as well as some mathematical self-similar ones, are stochastic, or random; they thus scale in a statistical sense.

Another key characteristic of a fractal is a mathematical parameter called its fractal dimension. Unlike Euclidean dimension, fractal dimension is generally expressed by a noninteger—that is to say, by a fraction rather than by a whole number. Fractal dimension can be illustrated by considering a specific example: the snowflake curve defined by Helge von Koch in 1904. It is a purely mathematical figure with a six-fold symmetry, like a natural snowflake. It is self-similar in that it consists of three identical parts, each of which in turn is made of four parts that are exact scaled-down versions of the whole. It follows that each of the four parts itself consists of four parts that are-scaled down versions of the whole. There would be nothing surprising if the scaling factor were also four, since that would be true of a line segment or a circular arc. However, for the snowflake curve, the scaling factor at each stage is three.

Fractal geometry with its concepts of self-similarity and noninteger dimensionality has been applied increasingly in statistical mechanics, notably when dealing with physical systems consisting of seemingly random features. For example, fractal simulations have been used to plot the distribution of galaxy clusters throughout the universe and to study problems related to fluid turbulence. Fractal geometry also has contributed to computer graphics. Fractal algorithms have made it possible to generate lifelike images of complicated, highly irregular natural objects, such as the rugged terrains of mountains and the intricate branch systems of trees.

Details

In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.

One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension. This power is called the fractal dimension of the geometric object, to distinguish it from the conventional dimension (which is formally called the topological dimension).

Analytically, many fractals are nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still topologically 1-dimensional, its fractal dimension indicates that it locally fills space more efficiently than an ordinary line.

Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century.

There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, darn hard, increasingly useful. That's fractals." More formally, in 1982 Mandelbrot defined fractal as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole." Still later, Mandelbrot proposed "to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants".

The consensus among mathematicians is that theoretical fractals are infinitely self-similar iterated and detailed mathematical constructs, of which many examples have been formulated and studied. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural media and found in nature, technology, art, and architecture. Fractals are of particular relevance in the field of chaos theory because they show up in the geometric depictions of most chaotic processes (typically either as attractors or as boundaries between basins of attraction).

Etymology

The term "fractal" was coined by the mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus, meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.