Fractals have existed since the beginning of time. There are haven been patterns in churches, architecture, and nature that can now be described mathematically as fractals. They first began to be used in the 1800’s as counter examples to the ideas of continuity that had grown from calculus concepts, as was evidenced by the graphs of Bernhard Riemann in 1861 and by Karl Weierstrass in 1872. Geometric figures came next when introduced by Helge von Koch in 1906 and by Waclaw Sierpinski and Karl Menger in 1926 and Paul Levy in 1937. For many years these shapes, from the point of view of mathematics, tended to be viewed as rare and pathological examples with no general significance. However in late 1960’s and early 1970’s Benoit Mandelbrot developed the idea that nested shapes can be identified in a great many natural systems and in several branches of mathematics. He is now known as the father of fractal geometry to some. Here is a chronological list involving some of the major breakthroughs involving fractal geometry:

The Monsters (complex, irregular objects)
1872: The Cantor Set
1875: Weierstrass’s continuous nowhere differentiable curve (du Bois)
1906: Brownian Motion (Perrin) and Koch Snowflake (von Koch)

Scaling Behaviour
1919: The Hausdorff dimension of complex geometric objects
1951: Hurt’s Law for scaling behaviour of Nile River discharges
1956: The Gutenberg-Richter law for the distribution of earthquake magnitudes
1961: Richardson’s scaling laws in the measurement of complex natural curves such as coastlines
1963: The Stommel diagram describes spatial and temporal scales for ocean dynamics in space and time
1968-9: Mandelbrot, Van Ness, and Wallis extend Hurt’s work in hydrology

The Science of Fractals (except for dynamics)
1975: Mandelbrot coins the word ‘fractals’
1977: Mandelbrot’s Fractals: Form, Chance and Dimension
1980: The Weierstrass-Mandelbrot fractal function- the natural geometry of the Weierstrass monster of 1875
1982: Fractal models applied to ecology (Hastings et al.) and cloud patterns (Lovejoy). Mandelbrot’s revised monograph The Fractal Geometry of Nature.
1986: Iterated function systems (Barnsley et al.)

Fractals and Dynamics
1981: Witten and Sander introduce diffusion-limited aggregation
1983: Hentschel and Procaccia relate fractals and strange attractors
1984: Wolfram’s dynamics of cellular automata
1987: Bak, Tang and Weisenfeld’s self organized criticality
1991: Schertzer and Lovejoy’s book on fractals and multifractals in geophysics

The Present
1991: 400 publications involving fractals
1992: Fractals applied to answer open questions ranging from the origin of the structure of the universe to the distribution of earthquakes

Source: Fractals by Harold Hastings and George Sugihara