The mathematics behind fractals began to evolve 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 a self-similar outcome. 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. A class of examples is given by the Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, and Koch curve. In 1904, Helge von Koch demonstrated his dissatisfaction with Weierstrass's very abstract and analytic definition by defining a more geometric definition of a similar function, which is now called the Koch curve.
To create a Koch snowflake, one begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral bump, or three Koch curves. With every iteration, the perimeter of this shape increases by one third of the previous magnitude. The Koch snowflake is the result of an infinite number of these iterations, and has an infinite magnitude, while its area remains finite. The image below illustrates the Koch snowflake and similar constructions that were sometimes called "monster curves."
Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups. Fractals are deterministic, as are all the above, or stochastic and non-deterministic. For example, the trajectories of the Brownian motion in the plane have a Hausdorff dimension of 2.
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. The animated construction of a Sierpinski Triangle below iterates nine generations of infinite possibilities.
In 1918, Bertrand Russell recognized a "supreme beauty" within the emerging mathematics of fractals. The idea of self-similar curves was taken further by Paul Pierre L¨évy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, the L¨évy 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 Poincar¨é, Felix Klein, Pierre Fatou and Gaston Julia. Without the aid of modern computer graphics however, they lacked the means to visualize the beauty of many of the objects that they had discovered.
In the 1960s, Beno?t 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 convincing 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."
Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals as in an attractor. Geometrically, an attractor can be a point, a curve, a manifold, or even a complicated set with a fractal structure known as a strange 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 is 1, a result proved by Mitsuhiro Shishikura in 1991. Closely related to the Mandelbrot fractal set is the Julia fractal set, as illustrated below.