Fractals are plots of nonlinear equations (equations in the result is used as the next input) which can build up to astonishingly complex and beautiful designs. Typical of fractals is their
scale invariance, which means that no matter how you view them, zoom in or zoom out, you will find selfsimilar and repeating geometric patterns. This distinguishes from most natural patterns (though some are fractallike, such as mountains or the branches of a tree, stars in the sky), in which as you zoom in or out completely changes what you see  e.g., galaxies to stars down to atoms and subatomic nuclei and so forth). Nevertheless, what is most (to me) fascinating about fractals is that they allow us to simulate reality in so many ways.
The Mandelbrot set shown above is the most famous example of fractal design known.
The
Mandelbrot set is a
mathematical set of
points whose
boundary is a distinctive and easily recognizable twodimensional
fractal shape. The set is closely related to
Julia sets (which include similarly complex shapes), and is named after the mathematician
Benoit Mandelbrot, who studied and popularized it.
Mandelbrot set images are made by sampling
complex numbers and determining for each whether the result tends towards infinity when a particular
mathematical operation is iterated on it. Treating the real and imaginary parts of each number as image coordinates, pixels are colored according to how rapidly the sequence diverges, if at all.
More precisely, the Mandelbrot set is the set of values of
c in the
complex plane for which the
orbit of 0 under
iteration of the
complex quadratic polynomial

remains
bounded.
^{[1]} That is, a complex number
c is part of the Mandelbrot set if, when starting with
z_{0} = 0 and applying the iteration repeatedly, the
absolute value of
z_{n} remains bounded however large
n gets.
In general, a
fractal is a mathematical
set that typically displays
selfsimilar patterns, which means they are "the same from near as from far".
^{[1]} Often, they have an "irregular" and "fractured" appearance, but not always. Fractals may be exactly the same at every scale, or they may be
nearly the same at different scales.
^{[2]}^{[3]}^{[4]}^{[5]} The definition of fractal goes beyond selfsimilarity
per se to exclude trivial selfsimilarity and include the idea of a
detailed pattern repeating itself.
^{[2]}^{:166; 18}^{[3]}^{[6]}
One feature of fractals that distinguishes them from "regular" shapes is the amount their spatial content
scales, which is the concept of
fractal dimension. If the edge lengths of a square are all doubled, the
area is scaled by four because squares are two dimensional, similarly if the radius of a sphere is doubled, its
volume scales by eight because spheres are three dimensional. In the case of fractals, if all onedimensional lengths are doubled, the spatial content of the fractal scales by a number which is
not an integer. A fractal has a
fractal dimension that usually exceeds its
topological dimension^{[7]} and may fall between the integers.
^{[2]}
As mathematical equations, fractals are usually nowhere
differentiable.
^{[2]}^{[5]}^{[8]} An infinite fractal curve can be perceived of as winding through space differently from an ordinary line, still being a
1dimensional line yet having a fractal dimension indicating it also resembles a surface.
^{[7]}^{:48}^{[2]}^{:15}
The mathematical
roots of the idea of fractals have been traced through a formal path of published works, starting in the 17th century with notions of
recursion, then moving through increasingly rigorous mathematical treatment of the concept to the study of
continuous but not
differentiable functions in the 19th century, and on to the coining of the word
fractal in the 20th century with a subsequent burgeoning of interest in fractals and computerbased modelling in the 21st century.
^{[9]}^{[10]} The term "fractal" was first used by 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.
^{[2]}^{:405}^{[6]}
There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals."
^{[11]} The general consensus is that theoretical fractals are infinitely selfsimilar,
iterated, and detailed mathematical constructs having fractal dimensions, of which many
examples have been formulated and studied in great depth.
^{[2]}^{[3]}^{[4]} Fractals are not limited to geometric patterns, but can also describe processes in time.
^{[1]}^{[5]}^{[12]} Fractal patterns with various degrees of selfsimilarity have been rendered or studied in images, structures and sounds
^{[13]} and found in
nature,
technology,
art, and
law.
Much of this was taken from Wikipedia
Mandelbrot Set and
Fractal.