This is the second 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.

Note: Even though the articles are only loosely connected to each other, they are, nonetheless, parts of a series and it is recommended that you read them in the order that they were published for a more complete understanding.

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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The History Of Fractals, Pt 1

This article, especially the second part where I will introduce some early and simple fractals and fractal sets found by certain people, will require a little more than just basic mathematical knowledge (i.e. some calculus). I will try to simplify as much as possible, but I’m not a teacher and have not exactly mastered the mathematics of fractals, so it might sound a little too complicated, so be warned . If you have any questions about the parts that I couldn’t explain well, please do ask and I’ll gladly try to answer them…

Fractals of Nature

As mentioned in the first part, fractals are the nature’s geometry, so it’s only appropriate to start this article with a few examples of fractals from nature. One classical example is a tree:

---> --->

As you can see, from roots to branches to the tiny veins, the tree is a natural fractal that shows the same branching pattern at all scales. Another popular example from Kingdom Plantea is the fern:

(early growth period) (fully grown fern)

As a matter of fact, whenever you’re thinking of a branched structure in nature (veins, neurons, streaks of lightning etc.) you’re thinking of a fractal due to the natural, random distribution. Two relatively different examples of natural fractals are;
Snowflakes , and the Romanesco (that’s an edible fractal! )

Of course, the fractals of nature are limited when it comes to infinity, since infinity is only a theoretical concept and doesn’t exist in practice, so the fractal structure breaks down at a certain point (the atomic scale if not before). As a matter of fact, the computed fractals are never really infinite either, since theoretically, it would take an infinite amount of time to calculate and display infinity. But I digress…

The Weierstrass Function


The Weierstrass Function was one of the first fractal discoveries, even though Mr. Weierstrass didn’t know it then. This function presents a challenge to the idea that every continuous function is differentiable except on a set of isolated points, a basic rule of differentiability. As can be seen in the graph, the function is self similar; the corners, when zoomed in, have a pattern similar to that of the overall function. No matter the magnification, the function never becomes “smooth” and is made up of “corners”, which means that it’s not differentiable at any point, a good reason for why some mathematicians would refer to the group of Weierstrass Functions as “ Pathological functions”.

For those who are interested, the function is

where “a” is any constant. The function was originally defined for a=2, but it has the same properties for any “a”. If you have a graphing calculator, you can easily try this. Just manually write the summation up to k=7 or something and try zooming around the graph of the function. I tried it, real fun

The Cantor Set



The Cantor Set is one of the most simplistic fractals, introduced by Georg Cantor in 1883. It is formed by removing the middle one third of a straight line and repeating the process for the resulting lines for infinite times (of course, as mentioned before, this is only the theoretical Cantor Set due to infinite iterations being impossible).

For those who are interested, stretch your minds and think about what the phrase “the Cantor Set consists of holes” means and how you can prove it. If you can’t figure it out, go check the comments for an explanation

In the previous article, I had mentioned that the Cantor Set has a dimension of 0.63. Here’s a simplified way to get there;

Fractal dimension = log(n) / log (1/l) where n is the number and l the length of the lines acquired by the first iteration. (a simplified version of the original formula – you can’t calculate the dimensions of more complicated fractal forms as easily (i.e. the Mandelbrot Set))

For the Cantor set: n=2, l=1/3. Therefore:
log(2) / log (3) = 0.63...

The Koch Curve



The Koch Curve is actually very similar to the Cantor Set. The only difference is that, to form the Koch Curve, after removing the middle third of a straight line an equilateral “tent” is added in its place. This is, obviously, repeated infinitely to form a perfect Koch Curve.

Now you can go figure out what the dimension of the Koch Curve is! (check the comments for the right answer)

The Sierpinski Triangle


To form a Sierpinski Triangle, the midpoints of the three sides of a triangle are connected to form a new triangle in the middle, which is then removed. The step is repeated infinitely. The Sierpinski is used in fractal art much more often than the aforementioned, simpler fractal forms. Just browse for “Sierpinski” in the fractal art gallery to see some lovely uses of this rather simple form…

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Useful Links:

A thorough analysis of fractal structures found at the grand canyon.

An interesting article about the fractal structure of the Romanesco

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