Devlin's Angle

November 1999

When mathematics is plain sailing

In the ocean waters off New Zealand, an intense mathematical olympiad has just begun: The America's Cup. What's that you say, isn't that a yacht race? Yes it is. Indeed, it's the premier international event in ocean sailing. Competition is fierce. The technical challenges are enormous. And the costs are huge, with teams spending up to $15 million on the design and construction of a single boat.

The mathematics comes in because it can provide the crucial innovation that can mark the difference between winning and losing. John Marshall, who has been designing America's Cup boats for over twenty years, lays it on the line: "This sport [America's Cup racing] would not be possible today without mathematics." He explains: "To cross the line first, a boat must be skippered by a strategic genius, and the crew must be honed to a finely tuned machine. But the best skipper and crew can't compete successfully without a winning boat."

In 1995, Marshall headed the syndicate whose boat went up against New Zealand, and lost. He now runs a syndicate called PACT 2000 -- Partnership for America's Cup Technologies 2000 -- that is trying to win it back. In their search for the winning boat, Marshall and his design team rely on computer simulations and mathematics. He says, "In this day and age, it costs too much and the design problems are too complex to build a lot of boats and test them in the water. Boats are modeled and tested on a computer long before we start pouring the fiberglass."

Like all the other designers, Marshall looks for minute increases in efficiency. "A one percent increase in speed may not seem like much," he admits. "But that translates into an eight minute advantage in races that are often won by seconds. So the difference between winning and losing physics is not very much. That's why design is so crucial."

And that's why the mathematics is crucial. An America's Cup boat has a complex shape, and it moves through two mediums at once, air and water. It's the mathematics of fluid flow with a vengeance. Marshall remarks, "A lot of the mathematics and technology we used to create the boats was developed in the cold war to create weapons."

Ted Brown, the construction manager for AmericaOne, one of four other American entries besides Marshall's, describes the task this way: "This is as high tech as you can get and probably the biggest challenge of anyone's boat-building career."

In fact, not only is mathematics at the heart of America's Cup racing when it comes to boat design, it may be the only sport that is actually defined by a mathematical formula. In order to ensure fair competition, the International Racing Committee has established a rule the limits the sail area and hull size (or displacement). The bigger the hull, the smaller the allowable sail area, and vice versa. The rule is stated in the form of a mathematical formula. A typical entry is between 75 and 78 feet long and weighs 45,000 to 48,000 pounds. The principal goal facing designers is to find the optimal relation between those three key parameters: sail area, hull size, and keel. But they look for every little advantage they can find. For example, the cylindrical masts of yesteryear, made of hollow aluminum, have given way to sleek, aerodynamically designed shapes like aerofoils.

"The difficulty is trying to optimize a problem that has many parameters that trade off with each other," says Marshall. "For instance, you don't simply take all the sail area you can get, because you have to pay for it in some way. That kind of optimization problem is not unique to sail boat design. It's common to economics problems, to business management problems, to a vast array of real life problems. What you do is construct a mathematical model that includes all the important variables and all the equations that relate them."

"Take the width of the boat, for example. We can write equations that relate the width of the boat to the stability of the boat. We can also write equations that relate the stability of the boat to its performance in varying speeds of winds -- essentially unimportant in very light wind, critically important in heavy wind. So now you have a relation between the width of the boat and a performance parameter which in turn is related to wind speed. Then we go back to the width of the boat and look at its other effects on performance, say the wetted surface or frictional drag."

"So, gradually you build up a series of equations that are all interlinked, and which describe the entire physical system. ... A designer would now be able to select a set of parameters for the boat, choosing a number for each one, a length, a weight, sail area, and so on, and get a quantitative prediction of performance."

The entire design process, which begins the moment the previous America's Cup is over, involves not just computer simulations of single boats, but simulations of entire races, where one design is pitted against another to see which is best. At this stage, designers also take account of the weather conditions that are likely to prevail when the races will be run. Thus, the race simulations include wind and wave models. Getting it right isn't easy. In the last America's Cup in 1995, held in San Diego, everyone predicted light winds and small waves. But in the event, the winds were strong and the water was choppy. Australia's boat wasn't built to take the stress and it fell apart and sank. The winning New Zealand boat was designed for the rougher conditions that prevailed.

The final series of nine races to decide the winner of America's Cup 2000 begins in February. When the final race is over and the winner declared, it will be as much a triumph of the victorious team's mathematics as their sailing prowess.


Much of the above account is abridged from my book Life by the Numbers , published by John Wiley in 1998.


Devlin's Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary's College of California, in Moraga, California, and a Senior Researcher at Stanford University. His latest book InfoSense: Turning Information Into Knowledge, has just been published by W. H. Freeman.