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.

Benoit Mandelbrot
Benoit Mandelbrot

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 Cantor Set
The Cantor Set
The Koch Snowflake
The Koch Snowflake
A Julia Set
A Julia Set

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.

Fern
Fern
Sea Shell
Sea Shell

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.


Snowflake
Snowflake
Electrical Discharge
Electrical Discharge

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.


Fractal-generated mountain landscape
Fractal-generated mountain landscape

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.

Comments 5 comments

ryancarter profile image

ryancarter 6 years ago from Falkville, Alabama

Excellent Hub! You have an amazing scientific-writing ability that really conveys your level of understanding of the subject. I had a Physics professor tell me that "you'll know you really understand this stuff when you can go and explain it to an Art major!"

I'm pretty sure anyone would get a better grasp on fractals from reading this Hub!


dosters profile image

dosters 6 years ago from Chicago Author

Thanks Ryan! I tried to make it accessible to everyone, so hopefully people can see and learn more about these beautiful forces of nature and math.


Jerry G2 profile image

Jerry G2 6 years ago from Cedar Rapids, IA

Great stuff. When you have me interested in geometry, you know you're doing something right. Is this in the same general field as the Fibonacci sequence? I've done a lot of work on that because of its trading implications.


dosters profile image

dosters 6 years ago from Chicago Author

Jerry-Thanks for the kind words. There are some relationships between fibonacci sequences and fractal relationships, in that the golden ratio can serve to make whatever it is acting upon as a self-similar form. In fact there are romanesco variety of broccoli is both fractal and fibonacci related! Cool huh?

http://roxiemike.files.wordpress.com/2010/01/roman...


mgt28 profile image

mgt28 2 years ago

Thank you, now I understand what is fractal dimensionality. It is easy to understand some concepts from such articles than from mathematics texts. The more reason why Web 2.0 is a great thing, we learn from unlikey sources.

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