## WHAT IS A FRACTAL

My approach to art: Soon after my artistic awakening in 2017, I was creating art using fractal geometry concepts even before I knew what a fractal was. I was mainly engaging in “self-similarity” which involved using the same shape at different scales. My art on the left “Arabian Nights” is a perfect example of “self-similarity.” I found an amazing amount of comfort in the controlled regularity and patterns. Patterns are very soothing to me.

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.

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Fractal patterns repeat themselves at different scales - this is called “self-similarity.” They can be found in branching (like the branches on a tree), through spirals (think of a nautilus shell), and geometric (like the Sierpinski Triangle which is made by repeatedly removing the middle triangle from the prior generation. The number of colored triangles increases by a factor of 3 each step, 1,3,9,27,81,243,729, etc.

Algebraic fractals use a simple formula that repeats and repeats. The Mandelbrot Set is probably one of the most familiar fractal equations.

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HOW DOES THE MANDELBROT SET WORK?

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We start by plugging a value for the variable ‘C’ into the simple equation below. Each complex number is actually a point in a 2-dimensional plane. The equation gives an answer, ‘Znew’ . We plug this back into the equation, as ‘Zold’ and calculate it again. We are interested in what happens for different starting values of ‘C.’ Generally, when you square a number, it gets bigger, and then if you square the answer, it gets bigger still. Eventually, it goes to infinity. This is the fate of most starting values of ‘C.’ However, some values of ‘C’ do not get bigger, but instead get smaller, or alternate between a set of fixed values. These are the points inside the Mandelbrot Set, which we color black. Outside the Set, all the values of ‘C’ cause the equation to go to infinity, and the colors are proportional to the speed at which they expand.

## SELF-SIMILARITY

## FRACTAL RESOURCES

You can read about the Julia Set and more in depth at the Fractal Foundation. I am attaching an educator’s guide that is 20 pages long and explains the concepts perfectly. https://fractalfoundation.org/fractivities/FractalPacks-EducatorsGuide.pdf

Fractals are a combination of Science, Math and Art.

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Other resources:

A wonderful article from IBM about Benoit Mandelbrot and fractal geometry.