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### Sid Kemp says

Nettlemere's answer is beautiful, but . . .

Chaos theory is a field of study of mathematics related to a particular set of equations known as fractal equations. These equations generate extremely complex and unpredictable patterns when graphed. The patterns mimic nature amazingly well. In fact, some natural phenomena, such as the creation of snowflakes - where every snowflake is different, but all are hexagonal, have been modeled with chaos theory and fractal mathematics.

One of the most useful results has been that, by understanding the fractal nature of the electrical nerve impulse that causes the heart to beat, we now have better defibrillators to restore life after a heart attack.

Artificial landscapes can be created by fractal mathematics. The first major use of this was the world created by the Genesis Device in the movie Star Trek II: The Wrath of Khan.

### Rod Martin Jr says

I might just add to SidKemp's excellent answer that Chaos deals with the concept of small changes that generate large changes in complex systems that are locally unpredictable but globally stable.

The classic example of this deals with the complex system we know of as weather. A tiny change in one part of the system can affect the entire system in major ways. Like the beating of a butterfly's wings in China can change the direction of a hurricane in the Atlantic a month later. If the butterfly had not been there, the hurricane hits Miami; but with the butterfly, the hurricane hits Houston.

Another revelation from the study of chaos involves communications and the early attempts to overcome noise (static) by raising the amplitude of the signal. What resulted was not more signal, but more noise. This was an effect of chaos in something that is called a bifurcation of states. Applying more energy to the system generates other states above and below the new amplitude. The more energy that is applied the greater the bifurcation until there are countless states occurring, effectively eliminating the gains from boosting the signal. The same type of bifurcation occurs in ecosystems, like fish populations in a lake.

Fractals are fractional dimensions. We normally think of only integer dimensions, like 1st, 2nd, 3rd and 4th dimensions. But fractals deal with dimensions between these. A pattern that entirely fills a plane (like a sheet of paper) would be considered 2-dimensional. But a fractal pattern might fill only half the plane and thus would be a 1.5-dimensional object. The fractional dimension thus describes the scope of the fractal image generated and might give an indication of the character of that image. The frond of a fern might be a 1.375298 dimension object, while a cumulo-nimbus cloud might be a 2.947521 dimension object.

### Nettlemere says

I'm tempted to say a good example of it is several kittens with a dozen balls of wool, but perhaps that's string theory!

Great humor! I love it.