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Chaos, Nature, and Fuzzy Fractals

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How long is the coastline of North America? How red is a red apple? What will the weather be like tomorrow?

Until quite recently, scientists sought to find definite answers to all of the questions about nature and the human world. Scientists saw the world as consisting of imperfect geometrical shapes, following the Platonic model.

More recently, however, scientists' views of the world have changed. They have realized that North America is not an approximation of a triangle, and the ideal body temperature is not exactly 98.6°F. They have realized that the world is much more complex, and much more interesting, than traditional geometry allowed for. This philosophical revolution came about as a result of the rise of the parallel disciplines of Fractal Geometry and Fuzzy Logic.

How long is the coastline of North America?

Take out your atlas. Turn to the world map. Take a ruler marked off in eighths of an inch, and measure the coastline of North America. After some hard work, multiply your result by the scale and you will arrive at an answer. Now turn to the page with just North America and repeat the process. Repeat it again using the various small-scale regional maps. You will find that your result increases dramatically as you are able to measure around small and smaller curves in the coastline.

Now take a plane to Calais, Maine. Take out your ruler, and work your way around North America, measuring the entire coastline. Repeat the process once more, using a millimeter ruler. How much larger is your result?

As you use a finer and finer measuring instrument, the measured size of a given coastline will increase. How about a straight coastline, like that of western Manhattan, along the Battery Park seawall? If you use a fine enough ruler, you will be measuring around irregularities even there. But that is an artificial situation. There is no natural coastline that approaches its degree of straightness. A close look at the rest of nature confirms the following axiom of fractal geometry:

There are no straight lines in nature.

Not only are there no straight lines in nature, there are no traditional geometric shapes at all. A tree is not a cylinder, a mountain is not a cone, and the Earth is not a sphere. Traditional thought contends that these objects are approximations of these shapes. In fact, the opposite is true, leading to another axiom:

Geometric shapes are approximations of nature, not the other way around.

How does this relate to other fields?

How red is a red apple?

Go to the supermarket and buy some apples. Buy a few Red Delicious, a few BC Macintosh, and some New Zealand Granny Smith. When you get home, divide these apples into two groups: red and not red. You will classify some apples easily; some of the Red Delicious will be completely red, while the Granny Smith will probably have no red on them. But what do you do about the Macintosh apples that are part red and part yellow-green?

Some people may classify any apple with the tiniest bit of red as a red apple. Others may say that if it is not completely red, then it is a not-red apple. Still others may give up and just eat the apples that refuse to be classified. Who is right? Everyone is. And no one is. This is the domain of fuzzy set theory.

Traditional set theory holds that every object must have a definite classification. An ion is positive or negative. A test can be passed or failed. Fuzzy set theory eliminates the heavy line down the middle. A sodium ion (1+) is more positive than a chlorine ion (1-), but it is more negative than an aluminum ion (3+); it is positive to a certain degree. You may have received a passing grade on the test, but if you can't remember the material, you have failed to a certain degree. And most of your apples are red and not red—to a certain degree.

Traditional set theory sought to squeeze the world into a few neat little boxes. Fuzzy set theory gives us a clearer picture of the world by allowing for the objects that will not fit into a definite class. But are there really enough of those objects to worry about? Let's go back to our apples. Look at the reddest of your Red Delicious apples. Is it perfectly red? Is there even the tiniest spot of not-red on it? It turns out that if you're looking for perfection, your 'red' box will contain no apples at all. Perfectly red apples are a limiting case, just like a perfectly tall person (who is as tall as a person can possibly be) or a perfectly round ball. Which leads us to our next axiom:

Limiting cases do not exist in nature.

We have seen how fractal geometry and fuzzy set theory apply to static objects. How do they apply to systems as they move through time?

What will the weather be like tomorrow?

Forecasting the weather has to be the only profession where you can be completely wrong every day and still keep your job. Why is it so difficult?

To state it simply, weather is caused by the interaction of wind and water. During the day, the sun warms dry land more than it warms the oceans. The warm air over land rises, and the cooler air over the ocean moves onshore to fill the void. If there are mountains near the coast, the moist ocean air rises, cools, and drops its moisture as rain. If it is cold enough, snow falls instead of rain.

What's so difficult about that?

On a geometrically perfect world, with straight coastlines, perfect ridge-shaped mountains, and constant solar energy input, it might be easy to predict the weather. But coastlines are not straight, so parts of an air mass will reach land and start to rise before other parts. Irregularities in the mountains will deflect the wind, and the deflected wind will affect the movement of other parts of the air mass. The sun's energy fluctuates, adding another unstable element to the weather. And the rotation of the Earth causes air masses to spin, hitting coastlines and terrain at unpredictable angles.

The weather is a complex dynamic system, one in which a number of factors influence the moving components of the system. Some of the main factors that influence the weather are the flatness of the land, the temperature difference between land and water, and the amount of sunlight. But small factors can have a large influence on the weather.

In 1991, Mount Pinatubo erupted in the Philippines. The volcano erupted powerfully for about a week, spreading ash and pyroclastic lava over an area of Luzon, the northern island of the Philippines. The major effects may have been quite local, but the volcano had a minor effect on the whole world for the better part of the next year. Global temperature was lowered by about a degree Celsius. Most of North America experienced spectacular sunsets.

Are global systems like weather the only complex dynamic systems?

Far from it. Try this: go up to your wife and tell her you've been having an affair. (Women: switch genders where appropriate.) Can you tell exactly what her reaction will be? If you've lived with her for a long time, you can probably make a pretty good guess. (The next thing you'll have to figure out is where you will sleep that night.) Now tell her you've been having an affair with another man. Will her reaction still be predictable?

The human brain is a complex dynamic system. Even I cannot predict exactly what I will write next, although I have mapped out a general outline for this essay. Groups of people are predictably less predictable than individuals.

But the brain is not our only complex dynamic system; all of our body systems are. When you get a cold, you cannot look at your watch and predict exactly when it will go away. Your immune system's response to a virus is different every time. Of course, so is the virus.

Viruses are merely fragments of DNA enclosed in a protein jacket, but even they are complex dynamic systems. They mutate unpredictably, evolving into new viruses that cause entirely new diseases. Such is what happened when Simian Immunodeficiency Virus (SIV, which affects chimpanzees) mutated into HIV, the virus that causes AIDS. Human and simian DNA are so closely related that only a small mutation was needed to make the jump between species.

This brings us to Jurassic Park

In the movie, John Hammond (Richard Attenborough) proudly shows his guests around his dinosaur theme park while the brooding mathematician (Jeff Goldblum) insists that he is meddling with powers he cannot understand. Goldblum's character insists that Hammond cannot control the dinosaurs the way Disney controls its attractions. ("If something goes wrong, the Pirates of the Caribbean don't eat the guests.") And he turns out to be right. Putting the wrong plants in a particular paddock makes a triceratops sick. The amphibian DNA mutates unpredictably, spontaneously creating male dinosaurs. And the computer whiz has an alternate agenda, and throws the whole park into chaos.

The premise of Michael Crichton's story is that we cannot predict outcomes in a complex dynamic system, and it is dangerous to assume we can. The scientists working for the park shared the traditional view that outcomes can be predicted as long as all contributing factors are properly classified and accounted for, instead of the fuzzy logic view that these factors cannot be classified with an adequate degree of certainty. When they used amphibian DNA to splice together the missing gene sequences, the dinosaurs were no longer simply dinosaurs; they were dinosaurs and not dinosaurs. Even in nature, the animals we call dinosaurs were a more diverse group than today's mammals. The scientists fell into the common trap of assuming too much knowledge. (For another example of scientists assuming too much knowledge, see Big Bang: You're Dead! On this website.)

How large does a system need to be in order to be complex?

The following is an excerpt from "The Big Bang Never Happened", by Eric Lerner:

…T. Petrosky of the University of Texas, used a computer simulation to predict the number of orbits…a comet would make before being expelled from the solar system, consisting, in the model, only of the sun and Jupiter. He varied only the accuracy with which the orbit was calculated. If the velocities were calculated to a precision of one part in a million, the model showed that the comet would stick around for 757 orbits. When the accuracy was improved to one part in ten million the prediction was 38 orbits; one part in a hundred million, 235 orbits, and so on, down to one part in 10^16, 17 orbits. There was no tendency whatsoever for the predictions to approach a single solution with increasing accuracy…

The system used in the simulation consisted of the sun, Jupiter, and a comet; just three objects in the simulated universe. But without infinitely precise knowledge of the objects' motion, Petrosky could not predict certainly when the comet would be ejected from the system. This leads us to the final axiom in this essay:

All natural systems are complex dynamic systems.

But surely they can't be! We have plotted the orbits of the planets with a high degree of accuracy. We have some ability to predict the weather a day in advance. The universe does follow some rules.

A complex dynamic system is not completely unpredictable. In fact, simple systems are predictable to a large degree. Chaos theory states that systems with more components are less predictable, and that no system is perfectly predictable. Your result will depend on your measuring instrument. Having a finer measuring instrument does not necessarily lead you closer to the 'Truth'; certain measurements will be reasonable for certain purposes.

So go ahead and say that it will rain tomorrow. Here in Vancouver, you'll probably be right. Go ahead and say that Microsoft's legal problems will have a detrimental effect on the stock market. It might happen. And feel free to say that Canada has a longer coastline than the continental USA, as long as you define your measuring instrument.

Chaos theory is merely an extension of traditional mathematics, one which allows natural object to be described more accurately (fractal geometry), and sets limits as to the predictability of natural systems. The world is a fuzzy place, and an interesting one. And remember: "There are more things in heaven and Earth, Horatio, than are dreamt of in your philosophy."

 

References:

Michael Crichton. Jurassic Park. Movie by Steven Spielberg.

Bart Kosko. Fuzzy Thinking: The New Science of Fuzzy Logic. New York: Hyperion, 1993.

Eric Lerner. The Big Bang Never Happened. New York: Random House, 1991.

Benoit Mandelbrot. The Fractal Geometry of Nature. New York: W.H. Freeman and Company, 1983.

Mount Pinatubo. http://parallel.park.org:8888/Philippines/pinatubo/

William Shakespeare. Hamlet, Prince of Denmark.

Chaos, Nature, and Fuzzy Fractals, copyright 2000 by George Beckingham