Home: Scrying: Riding the waves of three

Riding the waves of three

March 03, 2006

I always enjoy reading the “Ask the Experts” section of Scientific American, and the question posted on the site this week was no exception:Waves in groups, water or chaos

Why do waves always break in odd-numbered groups?

Meteorologist John Guiney, who answered the question, suggested they don’t necessarily break in odd numbered groups. On the other hand, he pointed out that a group always contains at least three waves:
Over distance and time, waves that move at nearly the same speed keep pace with one another and form a group. Wave measurements usually show a tendency for large waves to group together—often referred to by scientists as “groupiness.” Normally, the number of waves in a group range anywhere from three to 15 or more, and it typically consists of smaller waves in the lead, larger waves in the middle and smaller waves again at the rear. This is because waves in the rear tend to move forward, build in size and then diminish as they reach the front.

The rest of the answer was quite interesting, but this was the part that stuck with me all week. It had a ring of familiarity—I can’t count the times I’ve heard that chaos always happens in threes. It doesn’t necessarily, but any less than three coincidental (aka, pain-in-the-ass) unexpected events do not constitute chaos. On the other hand, three is all it takes.

Look at Henri Poincare’s work on the three-body problem. In the middle of the 19th century, scientists struggled to predict the motion of planetary bodies. It was simple enough to calculate the trajectory of one or two planets, but when a third was added into the mix, the equations become complex and incomprehensible. Three was all it took.

When Poincare went to tackle the problem, he ended up with one of the first fractals. He wrote:

When we try to represent the figure formed by these two curves and their infinitely many intersections, each corresponding to a doubly asymptotic solution, these intersections for a type of trellis, tissue, or grid with infinitely fine mesh. Neither of the two curves must ever cut across itself again, but must bend back upon itself in a very complex manner in order to cut across all of the meshes in the grid an infinite number of times….

I shall not even try to draw it, [yet] nothing is more suitable for providing us with an idea of the complex nature of the three-body problem.


With the invention of computers, mathematicians were finally able to draw out Poincare’s figures. Here’s an example:

An example of a result of the three-body problem

So there you have it… When things come in groups of at least three, you get chaos. Whether ocean waves or waves of chaos, three is the necessary threshold. Without a bit of chaos, however, life just wouldn’t be worth it. Imagine the ocean with no waves, or a sky with no stars and planets. Chaos may be odd, true, but like a fractal, can also be strangely beautiful.

 Ride the waves….

Note: Wave image via NASA’s Sci Files; Three body fractal via Oliver Junge at the Paderborn Institute for Scientific Computation (the page also includes animations of the fractal being drawn.) Poincare quote from page 74, Einstein’s Clocks, Poincare’s Maps by Peter Galison. W. W. Nortan & Company, New York: 2003.

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