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“A fractal is a way of seeing infinity.”
~ Benoit Mandelbrot (1924-2010), the "Father of Fractals"
Fractals are objects in which the same patterns occur again and again at different scales and sizes. In a perfect mathematical fractal, this “self-similarity” goes infinitely deep: each pattern is made up of smaller copies of itself, and those smaller copies are made up of smaller copies again, forever. Many natural phenomena are fractal to some degree - eroding coastlines, snowflake geometry, and can even be found in partly random or chaotic phenomena such as crystal growth, fluid turbulence, and galaxy formation. Fractals are also found in human pursuits, with patterning in music, painting, architecture, and even stock market prices! The Mandelbrot set (named for a pioneer of fractal geometry, is a set of complex numbers that can be represented pictorially, showing increasingly intricate detail the closer one magnifies the image, referred to as a "zoom". This tartan echoes the beginning of a deep zoom by its visual replication of the tartan pattern, creating an illusion of deeper, smaller dimensions. Today's computer zooms allow you take a virtual trip beyond the distance equivalent of the known universe - from the outside edge down through the realm of the atom and beyond. For a "deep purple" zoom of this kind, dive in here: https://www.youtube.com/watch?v=N2cDdJpneWo ⚛️
Benoit B. Mandelbrot (1924 – 2010) was a Polish-born, French and American mathematician with broad interests and contributions in the practical sciences.
He is particularly recognized for his contribution to the field of fractal geometry, including coining the word "fractal", as well as for developing a theory of "roughness and self-similarity" in nature. He referred to himself as a "fractalist."
A fractal is a natural phenomenon or mathematical set that exhibits a repeating pattern that displays the same at every scale - also known as expanding symmetry or evolving symmetry. Fractals are not limited to geometric patterns, but can also describe processes in time.
Phenomena known to have fractal features include:
Mountain Goat horns
Animal coloration patterns
Because of his access to IBM's computers, Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his discovery of the "Mandelbrot set" in 1979. A Mandelbrot set is a series of images which exhibit an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications or "zooming in."
Mandelbrot set zoom videos are popular exercises in computer mathematical visualization.
This tartan, by designer Carol A.L. Martin, gives the impression of a pattern within a pattern, and the sense of peering into a moment of a Mandelbrot set zoom.
Click the snapshot of the Mandelbrot Set for an impressive video showing the correlation between this and the Julia Set (a different application of the same formula as the Mandelbrot set).