Fractal Geometry :: Fractals in Nature
What are Fractals?
A fractal is defined as a "rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," according to their "discoverer" Benoit Mendelbrot (Nov. 20, 1924 – Oct. 14, 2010). In plain English this means that something is "self-similar," or at least generally like a zoomed in/out iteration of itself.
For instance, the tree on the right has fractal characteristics. If you were to zoom in on one of the large branches, it would look almost exactly like the whole tree. If you were to zoom in on one of the branches of that branch, it would also look similar to both the tree as a whole and the branch to which it is attached.
History of Fractals
The term "fractal" was coined by Benoit Mandelbrot in 1975, after much work in applied areas such as measurement of coastlines. Until then, the measurement of coastlines varied depending on the "size of your ruler." That is, the more exact you tried to measure the coastline, the "longer" it became, because you found more inlets and small features to measure. An example of the idea from Britain is shown to the right. Mandelbrot saw this problem and supposed that if one could measure the "roughness" of the shore, one could get a better measurement. He said that a line is 1 dimensional, and a shape is 2 dimensional. If there was something in between, say, 1 and 2, then maybe that could measure the "roughness of the shoreline." Mandelbrot proposed that it could be measured more precisely using self-similar iterated functions. This was also the beginnings of what came to be known as "fractal dimensionality."
He also encountered self-similarity when he as working at IBM and analyzing electronic "noise" in their communication equipment. He noticed that despite the time interval being studied (one day, hour, second) that the relative spacing of the noise was similar.
The Original Fractals
Before Mandelbrot really kicked the door open on fractals, there were a few mathematical problems that had been colloquially called "monsters" in the mathematics community. The first is known as the Cantor Set after German mathematician Georg Cantor. It is a line segment in which with each iteration, the middle third of the segment is removed. In standard Euclidean geometry and calculus, the whole line is removed, yet at the same time that can not be true. For example, if the line segment was measured between 0 and 1, no matter how many iterations are carried out, the value 1/4 is never eliminated.
Another early notable fractal is known as the Koch Set or Koch Snowflake. The fractal was made by German Helge von Koch in 1904. In a similar way to the Cantor Set, it presents a mathematical paradox. The snowflake starts as an equilateral triangle. This obviously has a finite perimeter. Each side then has the middle third of it removed and the other two sides of an equilateral triangle are added in. This is repeated infinitely. The paradox is that a clearly enclosed shape as a mathematically infinite perimeter. The first 5 iterations of the snowflake are shown twice in the video to the right.
The third pre-Mandelbrot fractal is the Julia Set, named after the French mathematician Gaston Julia. While it involves spme complex math that I won't go into, a Julia set is basically a recurring iterated function. It was Julia's work with iterated functions that gave rise to the Mandelbrot set. The Mandelbrot set (seen at the beginning of the article) is a particular kind of Julia set and also the most famous mathematical fractal.
Fractals in Nature
Fractals are found in every corner of nature. They are found in the furthest reaches of the universe and the most minute details of our planet. Many different plants follow a fractal pattern for their growth. This has been found to be incredibly efficient in terms of allowing a relatively small amount of code to carry out incredibly complex and intricate tasks.
It is not just living things that can be represented by fractals. Fractals can be used to describe inorganic activity as well. Due to the geometry of water molecules, when water freezes into a snowflake, it invariable forms in a hexagonal arrangement. This arrangement is then repeated on each of the six arms or points, and so on. Electrical impulses follow a fractal pattern as well. Like the tree in the first illustration, the electrical impulse spreads like a tree, and each branch looks almost exactly like the pulse as a whole.
Another common natural element that is easily represented by fractals are mountains. These natural landscapes can be generated on a computer by relatively simple recursive programs. Most start with a triangle, then split that triangle into more triangles. The depth is adjusted randomly for each triangle, and is then split into more triangles. As the program repeats, the picture of a mountain emerges.
Fractals in the Rainforest-Start at 5:00
3:16-Fractals in Tornadoes
Fractals in Technology
In the 1990s, fractals were recognized for another ability. They could be used to receive multiple band widths of frequency in a very small enclosed space. Before this time, each band would have had to have a special antenna. However, the growing popularity of various features in cell phones made a multiple-antenna option unfeasible. The fractal antenna (one type seen to the right) solves the problem.
Books on Fractals
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