In mathematics, a fractal is any complex geometric shape
that exhibits self-similarity. Fractals are very different from the simple
figures of Euclidean geometry -- the square, the circle, and the sphere,
for example. They can describe the many irregularly shaped objects or spatially
strange phenomena in nature that cannot be explained using Euclidean geometry
[i.e. the growth pattern of ferns or the freezing pattern of water].
The term fractal, formed from the Latin word fractus ("fragmented"
or "broken"), was coined by the Polish-born mathematician Benoit B. Mandelbrot.
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 not only on
mathematics but also on such diverse fields as physical chemistry, physiology,
and fluid mechanics.
Although not all fractals are totally self-similar, most
are. A self-similar object is one whose parts resemble the whole object.
This pattern of irregular details or patterns occurs at progressively smaller
scales and can continue indefinitely, so that each section of the fractal,
when magnified, will look like the fractal as a whole. Basically, a self-similar
object remains essentially the same despite scale changes -- called scaling
symmetry. In nature, fractal phenomena are observed in such objects
as snowflakes and tree barks.
Another key characteristic of a fractal is a mathematical parameter
called fractal dimension. This is the one characteristic of a fractal that
remains the same regardless of how much the object is magnified or how much
the object is rotated. Unlike traditional Euclidean dimensions, fractal
dimensions are usually expressed with fractions -- big surprise. Fractal
dimension can be understood by considering a fractal curve. In a fractal
curve, the degree of curvature increases from the previous degree according
to the ration 4:3.
Fractal geometry, with its concepts of self-similarity and
fractional dimensionality, has been applied increasingly in statistical
mechanics, such as examining physical systems that seem randomly organized.
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 much 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.
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