Fractal

A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.

Abstract fractals – such as the Mandelbrot Set – can be generated by a computer calculating a simple equation over and over.

When looking around nature, you might have noticed intricate plants like these:

fern leaf
This Fern consists of many small leaves that branch off a larger one.


romanesco

This Romanesco broccoli consists of smaller spiralling around a larger one.

What is so special about Fractals?


Benoit Mandelbrot is generally considered to be the father of fractals. He coined the term fractal to describe curves, surfaces and objects that have some very peculiar properties. You learned in school that simple curves, such as a line,  have one dimension. Squares, rectangles, circles, polygons, etc. have two dimensions, while solid objects such as a cube, have three dimensions. The three dimensions define space. Time can be considered a fourth dimension. We normally think of dimensions as integers: 1, 2, 3, . . .

What is so peculiar about about fractals is that they have fractional dimensions! A fractal curve could have a dimensionality of 1.4332, for example, rather than 1. Fractals are not just a mathematical curiosity. Most natural objects are fractal by nature, and can be best described using fractal mathematics. Clouds, leaves, the blood vessel system, coastlines, particles of lint, etc. have fractal shapes. 

Fractals are generated by an iterative process - doing the same thing again and again. Fractals also have the property that when you magnify them they still look much the same. This is called self-similarity.

So how does Mandelbrot Set get into this discussion?


The term Mandelbrot set is used to refer both to a general class of fractal sets and to a particular instance of such a set. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected and not computable.

"The" Mandelbrot set is the set obtained from the quadratic recurrence equation

equation_m_set

with z_0=C, where points C in the complex plane for which the orbit of z_n does not tend to infinity are in the set. Setting z_0 equal to any point in the set that is not a periodic point gives the same result. The Mandelbrot set was originally called a mu molecule by Mandelbrot. J. Hubbard and A. Douady proved that the Mandelbrot set is connected.



Photo Credits: Wolfram Research, Inc.

It's a ubiquitous badge of mathematical pride.

Does it have any application? Does it mean something?

Mathematicians are still working on this :)

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