Fractal
In mathematics a fractal is an abstract object
used to describe and simulate naturally occurring objects. Artificially
created fractals commonly exhibit similar patterns at increasingly
small scales. It is also known as expanding symmetry or evolving symmetry. If the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger sponge. Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set. Fractals also include the idea of a detailed pattern that repeats itself.:166; 18
Fractals are different from other geometric figures because of the way in which they scale. Doubling the edge lengths of a 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 dimension of the space the polygon
resides in). Likewise, if the radius of a 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 dimension that the sphere resides in). But 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. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension.
