The
mathematics behind fractals began to take shape in the 17th century when mathematician and philosopher
Leibniz considered
recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense). It took until 1872 before a function appeared whose
graph would today be considered fractal, when
Karl Weierstrass gave an
example of a function with the non-
intuitive property of being everywhere
continuous but
nowhere differentiable. 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 curve. (The image at right is three Koch curves put together to form what is commonly called the
Koch snowflake.) In 1915,
Waclaw Sierpinski constructed his
triangle and, one year later, his
carpet. Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions. In 1918,
Bertrand Russell recognized a "supreme beauty" within the emerging mathematics of fractals.
[2] 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 also 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 were 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, which built on earlier work by
Lewis Fry Richardson. Finally, 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".
[edit] Examples
A
Julia set, a fractal related to the Mandelbrot set
A class of examples is given by the
Cantor sets,
Sierpinski triangle and
carpet,
Menger sponge,
dragon curve,
space-filling curve, and
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 a Hausdorff dimension of 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 a Hausdorff dimension equal to its topological dimension of 2—but what is truly surprising is that the
boundary of the Mandelbrot set also has a Hausdorff dimension of 2 (while the topological dimension of 1), a result proved by
Mitsuhiro Shishikura in 1991. A closely related fractal is the
Julia set.
[edit] Generating fractals
