Introduction to Chaos and Fractals

By the end of the renaissance, scientists felt that the universe could be understood in terms of a small number of physical laws. The solar system was like a clock, with the planets orbiting the Sun, the moons orbiting their planets and so on. In principle, the motions of the planets could be predicted infinitely far into the future -- the only restriction was our ability to do the computations. The philosophy of "determinism" pervaded all of physical science -- any event or observation could be predicted from preceeding events, provided the physical laws were known.

The solar system was thought to be so clocklike that the Earl of Orrery (Charles Boyle, 1676-1731) built elaborate mechanical models of the solar system which were used to predict the positions of the planets. We now call such models "Orreries". Here is a picture of one.

Modern-day "Orreries" are computer programs that predict the positions of the planets.

The determinalistic view of science changed in 1960 when a meteorologist named Edward Lorenz pointed out that for some physical systems, it is impossible to predict the future, even if we know all the relevant physical laws. This is because sometimes the calculation of what will happen depends on minute details of the present configuration of the system -- what mathemations call the "initial conditions". An example is the weather -- we are all familiar with the fact that it is possible to predict what the weather will be like for the next few days in a general way, but after that, weather forecasts tend to be fairly useless. The famous statement he made was that if "a butterfly flaps its wings" in China the initial conditions of the weather are sufficiently changed that the long-term weather in New York would be affected. Lorenz called systems like the weather "CHAOTIC" physical systems. You would need to know the initial conditions infinitely accurately in order to predict the weather, which is impossible.

For a simple explanation of what CHAOS means, read " What is Chaos? a five part online course for everyone". Don't worry, each part is only a few sentences long.

Since Lorenz, many examples of CHAOS in nature have been suggested, from the solar system to the stock market.

In 1987, astronomers at MIT used a computer program called the 'Digital Orrery' to calculate the orbit of Pluto for the next 40 million years at 40 day intervals. This is enough time for Pluto to complete 15,000 orbits of the Sun. They found that the orbit of Pluto is chaotic, that is, its long-term shape depends very sensitively on the exact input parameters used to start the calculations. Not knowing where Pluto is today to a precision of less than one kilometer adds up over thousands of orbits to make the predictions vary over a wide range of possibilities.

More recently, astronomers using a similar computer program found that the orbits of Mercury, Venus, Earth and Mars may not be stable over the timescale of a few billion years. Depending on the exact initial conditions, the cumulative influences of Jupiter and Saturn eventually cause the inner planets to have more elliptical orbits, and in some simulations, Earth, Mars or Venus are actually ejected from the solar system. This result is controversial, but even if such drastic things don't happen, it still is impossible to specify the positions of the planets today infinitely accurately -- and so the exact positions of the planets several billion years now cannot be calculated.

FRACTALS are closely related to chaotic systems. A fractal is a geometrical shape which looks the same when you magnify it an arbitrary number of times. The interesting thing is that many fractals are very similar in appearance to patterns in nature -- like ferns, branching trees, coastlines, even cauliflower. Some famous fractals are the Mandelbrot set and the Julia set. The Mandelbrot set is shown at the top of this page. On the web there are sites that allow you to "explore" the Mandelbrot fractal, by putting your cursor on any part of the image and zooming in -- for example, check out The Beauty of Chaos web site. Althernatively, you can have a look at some animated gifs by Strumia and Alden by clicking here .

Fractals can be generated by specifying simple rules and repeating them many times. Examples of how this works are the Sierpinski Triangle, the Jurasic Park fractal, and the Koch snowflake . Click on these sites and then click on the black arrow pointing right to generate the fractal.

The appearance of fractals made with the same rules for generation can be very different if the starting points are different.

Here is a picture of the Mandelbrot Set:

Check out this movie about the mandelbrot set and its properties.

Fractals in nature: