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Fractals are complex patterns that are self-similar, and therefore exhibit similar patterns at every scale. Fractals can be patterns or shapes that are non-regular and differ from traditional geometric shapes, but occur very commonly in nature, such as clouds, mountains, trees and snowflakes. The most well-known illustration of fractals is the Mandelbrot set, which when magnified simply shows repetitions of the same pattern, making it hard to determine the level of magnification due to the recurring patterns.
Fractal geometry is considered a special field in mathematics simply because fractals have very different mathematical equations than regular geometry. The phenomena has been studied for hundreds of years, but fractals have largely been ignored as "mathematical monsters" because of unfamiliarity, being very different from established geometry. The mathematics behind fractals started in the 17th century when mathematician Gottfried Leibniz started studying recursive self-similarity and used the term "fractional exponents" to describe them, but it was not until 1872 that Karl Weierstrass presented the first definition of a function with a graph that can be considered a fractal by today's definition.
Another milestone in fractal geometry came when Helge von Koch gave a more geometric approach to the idea of fractals with a hand-drawn image which is now called the Koch snowflake. The Koch snowflake fractal starts out as an equilateral triangle and then iteratively replaces the middle third of every line with another equilateral triangle, albeit smaller because each side would only be as long as 1/3 of the original line it is on. This can go on infinitely or as long as it is physically possible in the media where it is illustrated, which when modeled using a computer can practically stretch into infinity. The term fractal was coined by Benoit Mandelbrot in 1975.
Today, fractal studies are essentially computer-based because of their nature and see use in general mathematics, computer simulations, imaging and graphics processing. Researchers postulated that because there were no computers in the past, early investigators of the phenomena were very limited in the ways that they could depict fractals, hence they lacked the means to truly visualize them and appreciate their implications.