Chaos Theory

Note: Thorough coverage of chaos theory involves many complicated mathematical and geometrical expressions.  Its coverage in Fractals Rock is designed to be comprehensible in layman’s terms, and so does not touch on these complex expressions.  This is simply an overview of the broad mechanics of chaos theory for the purpose of gaining a deeper understanding of how it is used in fractal design.  As such, the following descriptions have been simplified considerably, and many details of little direct importance to fractal design have been left out.  For more in-depth information on chaos theory and its related concepts, type any of the headings you see below into your favorite search engine.  To achieve the best and most precise results, remember to enclose your search string in “parentheses”.   

Dynamical Systems
The appearance of a fractal is controlled by mathematical scripts.  The syntax of the script itself is unique to the software that is used to run it.  To understand the effects achieved by these scripts, one must first understand that a core portion of these scripts is dedicated to creating a self-contained “dynamical system”. 

Dynamical systems are difficult to describe in simple terms.  One way to think of a dynamical system in terms of fractal design is being the study of the evolution and movement of a point, given a dynamic course over a specific period of time.  While some systems are designed to show points at specified moments in time, the result is most often displayed as a continuous line or curve. In order for this evolution to be measured, several mathematical factors must be set.  The point obviously must have a starting location (most often set to a program-specific default), it must have movement (i.e. velocity, often written as a progressive or digressive equation), and it must have an element or series of elements that affect the movement of the point over time.  These elements are known as “attractors”.

An “attractor” is a point in space that affects the course and velocity of an object (in this instance, that object would be a separate point in motion).  In most cases, the attractor represents a point or area to which an object is “attracted”, hence the name.  Ideally, the object’s velocity and direction would degrade over time until it came to rest on the attractor.  The dimensions of “attractors” are specified in integers, and generally have a relatively predictable effect in a dynamical system.  In a chaotic dynamical system, however, one or more of these attractors is a “strange attractor”.

Strange Attractors
A strange attractor is an attractor that is measured in something other than integers (i.e. decimals or even complex equations), and therefore produces chaotic and often unpredictable results.  The effects of a strange attractor tend to fluctuate.  It may pull an object toward it or push it away, or cause an increase or decrease in velocity.  What makes these elements so interesting is that the fluctuations caused by the strange attractor(s) often form distinct visual patterns when observed in a dynamical system.  The visual representation of these patterns is, essentially, the very definition of a fractal, and is the core dynamic of fractal design.

The "Butterfly Effect"
Contrary to popular belief, this is not the process of taking notes on blackout periods in ones life for the purpose of time travel, as seen in the 2004 movie, “The Butterfly Effect”. There is a similarity, however. This movie is based on a phenomenon of quantum physics that gets its origins from the "butterfly effect" concept of mathematical chaos theory.  This concept, coined by Edward Lorenz in 1961 (see side panel), gets its name from the theory that the flapping wings of a butterfly could have such a profound effect on its surroundings that it could completely alter the weather patterns of the surrounding area.  In chaos theory as it relates to fractals, a minor change in the mathematical equation used to generate the fractal will undoubtedly have a dramatic effect on its subsequent appearance.  What this means is that even the most subtle changes in any area of a fractal script could have an infinite number of unique visual outcomes.  Consider the implications.  It is possible for a single fractal script to be used by a multitude of artists with no two designs ever looking exactly the same.  Now consider that many fractal artists create their own unique scripts, which could also be adjusted an infinite number of ways.  In this light, it is easy to see why there is such a wide variety of unique and beautiful fractals in the art world today.

Edward Lorenz
Edward Lorenz

American mathematician, Edward Lorenz (May 23, 1917 - April 16, 2008), was one of the key minds in the development of the chaos theory.  He coined the term “butterfly effect” to explain the dramatic effects in the outcome of a weather prediction system that occurred because of an error in input involving a mere 1/10,000 of a decimal (a variation that could theoretically be caused by something as minute as a butterfly’s wing beat).  While applying his mathematical skills in the field of meteorology, he observed a glowering problem with the use of linear mathematics to predict the weather, owing to the fact that many of the conditions involved in weather have intrinsically chaotic characteristics.  His award-winning work in the fields of mathematics and meteorology led to the discovery of the “strange attractor” and the development of many other related principals, the culmination of which is known today as the “chaos theory”.