This is the first part of a series of articles about fractals, aiming to introduce the mathematical side of fractals to the DA community, who are familiar with the artistic aspects of them. The information given will be very basic and won't require anything beyond basic mathematical knowledge.

Part 1: What Are Fractals? : An Introduction
Part 2: The History of Fractals - 1 : Fractals before Gaston Julia
Part 3: The History of Fractals - 2 : Julia and Mandelbrot Sets
Part 4: Fractals and Computers : IFS and Escape-Time Fractals
Part 5: Fractals and Art : Mathematics Meet Art

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What Are Fractals – An Introduction

This question is one of the killer questions you can ask a fractal artist, because the answer is likely to be either too complicated or not explanatory enough. So, in order to help the artists remain sane, read this article for the answer to that question.

The American Heritage Dictionary defines “fractal” as “a geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry”. Fractal is a word that was first used by Benoit Mandelbrot to define the irregular and fragmented geometry of the nature that the Euclidean geometry was unable to describe.

"Many important spatial patterns of Nature are either irregular or fragmented to such an extreme degree that ... classical geometry ... is hardly of any help in describing their form. ... I hope to show that it is possible in many cases to remedy this absence of geometric representation by using a family of shapes I propose to call fractals -- or fractal sets." [Mandelbrot, "Fractals," 1977]

The main properties and features of fractals are;

• They are self similar, which means that the smaller elements of the shapes resemble the original shape. This property isn’t seen in Euclidean shapes. For example, one side or a vertex of a triangle isn’t a triangle itself and neither is an arch a circle. However, when we look at a tree, a natural element which the fractal geometry is able to represent, we can see that the structure of a tree resembles that of its branches, which resembles that of the veins on a leaf and so on.

• They don’t lose structure even in extremely small scales. Look at the circle below;



As we zoom in to the edge of the circle, a non-fractal shape, we see that it starts to become almost linear, losing its curved and circular structure. This doesn’t happen with fractals. Let’s have a look at the Mandelbrot Set;

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(Those have the magnifications of x1, x128000 and approximately x10billion (10^10 that is), respectively.)
As we zoom in to the edges, or borders, of the Mandelbrot Set, which is one of the most commonly used fractals, we see that, instead of losing structure, it actually becomes more complex and increases in detail.

• They have non-integer dimensions, as opposed to integer dimensions of regular geometry. While a line has only 1 dimension and a circle only 2, a Cantor Set, (formed by removing the middle one third of a straight line and repeating the process infinitely) has a dimension of 0.63 and a  Koch Curve, (formed by replacing the middle one third of a straight line with an equilateral tent, again repeated infinitely), has a dimension of 1.26. I will show how they're calculated in part 2, just for fun. This is a rather difficult concept to get the hang of, but I'll explain it with more examples as we move on through the history of fractals.

Useful Links:

To zoom into Mandelbrot Set and see the wonders of it with your own eyes: [link]

For more basic information on these subjects (and more): [link]

For more advanced information on fractals and chaos: [link]

More links will come with each article, so keep an eye on this links section