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What is a Mandlebrot Set?

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  • Written By: Michael Anissimov
  • Edited By: Niki Foster
  • Last Modified Date: 25 November 2016
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    Conjecture Corporation
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A Mandlebrot set is a fractal which can be plotted using an iterative complex function. A fractal is a mathematically generated image that is rough, irregular and complex. A fractal also possesses self-similarity on many levels of magnification, so that tiny parts of the fractal resemble larger parts. Fractals continue to appear complex no matter no much you magnify them, leading some to say that they have infinite complexity. The Mandlebrot set is the most famous example of a fractal, consisting of a cardoid, a circular object with a dimple on one side, surrounded by progressively smaller arrangements of near-circles and interesting spiral patterns, all tangent to one another.

The underlying mathematics of the Mandlebrot set were devised in 1905 by Pierre Fatou, a French mathematician exploring the field of complex analytic dynamics. He enjoyed studying the behavior of recursive processes, functions whose outputs were fed back into their inputs. Fatou attempted to plot some of his complex sets by hand, but too many calculations were required for the full image of certain sets (including the Mandlebrot set) to appear. It was not until the distribution of desktop computers that plotting this set became practical.

The Mandlebrot set was first plotted by Professor Benoît Mandlebrot, a mathematician who coined the term fractal and popularized the idea in a 1975 book entitled, Fractal Objects: Form, Chance and Dimension. Before being called fractals, these structures were referred to as "monster curves."

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Mandlebrot saw connections between fractals like his Mandlebrot set and real-world phenomena, prompting him to study the connections in detail. Fractal-like structures can be found in nature, for example in the arrangement of petals on certain flowers. Mandlebrot pointed out that real shapes in nature never have the bland regularity of Euclidean geometric structures, but in fact more closely resemble fractals. Other examples include shapes found in coastlines and rivers, plants, blood vessels and lungs, galaxy clusters, Brownian motion, and patterns in the stock market.

Because the Mandlebrot set is so complex and shows such variation, hobbyists have devoted thousands of hours to locating unique structures within the set, color-coding them, and sharing them with others. Structures similar in appearance to the entire set can be found on the smallest scales, sometimes connected to the main set only by tiny tendrils. The apparent complexity of the set actually increases with magnification. Today, good software applications are available for hobbyists to plot the Mandlebrot set and other fractals and to study their appearance.

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