1999 POSTS

JANUARY 1999

I have no file for a January 1999 post. It’s possible I did not post one. KD

FEBRUARY 1999

Stardust Equations

On Sunday, February 7, at 4:04 PM, a Lockheed Martin Delta II rocket lifted off from Cape Canaveral carrying an unmanned spacecraft about the size of an office desk, having the romantic name of Stardust.

If all goes well, on January 15, 2006, the spacecraft will return to Earth and release into the atmosphere a small capsule, measuring 30 inches across and 20 inches deep, which will parachute down onto the salt flats in the Utah desert, landing at 3:00 AM.

Inside the capsule, buried in two pieces of an ultra light, sponge-like, synthetic material called aerogel, scientists hope to find the precious substances the spacecraft was sent up to obtain: dust from distant stars (the “stardust”) and minute particles from the tail of a comet. Expected total weight of the cargo: about one-thousandth of an ounce.

During its seven year journey, Stardust will travel some 3.1 billion miles through the solar system at an average speed of 48,000 miles per hour.

One major dust collection operation will take place over a period of a few minutes on January 2, 2004, when Stardust flies through the streaming tail of the comet Wild 2 (it’s pronounced “vihlt”). In addition, between March and May 2000 and again between July and December 2002, while en route to the encounter with Wild 2, a second aerogel collector will collect interstellar dust from a recently discovered beam of particles streaming into the solar system from other stars in outer space.

[Click here to see an artist’s illustration of Stardust’s encounter with Wild 2, displayed on the NASA Stardust website.]

Apart from the two dust collections, the only other activity of note that the spacecraft will perform is to take close-up photographs of the comet and a few tourist snapshots during the journey.

With manned space flights getting most of the media attention—especially when they involve septuagenarian American legends—an unmanned mining operation costing a mere $200 million is unlikely to cause much of a stir. And yet, if you stop to think for a moment, a voyage of 3 billion miles culminating in the return to Earth at a specific location, at a specific time on a specific day seven years later, is a powerful reminder of the predictive power of mathematics. As much as being a triumph for engineering, a mission such as Stardust is a significant mathematical feat.

To mount the Stardust mission, both the orbit of the comet and the trajectory the spacecraft will follow on its seven year journey have to be calculated in advance, with great precision, so that the spacecraft will come within 75 miles of the comet’s main body before heading back for Utah.

That would be difficult enough—though essentially a routine application of Newtonian mechanics—if the spacecraft followed the most direct route to its encounter with the comet. But that’s not how NASA plans to carry out the mission. Although the Stardust-comet encounter will take place a mere 242 million miles from Earth, Stardust will have traveled 2 billion miles through space to get there, and will travel a further billion miles on the return journey. Clearly, the spacecraft will not be taking the most direct route, to say the least! Why?

Fuel economy, that’s why. For Earthly travel, the longer the journey, the more fuel is required, but that’s not the case for space travel. Without air resistance to slow it down, a spacecraft can make use of the gravitational pull of the Earth, the moon, or other planets, in order to propel it on its way. It’s all a question of taking the right path.

The fuel-efficient stairways to heaven

In NASA-speak, Stardust will be making use of an “Earth Gravity Assist”, or EGA, in order to meet up with the comet. The idea behind gravity assist is to use the gravitational force of a planet as a source of energy, either to change the direction of flight of a spacecraft or to increase its speed, or both.

For example, in 1970, the world watched breathlessly as NASA used a Moon gravity assist to rescue the Apollo 13 astronauts after an on-board explosion had severely damaged their spacecraft as it set off for the moon. By using a relatively small amount of fuel to put the spacecraft onto a suitable trajectory, the NASA engineers and the astronauts were able to use the Moon’s gravity to turn around and head back for Earth. (A short—but accurately timed—rocket burn behind the moon ensured that the stricken craft did not go into permanent lunar orbit.)

In the case of Stardust, the Delta II rocket will launch the spacecraft into a wide, eccentric orbit around the sun, well outside the Earth’s own orbit. That initial trajectory will bring the spacecraft back into close proximity with the Earth two years later. As Stardust approaches the Earth, it will swing around the Earth under the Earth’s gravity. A small rocket burn will ensure that, when it starts to move away from the Earth, the spacecraft will follow a much wider heliocentric orbit (well beyond Mars and over half the distance to Jupiter) that will bring it to the Wild 2 orbit just as it (the spacecraft) completes its second circuit around the sun — at precisely the moment when Wild 2 itself reaches that location. In other words, NASA will use the Earth’s gravity to provide a kind of “slingshot” to fling the spacecraft in the right direction.

Then, after the brief, twelve-hour union of the two space travelers—one of natural origins, the other “Made in America”—Stardust will set off on its third and final wide loop around the sun, this time resulting in the reentry capsule’s separation and high-speed entry into the Earth’s atmosphere—28,000 miles per hour at first contact with the atmosphere, 70% faster than the space shuttle. (The main spacecraft itself will go into permanent solar orbit.)

If you click here, you’ll see a map of Stardust’s proposed trajectory, as shown on the NASA Stardust website. The map shows the three points where small guidance thrusters on the spacecraft will be fired to adjust the orbit, in March 2000, November 2001, and July 2003. These locations are carefully chosen so that a small force will lead to a significant change in the orbit. Apart from the guidance thrusters, gravity provides the craft’s propulsion, with solar panels being used to generate the electricity to power the on board equipment.

All in all, with its use of gravity for propulsion and solar power to generate electricity, Stardust is a decidedly ecological spacecraft—the NASA equivalent of organic farming. The same cannot be said of another planetary wanderer that will be in the heavens at the same time. Launched on October 15, 1997, Cassini was designed to explore Saturn and its moons. Besides being much larger (about the size of a 30-passenger school bus), and requiring a massive Titan IV rocket to send it on its way, Cassini is powered by nuclear energy. And it’s the 72 pounds of plutonium fuel on board that has caused—and continues to cause—controversy about the enterprise.

Prior to the mission, a number of individuals and groups attempted to prevent it, concerned at the consequences of a mishap during the launch. (Certainly, the release of 72 pounds of highly radioactive and intensely toxic plutonium into the atmosphere could be deadly to a large segment of the Earth’s population. Much of the debate centered on whether such an outcome were possible, given the design of the spacecraft and of its three nuclear power units.)

In the event, after two delays, the launch went ahead without problem. But that was not the end of the debate. It reemerged just after last Christmas, and is raging now, with the approach of the planned Earth-flyby of the spacecraft just 720 miles above the Earth’s surface, that will take place this coming August.

For, as with Stardust, Cassini makes use of planetary gravitation in order to propel itself to its destination. (Unlike Stardust, Cassini has two large engines, but they are not used to propel the spacecraft to Saturn; rather, their function is to guide it around the Saturn system for four years of exploration when it gets there.) In the case of Cassini, however, gravity assist is used primarily to accelerate the spacecraft. This means that its trajectory is much more complicated than for Stardust.

Cassini’s flight path to Saturn involves two swings by Venus—one last April 21, the second on June 20 of this year—followed by a close pass of Earth on August 16, and a close pass by Jupiter on December 30, 2000. Yes, that’s right, to get to Saturn, far out in the Solar System, Cassini starts off by heading for Venus, which is closer to the Sun than is the Earth. In order to reach an outer planet, the spacecraft starts out traveling in the “wrong” direction.

NASA refers to Cassini’s flightpath as a VVEJGA trajectory, for Venus-Venus-Earth-Jupiter Gravity Assist. [To see the entire trajectory, click here.] The spacecraft is scheduled to arrive at Saturn on July 1, 2004.

The current controversy concerning Cassini’s upcoming Earth swingby centers on the risk of the spacecraft accidentally re-entering the Earth’s atmosphere. According to the critics, with the spacecraft moving so fast (42,000 miles per hour), the 720 mile separation from the atmosphere is far too small, and a tiny error in computing the orbit or a sudden malfunction on the spacecraft could send it hurting toward Earth, with catastrophic consequences. NASA says that such an eventuality is so unlikely that there is no reason to call off the mission and destroy the spacecraft remotely as opponents are demanding. One thing both sides agree on however, is this: NASA has to make sure it gets its math right.

Once Cassini arrives in the vicinity of Saturn, it will begin a complicated sequence of orbits around the planet that will last four years, during which time various on board instruments will measure the atmosphere and magnetic field of Saturn and of several of its many moons. In addition, a separate probe, called Huyghens, will be released from the mother ship to carry out a special examination of Saturn’s largest moon, Titan, a body similar in size to the Earth. The Huyghens probe will eventually land on Titan’s surface, from where NASA hopes to receive pictures, relayed via Cassini.

In order to use gravity assist to increase the speed of a spacecraft, the idea is to use planets to provide “slingshots” that “hurl” the craft from one planet to another, its speed increasing with each slingshot. Put simply, here is how it works.

As the spacecraft approaches the planet, the planet’s gravity accelerates the space vehicle. If the spacecraft’s initial velocity is too low, or if it is heading too close to the planet, then the planet’s gravitational pull will simply suck it down to the planet’s surface. But if the initial speed is great enough, and if its orbit does not bring it too close to the planet, then the gravitational pull will just bend the spacecraft’s trajectory around, and the accelerated spacecraft will shoot right past the planet and start to head away from the planet.

If there were no other gravitational sources around, then the gravity from the planet just passed would start to slow down the spacecraft as it moved away. If the planet were stationary, the slow-down effect would be equal to the initial acceleration, so there would be no net gain in speed. But the planets are themselves moving through space at high speeds, and this is what gives the “slingshot” effect. Provided the spacecraft is traveling through space in the same direction as the planet (say, counterclockwise around the sun), the spacecraft will emerge from the gravity assist maneuver moving faster than before. (In theory, the speed of the spacecraft can be increased by the speed of the planet.)

In fact, the same planet can be used two or more times in succession, with the speed of the spacecraft being increased on each flyby. This is what NASA does with Cassini, sending it on two Venus flybys before it swings by Earth and then out towards the edge of the solar system.

Cassini provides a good example of a multi-planet gravity assist trajectory. By choosing a trajectory that brings the spacecraft successively into one gravitational field after another, it is possible to fling the vehicle across space at ever increasing velocities, much like the skier who goes from one mogul to another, the hang glider who soars from one air current to the next, or the surfer who rides the ocean waves, riding first one crest and then, as it starts to lose force, moving onto another.

For this to work, the planets have to line up in just the right fashion, so NASA has to time its gravity assist missions with considerable care. Some missions may only be possible once every hundred years or more. (It would be 175 years before another Cassini mission could be mounted.)

The only fuel required to surf space using gravity assist is the modest amount required for rocket propulsion to change direction at suitable moments. Gravity provides the main thrust. And once you are in space, gravity is free. You just have to find it. Like the secret to success in (some) business(es), the trick is to be in the right place at the right time, doing the right thing. Everything else follows automatically with no further effort.

NASA has been using gravity assist on a regular basis since the 1973 Mariner 10 mission, which flew past Venus on the way to Mercury. Among other missions that have made successful use of the technique since then were Pioneer 11 to Saturn, Voyagers 1 and 2 to the outer planets, Galileo to Jupiter, and Ulysses to the Sun. The most complicated gravity assist trajectory was that of Voyager 2, which swung by Jupiter, Saturn, and Uranus on its way to Neptune in August 1989, at which point it used a swing by of Neptune itself to help propel it out of the solar system and into deep space.

Wild in the heavenly streets

Even wilder than the idea of flinging spacecraft across the heavens using planets as slingshots (gravity assist) is a technique known as chaotic control, which NASA first used in the early 1980s. Gravity assist flights depends on the fact that, in the vicinity of a planet, mathematicians can write down and solve equations that describe the path of a spacecraft that goes there, enabling them to calculate the trajectory of the craft with considerable accuracy. All they need to know are the masses of the planet and the spacecraft and the direction and speed of the spacecraft’s initial approach toward the planet. The rest is straightforward. (In most cases you can get a good result using college calculus and Newtonian mechanics.)

Chaotic control, on the other hand, depends upon the fact that when a spacecraft is midway between two or more planets—more accurately, in the region where the gravitational forces from those planets cancel each other out—it can be virtually impossible to solve the mathematical equations that describe the spacecraft’s trajectory.

A simple way to think of the situation is in terms of taking a canoe trip. At the start of the trip (the launch), you have to expend some energy to get the canoe out into the river. But once you are in the middle of the river and pointing in the right direction, you can sit back and let the current take you along, with little more than an occasional stroke of the oars to keep you in the middle of the river (a course-correction thruster burn).

But then you come to a place where several rivers come together and several more flow out. Suddenly you find yourself in the middle of the rapids: a seething cauldron of intersecting currents, where you are buffeted from all sides by conflicting pressures. There’s no way that you can work out in advance how to negotiate these rapids to make your way to the outflowing river you want to take. Nevertheless, for a skilled canoeist, all it takes is one or two strokes of the oars at just the right moment, and the canoe heads in precisely the right direction. The expert canoeist is able to take advantage of the chaos to maneuver the boat in the desired direction. Practically all of the propulsive power comes from the water; all the canoeist does is provide the occasional nudge in order to take advantage of the chaos.

One thing to note is that, when your canoe is traveling down the middle of a fairly fast flowing river, it can require a lot of effort to change direction and go against the flow in some way. Though it might seem paradoxical at first, the chaos of the rapids can provide an opportunity to change direction using skill—brainpower—rather than raw physical power. This is precisely what NASA has learned to do to steer its spacecraft across the solar system, using mathematics rather than rocket fuel.

Although we can’t see gravitational forces with our eyes, mathematicians can “see” them by writing down equations (and graphing their solutions on a computer screen if they wish). What they find, more or less, is that most of the time the gravitational forces exerted by the planets in the solar system are like a vast system of rivers. (A better picture would be a vast ocean, with many currents, but rivers provide a more familiar setting for our canoeist example.) But there are some places where the gravitational rivers from different sources (planets) intersect, and in these regions you can get gravitational rapids. The theoretical existence of such points was first discovered by the 18th Century French mathematician Joseph-Louis Lagrange, and are nowadays known as Lagrange points. By directing a spacecraft to a Lagrange point, NASA can—by using a lot of mathematics but only a tiny amount of fuel—change the craft’s direction to send it toward the intended target. (Not all Lagrange points are like rapids. At some of them you get a region of calm. If you find yourself at such a point, it requires quite a lot of energy to escape. I’ll say more about Lagrange points in a moment.)

Before we go any further, I should say what mathematicians (and space scientists) mean by the word “chaos”. A more accurate term would be “unpredictable.” Both rivers and gravitational forces obey the laws of physics, and as such are entirely deterministic, i.e. not random. However, in both examples, situations can arise in which small causes can give rise to major effects. In such a situation, if the consequences of two or more of those small initial causes interact, the result can be virtually impossible to predict. You get what mathematicians call “chaos”.

The most oft-cited illustration of (mathematical) chaos is the “butterfly effect,” where the flapping of a butterfly’s wings in China can cause a hurricane in Florida two months later. In principle, this is indeed possible. In systems where different factors interact and effects can build up, such as the weather, a small initial action can indeed lead to a significant effect. In practice, however, so many factors influence the weather that it would be impossible to cause a hurricane intentionally, by, say, clapping your hands. When it comes to the weather, nature chooses which, if any, butterfly will have an effect.

In outer space, however, there are far fewer factors involved, and that opens up the possibility of being able to turn things around, and to make use of chaos to serve our own ends. In particular, a tiny nudge given to a spacecraft’s trajectory at just the right moment (e.g., at a chaotic Lagrange point) could result in a major—and planned—change in where it ends up some weeks or even months later.

The irony is, chaotic control is made possible by what mathematicians used to regard as an insurmountable problem: their inability to solve the equations of motion for a system of three or more objects moving freely in space under gravitational forces—the so-called “three body problem.” For two bodies, the solution is easy: the two bodies move around each other in elliptical orbits—or follow parabolic or hyperbolic paths in some special cases. But with three or more bodies, the relative motion can be highly erratic, with a tiny perturbation to the motion of one of the objects giving rise to wildly different behavior of all three.

In the case where one of the three bodies is a tiny spacecraft and the other two are large planets, the planets’ gravitational forces will dominate that of the spacecraft, of course, and hence the movement of the spacecraft relative to the two planets can be very unstable. The instability is particularly high near the “neutral points,” where the gravitational forces of the two planets cancel each other out. This enables space scientists to make use of the neutral points for navigational purposes.

In fact, things are a bit more complicated than that, since the relative rotation of the two planets produces a “centrifugal” force that also contributes to the position of any neutral points. This is the situation that Lagrange examined. He found that in a system where one planet orbits around a much larger one (such as the moon around the earth or the earth around the sun), there will be five points where all three forces cancel out (the two gravitational forces and the “centrifugal” force). These points are the Lagrange points, generally labeled L1, L2, L3, L4, and L5.

The L1, L2, and L3 points all lie on the straight line connecting the two planets, with L1 and L3 inside the smaller planet’s orbit around the larger and L2 outside that orbit. L4 and L5 lie on the orbit, located symmetrically one to each side of the line connecting the others. Click here for diagrams and further details about Lagrange points, displayed on NASA’s website.

[If you want to see some of the mathematics involved in locating Lagrange points, there are two excellent web sites you can consult, one at Stanford and the other at The Geometry Center. These sites are not for the mathematically faint-hearted.]

Because the gravitational and “centrifugal” forces all cancel out at a Lagrange point, you might expect that a small object such as a spacecraft that was brought to a halt at a Lagrange point would remain there indefinitely. But things are a bit more complicated than that.

Very little correction is required to maintain an object at either of the L4 and L5 Lagrange points. These points are stable: an object brought to rest there would, in theory, remain there forever unless subjected to a major new force. (In practice, the forces resulting from the movement of other bodies in the universe will mean that an occasional correction may be required.) As a result, the Earth-Moon L4 and L5 points have been suggested as possible locations for a future space station.

But the other three Lagrange points—L1, L2, and L3—are unstable: a tiny perturbation is all it takes to send a small object located there off onto a journey that can be all but impossible to predict.

The difference between the stable and unstable Lagrange points is a bit like the difference between standing at the bottom of a (sharp pointed) crater and the top of a (sharp pointed) mountain. Both places are “stable,” in that in your immediate vicinity you can walk around on the level. At the crater bottom, it can be very difficult to walk too far in any one direction, because gravity keeps dragging you back down. (Crater bottoms are like L4 and L5 Lagrange points.) But if you walk too far in any one direction from the mountain top, pretty soon you will find yourself tumbling downhill, away from the summit. (Mountain tops are like the L1, L2, and L3 Lagrange points.)

The mountain comparison is in fact quite a good one on two other counts. For one thing, in the case of a mountain summit, notice that a small difference in direction when you leave the summit can result in a huge difference in where you end up when you tumble down the mountain side. A few feet to the left or to the right of your intended direction and you can end up miles off course when you reach the valley bottom below. This means that a mountain top is a good place to change direction of travel, with little expenditure of energy. You just need to make sure your first few steps are in the right direction! This is the basis of chaotic control.

Another lesson we can learn from the mountain example comes from the observation that, if the summit is so sharp that you cannot stand there in comfort, then one way to achieve a good measure of stability there, with a minimal expenditure of energy, is to keep on circling around the summit, just a few feet below it. Since you are not descending, you remain close to the summit. But since you are not climbing, you are not expending much energy. This is exactly what NASA does when it wants to station a satellite at one of the unstable Lagrange points. It puts the satellite into a small orbit around the Lagrange point. Such orbits are called “halo orbits.” (The mathematics behind halo orbits is quite a bit more complicated than for regular orbits around planets. If you really want to know, you should consult an expert.)

For instance, Vice President Gore recently proposed that a satellite be put into orbit at an Earth-Moon Lagrange point to beam back live photographs of the Earth for constant display on the Internet. Neither of the two stable Lagrange points could be used, because they are too far away from Earth. In fact the only viable possibility is L1, the Earth-Moon Lagrange point closest to Earth. To make Gore’s suggestion a reality, NASA will have to put the satellite into a small halo orbit around L1.

To get back to chaotic control now, the idea is to find a way to use the instability of an L1, L2, or L3 Lagrange point change the direction of a spacecraft without expending much fuel. On its own, mathematics can’t do that—the very nature of an unstable Lagrange point is that the behavior of an object there is chaotic. But mathematics combined with some heavy duty computing can sometimes work. The first demonstration of the power of the math-computing alliance to achieve chaotic control was when NASA used it to coax a second, unplanned mission out of the International Sun-Earth Explorer 3 (ISEE-3) spacecraft, the third in a series of three missions to study the solar wind and the solar-terrestrial relationship at the boundaries of the Earth’s magnetosphere. Launched on August 12, 1978, ISEE-3 was placed into orbit around the Sun-Earth L1 point, some 235 Earth radii from Earth.

After four years of excellent service measuring plasmas, energetic particles, waves, and fields, NASA decided to try to recycle ISEE-3 to take a close-up look at the comet Giacobini-Zinner, which was making its way back into the inner Solar System. The problem was, hardly any of the spacecraft’s hydrazine fuel remained, so there was no possibility of any major rocket maneuvers to put it into a new trajectory. NASA scientists decided to see if they could make use of the gravitational instability in the region of, first the Sun-Earth L1 point and then the Earth-Moon L1 point to persuade ISEE-3 to go where they wanted it to. On June 10, 1982, they began a series of what turned out to be 15 small fuel burns to gradually nudge the satellite onto a trajectory toward the Earth-Moon L1 point.

Once they had the satellite in the Earth-Moon system, the NASA engineers then flew it past the Earth-Moon L1 point five times, giving it a tiny nudge on each lunar flyby, until they had put it onto a path that would lead it eventually to an encounter with Giacobini-Zinner. The fifth and final lunar flyby took place on December 22, 1983, when the satellite passed a mere 119 km above the Apollo 11 landing site. At that point, the spacecraft was renamed the International Cometary Explorer (ICE).

On June 5, 1985, ICE was maneuvered into Giacobini-Zinner’s tail, some 26,550 km behind the comet itself, and its instruments began to analyze the tail’s composition, eventually confirming the theory that comets are essentially just “dirty snowballs.”

Unexpectedly, ICE survived its passage through the comet’s tail, and in 1986 it was used to make distant observations of Halley’s Comet. The spacecraft will return to the vicinity of the Earth in 2014, when it may be possible to capture it and bring it back to Earth. NASA has already donated it to the Smithsonian Institute in case it is indeed recovered.

Following NASA’s initial, somewhat “trial and error” approach with ISEE-3, the mathematics of chaotic control began to be developed properly in 1990, starting with some work of Edward Ott, Celso Gregobi, and Jim Yorke of the University of Maryland. Their method, known after their initials as the OGY technique, involves the calculation of a sequence of “butterfly wing flaps” that will produce the desired effect—something not possible with weather on Earth but which can be achieved when it comes to maneuvering a spacecraft in space.

If we were ever to establish a space station on the Moon, the OGY technique could be used to get materials from Earth orbit to lunar orbit and back again in a relatively cheap fashion. The “standard” way to do this is to use a rocket burn to put the spacecraft into an elliptical trajectory that just touches the Earth at one end and the Moon at the other, called a Hohmann ellipse. In addition to the initial rocket burn to move the spacecraft out of, say, Earth orbit and onto the Hohmann ellipse, a second rocket burn is required to slow it down at the other end and put it into lunar orbit.

In 1995 Erik Bollt of the United States Naval Academy and James Meiss of the University of Colorado discovered a different way that is much more fuel efficient. To send a cargo from Earth orbit to lunar orbit (the same approach works in the other direction), the idea is to give the spacecraft a hefty nudge that knocks it out of Earth orbit and puts it into the first of a chaotic series of wider Earth orbits that each pass through the Earth-Moon L1 point. After the craft has gone through the L1 point a number of times (48 according to one calculation), with the occasional tiny nudge from the rockets from time to time, it slips into a chaotic series of wide orbits around the Moon, still passing through the L1 point. After a number of lunar orbits (10 by the calculation that gives 48 Earth orbits), it ends up automatically in a stable, close orbit around the moon.

It takes much longer to get from Earth orbit to lunar orbit (or back again) using this technique — about two years as opposed to three days the “standard” way. But it’s much more fuel efficient: you can carry 80% more materials this way. That’s not much use if you want to ship perishable food supplies to the crew of a lunar space colony, but fine for transporting building materials to the Moon and bringing mined minerals back to Earth.

NASA will use the latest version of chaotic control on the Genesis mission, scheduled for launch in January 2001 to carry out a two-and-a-half year “trawling” operation to collect charged particles in the solar wind and return them to Earth. Genesis will do its collection work while it orbits the Sun-Earth L1 point. But by the time it has finished, it will not have enough fuel for a direct return to Earth. Instead, it will first be sent onto a long detour to the L2 point, outside the Earth’s heliocentric orbit. Because of the flow of the “gravitational rivers,” from the Sun-Earth L2 point it can be brought back very economically to the Earth-Moon L1 point, where a few, cheap chaotic orbits of the Moon will eventually put it into a stable orbit of the Earth, from which its cargo capsule will be released to parachute down onto the Utah salt flats in August 2003.

What makes this possible is the existence of a “free ride” path from L1 to L2. (It’s a bit like finding a freeway from New York to Los Angeles that you can drive without using any gasoline.) It took a large dose of advanced mathematics to discover this path, but the resulting cost savings achieved by making the L1-to-L2 detour are huge. NASA will run the Genesis mission on a total budget of $216 million, a paltry amount by space-exploration standards.

To the outsider, modern space exploration might appear to be largely a matter of engineering and rocket science. But a key factor is the clever use of mathematics to make nature’s own gravitational forces do most of the work for us. The raw muscular power that sent the Apollo missions to the moon has given way to delicate movements more reminiscent of jujitsu. In the early days, rocket science was a mixture of propellant chemistry, engineering, and a lot of luck. These days, it’s a mixture of propellant chemistry, engineering, and a lot of mathematics.

– Keith Devlin

NOTE: Keith Devlin discussed the subject of this column in a nationally broadcast radio interview with Scott Simon on NPR’s Weekend Edition on Saturday January 30. Click here to listen to the interview. (This requires that your computer has RealAudio installed.)

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 The Language of Mathematics: Making the Invisible Visible, has just been published by W. H. Freeman.

MARCH 1999

New Mathematical Constant Discovered – Descendent of Two Thirteenth Century Rabbits

A recent mathematical result by Divakar Viswanath, a young computer scientist at the Mathematical Sciences Research Institute (MSRI) in Berkeley, California, has put the Fibonacci numbers back in the news and shown a connection between rabbits and the number 1.13198824 . . . .

The story begins in the early 13th century, when the great Italian mathematician Fibonacci posed the following simple problem. A man puts a pair of baby rabbits into an enclosed garden. Assuming that each pair of rabbits in the garden bears a new pair every month, which from the second month on itself becomes productive, how many pairs of rabbits will there be in the garden after one year? Like most mathematics problems, you are supposed to ignore such realistic happenings as death, escape, impotence, or whatever. It is not hard to see that the number of pairs of rabbits in the garden in each month is given by the numbers in the sequence 1, 1, 2, 3, 5, 8, 13, etc. This sequence of numbers is called the Fibonacci sequence. The general rule that produces it is that each number after the second one is equal to the sum of the two previous numbers. (So 1+1 = 2, 1+2 = 3, 2+3 = 5, etc.) This corresponds to the fact that each month, the new rabbit births consists of one pair to each of the newly adult pairs plus one pair for each of the earlier adult pairs. Once you have the sequence, you can simply read off that after one year there will be 377 pairs.

Most popular expositions of mathematics mention the Fibonacci sequence, usually observing that the Fibonacci numbers arise frequently in nature. For example, if you count the number of petals in various flowers you will find that the answer is often a Fibonacci number (much more frequently than you would get by chance). For instance, an iris has 3 petals, a primrose 5, a delphinium 8, ragwort 13, an aster 21, a daisy 34, and michaelmas daisies 55 or 89 petals—all Fibonacci numbers.

For another example from the botanical world, if you look at a sunflower you will see a beautiful pattern of two spirals, one running clockwise, the other counterclockwise. Count those spirals and you will find that there are 21 running clockwise and 34 counterclockwise—both Fibonacci numbers. Similarly, pine cones have 5 clockwise spirals and 8 counterclockwise spirals; and the pineapple has 8 clockwise spirals and 13 going counterclockwise.

One further example concerns the way the leaves are located on the stems of trees and plants. If you take a look, you will see that, in many cases, as you progress up along a stem, the leaves are located on a spiral path that winds around the stem. The spiral pattern is sufficiently regular that it leads to a numerical parameter characteristic for the species, called its divergence. Start at one leaf and let p be the number of complete turns of the spiral before you find a second leaf directly above the first, and let q be the number of leaves you encounter going from that first one to the last in the process (excluding the first one itself). The quotient p/q is the divergence of the plant.

If you calculate the divergence for different species of plants, you find that both the numerator and the denominator tend to be Fibonacci numbers. In particular, 1/2, 1/3, 2/5, 3/8, 5/13, and 8/21 are all common divergence ratios. For instance, common grasses have a divergence of 1/2, sedges have 1/3, many fruit trees (including the apple) have a divergence of 2/5, plantains have 3/8, and leeks come in at 5/13.

None of the examples I have given are numerological coincidences. Rather they are consequences of the way plants grow. (For instance, the leaves on a plant stem should be situated so that each has a maximum opportunity of receiving sunlight, without being obscured by other leaves.) The Fibonacci sequence is one of a number of very simple mathematical models of growth processes that happens to fit a large variety of real-life growth processes.

The Fibonacci numbers also arise in computer science—in database structures, sorting techniques, and random number generation, to name three examples. In this case, the explanation is that, in certain instances, the Fibonacci sequence models information growth.

Fibonacci numbers, the golden ratio, and the world of chance

In addition to its connections with the natural world, the Fibonacci sequence has a number of curious mathematical properties. Perhaps the most amazing is that, as you proceed along the sequence, the ratios of the successive terms get closer and closer to the famous “golden ratio” number 1.61803 . . . , the “perfect proportion” ratio much beloved by the ancient Greeks.

Another way to express the same result is that the Nth Fibonacci number is approximately equal to the Nth power of the golden ratio. This gives a way to calculate the Nth Fibonacci number without generating the entire sequence of preceding Fibonacci numbers: Take the golden ratio, raise it to the power N, divide by the square root of 5, and round off the result to the nearest whole number. The answer you get will be the Nth Fibonacci number.

Faced with such a result, most numerically minded citizens will nod appreciatively and move on to something else. But mathematicians ask “What if?” questions. For example, suppose that, when you generate the Fibonacci sequence, you flip a coin at each stage. If it comes up heads, you add the last number to the one before it to give the next number, just as Fibonacci did. But if it comes up tails, you subtract.

For example, one possible sequence you could get in this way is:

1, 1, 2 (H), 3 (H), -1 (T), 4(T), -5 (T), -1 (H), . . .

Another is:

1, 1, 0 (T), 1 (H), -1 (T), 2 (T), 1 (H), 1 (T), . . .

The random sign changes can lead to sequences that suddenly switch from large positive to large negative, such as:

1, 1, 2, 3, 5, 8, 13, 21, -8, 13, -21, . . .

as well as to sequences that cycle endlessly through a particular pattern, such as:

1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, . . .

1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, . . .

If you are like the character Rosenkrantz in Tom Stoppard’s play “Rosenkrantz and Guildenstern are Dead” and your coin keeps coming up heads every time, you can even get the original Fibonacci sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .

With such a variety of behavior, it’s not obvious that such sequences follow the nice kind of growth pattern of the Fibonacci sequence.

But they do. Last fall, Viswanath, who recently finished a Ph.D. in computer science at Cornell University in New York, showed that the absolute value of the Nth number in any random Fibonacci sequence generated as described is approximately equal to the Nth power of the number 1.13198824 . . . .

Actually, that’s not quite accurate. Because the sequences are generated randomly, there are infinitely many possibilities. Some of them will not have the 1.13198824 property. For example, the sequence that cycles endlessly through 1, 1, 0 does not have the property, nor does the original Fibonacci sequence. But those are special cases. What Viswanath showed is that if you actually start to generate such a sequence, then with probability 1 the sequence you get will have the 1.13198824 property. In other words, you can safely bet your life on the fact that for your sequence, the bigger N is, the closer the absolute value of the Nth number gets to the Nth power of 1.13198824 . . .

To give some idea of what this result says, the way the randomized Fibonacci sequence is generated is a bit like the daily weather at a particular location. Today’s weather can be assumed to depend on the weather the previous two days, but there is a large element of chance. The analog of the number 1.13198824 . . . for the weather would give a quantitative measure of the unpredictability of weather. It measures the rate at which small disturbances explode exponentially in time. It would tell you for exactly how many days high-speed computers can forecast weather reliably. Unfortunately, nobody knows this number for global weather, and probably never will.

Viswanath’s result brings to an end a puzzle that has its origins in 1960. In that year, Hillel Furstenberg (now at the Hebrew University) and Harry Kesten (at Cornell University) showed that for a general class of random-sequence generating processes that includes the random Fibonacci sequence, the absolute value of the Nth member of the sequence will, with probability 1, get closer to the Nth power of some fixed number. (The exact formulation of their result is in terms of random matrix products, and is not for the faint-hearted. See Viswanath’s paper — cited below — for an exact statement, or read the whole story in the book Random Products of Matrices With Applications to Infinite-Dimensional Schrödinger Operators, by P. Bougerol and J. Lacroix, published by Birkhauser, Basel, in 1984.)

Since Furstenberg and Kesten’s deep result applied to the randomized Fibonacci process, it followed that, with probability 1, the absolute value of the Nth number in any random Fibonacci sequence will get closer and closer to the Nth power of some fixed number K. But no one knew the value of the number K, or even how to calculate it.

What Viswanath did was find a way to compute K. At least, he computed the first eight decimal places. Almost certainly, K is irrational, so cannot be computed exactly. Viswanath presented his new result at a colloquium at MSRI last month.

Since there is no known algorithm to compute K, Viswanath had to adopt a circuitous route, showing that K equals e^P, where P lies somewhere between 0.1239755980 and 0.1239755995 (and, as usual, e is the base for natural logarithms). Since those two numbers are equal in their first eight decimal places, that meant he could calculate K to eight decimal places.

The process involved large doses of mathematics and some heavy duty computing. Since his computation made use of floating point arithmetic — which is not exact — Viswanath had to carry out a detailed mathematical analysis to obtain an upper bound on any possible errors in the computation. He describes the key to his new result this way: “The problem was that fractals were coming in the way of an exact analysis. What I did was to guess the fractal and use it to find K. To do this, I made use of some devilishly clever work carried out by Furstenberg in the early 1960s.”

And with that computation, mathematics has a new constant, a direct descendent of a pair of rabbits in thirteenth century Italy.

Yes, but what’s it good for?

Does Viswanath’s new result have any applications? Probably not—unless you count the fact that an easily understood, cute, counter-intuitive result about elementary integer arithmetic can motivate a great many individuals (your present columnist included) to take a look at an area of advanced mathematics full of deep and fascinating results that has perhaps not hitherto had the attention it deserves. Indeed, in that respect, the randomized Fibonacci sequence problem resembles Fermat’s Last Theorem. It too is easy to state and to understand, and yet it’s only a hairsbreadth away from some of the deepest and most profound mathematics of all time. Over the years, Fermat’s Last Theorem attracted many people—both professional and amateur—to learn about analytic number theory (including a young man in England by the name of Andrew Wiles, who would go on to solve the problem in 1994). When we talk about “applications,” we often overlook the very important application of attracting people to mathematics in the first place.

If it’s “real” applications you want, however, then you don’t have to go any further than the work of Furstenberg and Kesten that lays behind Viswanath’s recent result. Applications of that work have led to advances in lasers, new industrial uses of glass, and to development of the copper spirals used in birth control devices. The research which led to those advances earned the 1977 Nobel Prize in physics for the three individuals involved: Philip Anderson of Bell Laboratories, Sir Neville Mott of Cambridge University in England, and John van Vleck of Harvard.

The citation that accompanied the Nobel Prize to the three researchers declared it to be “for their fundamental theoretical investigations of the electronic structure of magnetic and disordered systems.”

“Disordered systems” exists within noncrystallic materials that have irregular atomic structures, making it difficult to theorize about them. The key starting point for their work was to realize the importance of electron correlation—the interaction between the motions of the electrons.

Anderson’s main contribution was the discovery a phenomenon known as Anderson localization, and this is where the random matrix multiplication comes in. Imagine you have a material, say a semiconductor, with some impurities. If you pass a current through it, you might expect it to get dispersed and diffracted in a random fashion by the impurities. But in fact, at certain energies, it stays localized. The first rigorous explanation of this used Furstenberg and Kesten’s work. (Moreover, the application came long after the mathematics had been worked out—yet another example of the practical importance of “pure, curiosity driven research.”)

A similar explanation shows why you can see through glass. The irregular molecular structure of glass—it’s really a “liquid”—should surely cause some of the incident light rays to bounce around in a seemingly random fashion, resulting in a blurred emergent image. But as we all know, that’s not what happens. Somehow, the repeated random movements lead to orderly behavior. Furstenberg and Kesten’s work on random matrix multiplication provides the mathematical machinery required to explain how this happens.

On a final note, Viswanath’s paper, in which he describes his new result, has just been accepted for publication in the journal Mathematics of Computation. His email address is divakar@msri.org. To download a Postscript file of his paper, go to the MSRI web site, click on “People”, then on “Current visitors”, and then on “Viswanath”.

– Keith Devlin

NOTE: Keith Devlin discussed the Fibonacci sequence in a nationally broadcast radio interview with Scott Simon on NPR’s Weekend Edition on Saturday February 27. Click here to listen to the interview. (This requires that your computer has RealAudio installed.)

Devlin’s Angle is updated at the beginning of each month.

APRIL 1999

What’s Going On During Mathematics Awareness Month?

Even in mathematics, inflation seems to take its toll, and what used to be Mathematics Awareness Week (MAW) has become—for this year at least—Mathematics Awareness Month (MAM), lasting for the whole of April.

MAM is organized by the Joint Policy Board for Mathematics (JPBM), which represents the three major organizations for professional mathematicians: The American Mathematical Society, which focuses on mathematical research, the Mathematical Association of America, whose main interest is university mathematics education, and the Society for Industrial and Applied Mathematics, which concentrates on the uses of mathematics in industry. Altogether, the three societies represent about 100,000 professional mathematicians, mostly in the US, although all three organizations have overseas members.

The JPBM organized its first Mathematics Awareness Week back in April of 1991. The motivation was to try to counteract many years of bad press for mathematics. There was a feeling that the major news media covered stories about advances in physics, medicine, chemistry, astronomy, and all the other sciences, but the only time math made the news was when another report came out saying how poorly American schoolchildren performed in math compared with other nations. The JPBM hired a Washington PR firm to put together a campaign to try to raise public awareness of mathematics, and the first Mathematics Awareness Week was set up as part of that campaign.

That proved to be a success, and as a result there has been a Mathematics Awareness Week every spring since then. Each one has had a special theme. In 1992 the theme was mathematics and the environment; in 1993 mathematics and manufacturing; in 1994 mathematics and medicine; in 1995 mathematics and symmetry; in 1996 uses of mathematics in decision making; in 1997 mathematics and the Internet; and last year the theme was mathematics and imaging—for example, how mathematicians use Hollywood film-making techniques to represent complex data in a visual form that is easy to understand.

This year’s theme is mathematics and biology. Two important examples of how mathematics is used in biology are: in developing computer models of the human heart, which are helping us to understand how the heart works and, more importantly, how it goes wrong, and the use of a branch of mathematics called knot theory to help us to understand how viruses attack the human body.

These examples show why mathematicians think that it’s important to raise public awareness of mathematics. After all, much mathematical research depends on public funding. It’s important work, but most of it is very invisible, and hard for a non expert to understand, so it doesn’t make the news very often. Mathematics Awareness Week—or this year Mathematics Awareness Month—is a deliberately created opportunity to say, “Hey look folks, most of the time you don’t know we’re here, and what we do usually doesn’t look very sexy, but it’s important stuff. Indeed, one day your life may depend on what we do.”

For the most part, all MAM activities are organized at a local level, by mathematicians at colleges, universities, and occasionally businesses and high schools all over the country. At the national level, the JPBM produces a poster that local organizations can display, sends out press releases to all the major national and regional news outlets, and prepares some background materials on the annual theme that it posts on its web site, hosted by Swarthmore College.

Local groups are free to do whatever they want. The idea is to organize events that give schoolchildren and students some idea of what mathematics is all about, and, if possible, might catch the attention of local media. Local groups are encouraged to develop activities that connect to the main theme, but they don’t have to.

In preparation for a piece on MAM for National Public Radio, I arranged for a message to be sent out to all local MAA liaisons on the MAA’s listserve, asking people to let me know what they were up to. Here is a sample of what I received back.

Students at Albion College in Michigan are building a three-dimensional fractal out of 66,048 business cards. (There’s a regularly updated photo of the object as it is being constructed on the Albion college web site.) What’s the connection with biology? you ask. Well, MAM activities don’t have to be directly connected to the main theme, and many are not. In this case, there is a link of sorts, in that there has been some interesting work using three-dimensional fractals to model the behavior of the human heart.

Folks at Pellissipi State Technical College in Knoxville, Tennessee, are organizing a contest for local middle school pupils to see who can design and build the best egg launching device. The idea is to build a craft that launches an egg and carries it the furthest, bringing it back to land without breaking. Biology? Well, it involves eggs!

Faculty and students at Marshall University in West Virginia are working with local elementary schools to build a Mars Rover out of Lego, that will have an on-board camera and be controlled from anywhere in the world over the Internet. The connection with biology is that essentially the same mathematics is required to investigate the interior of the human body using a laproscope.

The University of California at Davis is organizing an open day when members of the public can come along and try their hands at a whole range of mathematical puzzles.

Kennesaw State University in Georgia are organizing a day-long Mathematics Rodeo, where sixth, seventh and eighth grade students from local schools competed in various mathematical games, puzzles and challenges.

Many places organize pizza evenings where they show mathematics related videos, such as the PBS Nova documentary about the proof of Fermat’s Last Theorem, another one about the National Security Agency, some of the amazing computer-generated videos produced by the Geometry Center in Minneapolis, the PBS series Life by the Numbers, and many others. (I’m sure that some groups show the movies Good Will Hunting or Pi as well.)

Or there are evening public lectures about the mathematics of identification numbers such as driving license numbers, the mathematics of music, mathematics and architecture, logical puzzles, and more.

I heard of a professional psychologist who is organizing some special workshops to help people overcome math phobia.

There’s a lot going on in Kansas. The month began when the governor of the state declared April to be officially Mathematics Awareness Month in the State.

The University of Kansas is organizing a number of workshops for K-12 schoolchildren. Topics include: Math & Sea Shells, where students will see how to use mathematics to model patterns of sea shells on a computer; Fibonacci Numbers in Nature; The Mathematics of Bacteria and Viruses; The Mathematics of Population Dynamics; Mathematics and the Environment; and Mathematics and Epilepsy Research — this last one being presented by one of the university’s Ph.D. graduates, who, with a doctor from the KU Medical Center, has made a breakthrough in predicting epilepsy seizures.

They are also organizing contests for local K-4 schoolchildren designed around the theme “How Math and Biology Meet in Your Backyard”. One of those activities is to measure roly-polies to find their area when they are rolled up and comparing the answers to the area of body when they are stretched out. The students have been asked to use their data to see how much a roly-poly grows over a span of time. Apparently one enterprising seven-year old has found creative ways to keep the roly-polies laying still. As a result he has measured hundreds of roly-polies. (I don’t have any more detail, and I’m not sure I want to know!)

Among a whole series of Mathematics Awareness Month public lectures at the University of Kansas is one titled “Why Are Birds Blue? Mathematics and the Biophysics of Structural Colors in Birds.”

The University of Arizona is another institution that is really going to town this year. First, a member of the mathematics faculty and a USDA entomologist are working with a team of students at the University and teachers at Tucson High School to investigate mathematical models of honey bee populations. The results they produce will be used by the students at the Native American Summer Institute, who maintain an apiary. My informant at the University of Arizona tells me that, sadly, they are unlikely to harvest their first honey until after Math Awareness Month has finished, but they hope that the use of the mathematics will lead to a bumper harvest. (See their website for more details.) In addition to the bee study, the University of Arizona Mathematics Department is launching a series of websites dealing with the kinetics of pharmaceuticals, blood flow, flour beetle dynamics, and the cyclic behavior of mammal populations in the extreme north of Canada. As part of the official launch of the website, local high school students will be invited to visit the university’s biology labs where mathematics play a significant role in the labs’ research activities.

They are also putting on a public lecture on the mathematics of heart rhythms, a film festival, and a whole host of other events.

And that’s just a sample of the kind of activities that are going on around the nation throughout April. Given the brief—and generally misleading—exposure most people have to mathematics at school, raising the public awareness of mathematics will always be an uphill battle. But if you believe, as I do, that one of the main reasons why our country’s schoolchildren consistently perform poorly in international comparisons of mathematical ability is the attitude toward mathematics they pick up from society, then it’s a battle we should engage in.

– Keith Devlin

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. He worked on the “public awareness of mathematics” PBS television series Life by the Numbers and was the author of the official companion book of the same title, published in hardback in 1998 by John Wiley, and released in paperback later this month by the same publisher.

MAY 1999

The Greatest Math Teacher Ever

Each year, the MAA recognizes great university teachers of mathematics. Anybody who has been involved in selecting a colleague for a teaching award will know that it is an extremely difficult task. There is no universally agreed upon, ability-based linear ordering of mathematics teachers, even in a single state in a given year. How much more difficult it might be then to choose a “greatest ever university teacher of mathematics.” But if you base your decision on which university teacher of mathematics has had (through his or her teaching) the greatest impact on the field of mathematics, then there does seem to be an obvious winner. Who is it?

Having stated the problem, I should perhaps add that this column is really directed at mathematicians (and those simply interested in mathematics) under the age of 40. Most of us who have been in mathematics for over twenty years probably know the answer.

Let me give you some clues as to the individual who, I am suggesting, could justifiably be described as the American university mathematician who, through his (it’s a man) teaching, has had the greatest impact on the field of mathematics.

He died twenty-five years ago at the age of 91.

He was cut in the classic mold of the larger-than-life American hero: strong, athletic, fit, strikingly good looking, and married to a beautiful wife (in the familiar Hollywood sense).

He was well able to handle himself in a fist fight, an expert shot with a pistol, a lover of fast cars, self-confident and self-reliant, and fiercely independent.

Opinionated and fiercely strong-willed, he was forever embroiled in controversy.

He was extremely polite; for example, he would always stand up when a lady entered the room.

He was a pioneer in one of the most important branches of mathematics in the twentieth century.

He was a elected to membership of the National Academy of Science, as were three of his students.

The method of teaching he developed is now named after him.

If you measure teaching quality in terms of the product – the successful students – our man has no competition for the title of the greatest ever math teacher. During his 64 year career as a professor of mathematics, he supervised fifty successful doctoral students. Of those fifty Ph.D.’s, three went on to become presidents of the AMS – a position our man himself held at one point – and three others vice-presidents, and five became presidents of the MAA. Many more pursued highly successful careers in mathematics, achieving influential positions in the AMS and the MAA, producing successful Ph.D. students of their own and helping shape the development of American mathematics as it rose to its present-day position of world dominance.

In the last half-century, fully 25% of the time the president of the MAA has been either a student or a grandstudent of this man.

Other students and grandstudents of our mystery mathematician have served as secretary, treasurer, or executive director of one of the two mathematical organizations and have been editors of leading mathematical journals.

After reaching the age of 70, the official retirement age for professors at his university, in 1952, he not only continued to teach at the university, he continued to teach a maximum lecture load of five courses a year – more than most full-time junior faculty took on. He did so despite receiving only a half-time salary, the maximum that state regulations allowed for someone past retirement age who for special reasons was kept on in a “part-time” capacity.

He maintained that punishing schedule for a further eighteen years, producing 24 of his 50 successful Ph.D. students during that period.

He only gave up in 1969, when, after a long and bitter battle, he was quite literally forced to retire.

In 1967, the American Mathematical Monthly published the results of a national survey giving the average number of publications of doctorates in mathematics who graduated between 1950 and 1959. The three highest figures were 6.3 publications per doctorate from Tulane University, 5.44 from Harvard, and 4.96 from the University of Chicago. During that same period, our mathematician’s students averaged 7.1. What makes this figure the more remarkable is that our man had reached the official retirement age just after to the start of the period in question.

Who was he?

I’ll give you the answer next month.

Devlin’s Angle is updated at the beginning of each month.

JUNE 1999

The Greatest Math Teacher Ever, Part 2

In last month’s column, I gave a thumbnail sketch of a person I claimed was arguably the greatest university mathematics teacher ever and asked you to name the individual. If you have not read that article, please do so now, before I give the answer.

Done that?

Ready to proceed?

My nomination for the greatest ever university mathematics teacher is Robert Lee Moore. Born in Dallas in 1882, R. L. Moore (as he was generally known) stamped his imprint on American mathematics to an extent that only now can be fully appreciated.

If you measure teaching quality in terms of the product—the successful students—Moore has little competition for the title of the greatest ever math teacher. (Incidentally, he would hate to be described as a “math teacher”; he always insisted on using the word “mathematics” rather than the nowadays more prevalent abbreviation “math”.) During his long career as a professor of mathematics—64 years, the last 49 of them at the University of Texas—Moore supervised fifty successful doctoral students. Of those fifty Ph.D.’s, three went on to become presidents of the AMS (R. L. Wilder, G. T. Whyburn, R H Bing)—a position Moore also held—and three others vice-presidents (E. E. Moise, R. D. Anderson, M. E. Rudin), and five became presidents of the MAA (R. D. Anderson, E. Moise, G. S. Young, R H Bing, R. L. Wilder). Many more pursued highly successful careers in mathematics, achieving influential positions in the AMS and the MAA, producing successful Ph.D. students of their own—mathematical grandchildren of Moore—and helping shape the development of American mathematics as it rose to its present-day dominant position. That’s quite a record!

In 1931 Moore was a elected to membership of the National Academy of Science. Three of his students were also so honored: G. T. Whyburn in 1951, R. L. Wilder in 1963, and R H Bing in 1965.

In 1967, the American Mathematical Monthly published the results of a national survey giving the average number of publications of doctorates in mathematics who graduated between 1950 and 1959. The three highest figures were 6.3 publications per doctorate from Tulane University, 5.44 from Harvard, and 4.96 from the University of Chicago. During that same period, Moore’s students averaged 7.1. What makes this figure the more remarkable is that Moore had reached the official retirement age in 1952, close to the start of the period in question.

In 1973, his school, the University of Texas at Austin, named its new, two-winged, seventeen-story mathematics, physics, and astronomy building after him: Robert Lee Moore Hall. This honor came just over a year before his death. The Center for American History at the University of Texas has established an entire collection devoted to the writings of Moore and his students.

Discovery learning

Moore was one of the founders of modern point set topology (or general topology), and his research work alone puts him among the very best of American mathematicians. But his real fame lies in his achievements as a teacher, particularly of graduate mathematics. He developed a method of teaching that became widely known as “the Moore method”. Its present-day derivative is often referred to as “discovery learning”.

One of the first things that would have struck you if you had walked into one of Moore’s graduate classes was that there were no textbooks. On each student’s desk you would see the student’s own notebook and nothing else! To be accepted into Moore’s class, you had to commit not only not to buy a textbook, but also not so much as glance at any book, article, or note that might be relevant to the course. The only material you could consult were the notes you yourself made, either in class or when working on your own. And Moore meant alone! The students in Moore’s classes were forbidden from talking about anything in the course to one another—or to anybody else—outside of class. Moore’s idea was that the students should discover most of the material in the course themselves. The teacher’s job was to guide the student through the discovery process in a modern-day, mathematical version of the Socratic dialogue. (Of course, the students did learn from one another during the class sessions. But then Moore guided the entire process. Moreover, each student was expected to have attempted to prove all of the results that others might present.)

Moore’s method uses the axiomatic method as an instructional device. Moore would give the students the axioms a few at a time and let them deduce consequences. A typical Moore class might begin like this. Moore would ask one student to step up to the board to prove a result stated in the previous class or to give a counterexample to some earlier conjecture—and very occasionally to formulate a new axiom to meet a previously identified need. Moore would generally begin by asking the weakest student to make the first attempt—or at least the student who had hitherto contributed least to the class. The other students would be charged with pointing out any errors in the first student’s presentation.

Very often, the first student would be unable to provide a satisfactory answer—or even any answer at all, and so Moore might ask for volunteers or else call upon the next weakest, then the next, and so on. Sometimes, no one would be able to provide a satisfactory answer. If that were the case, Moore might provide a hint or a suggestion, but nothing that would form a constitutive part of the eventual answer. Then again, he might simply dismiss the group and tell them to go away and think some more about the problem.

Moore’s discovery method was not designed for—and probably will not work in—a mathematics course which should survey a broad area or cover a large body of facts. And it would obviously need modification in an area of mathematics where the student needs a substantial background knowledge in order to begin. But there are areas of mathematics where, in the hands of the right teacher—and possibly the right students—Moore’s procedure can work just fine. Moore’s own area of general topology is just such an area.

You can find elements of the Moore method being used in mathematics classes at many institutions today, particularly in graduate courses and in classes for upper-level undergraduate mathematics majors, but few instructors ever take the process to the lengths that Moore did, and when they try, they do not meet with the same degree of success.

Part of the secret to Moore’s success with his method lay in the close attention he paid to his students. Former Moore student William Mahavier addresses this point:

” Moore treated different students differently and his classes varied depending on the caliber of his students. . . . Moore helped his students a lot but did it in such a way that they did not feel that the help detracted from the satisfaction they received from having solved a problem. He was a master at saying the right thing to the right student at the right time. Most of us would not consider devoting the time that Moore did to his classes. This is probably why so many people claim to have tried the Moore method without success.”

Another well known mathematician who advocates—and has successfully used—(a modern version of) the Moore method is Paul Halmos. He says:

“Can one learn mathematics by reading it? I am inclined to say no. Reading has an edge over listening because reading is more active—but not much. Reading with pencil and paper on the side is very much better—it is a big step in the right direction. The very best way to read a book, however, with, to be sure, pencil and paper on the side, is to keep the pencil busy on the paper and throw the book away.”

Of course, as Halmos goes on to admit, such an extreme approach would be a recipe for disaster in today’s over-populated lecture halls. The educational environment in which we find ourselves these days would not allow another R. L. Moore to operate, even if such a person were to exist. Besides the much greater student numbers, the tenure and promotion system adopted in many (most?) present-day colleges and universities encourages a gentle and entertaining presentation of mush, and often punishes harshly (by expulsion from the profession) the individual who seeks to challenge and provoke — an observation that is made problematic by its potential for invocation as a defense for genuinely poor teaching. But as numerous mathematics instructors have demonstrated, when adapted to today’s classroom, the Moore method—discovery learning—has a lot to offer.

If you want to learn more about R. L. Moore and his teaching method, check out the web site www.discovery.utexas.edu.

But before you give the method a try, take heed of the advice from those who have used it: Plan well in advance and be prepared to really get to know your students. Halmos puts it this way: “If you are a teacher and a possible convert to the Moore method … don’t think that you’ll do less work that way. It takes me a couple of months of hard work to prepare for a Moore course. … I have to chop the material into bite-sized pieces, I have to arrange it so that it becomes accessible, and I must visualize the course as a whole—what can I hope that they will have learned when it’s over? As the course goes along, I must keep preparing for each meeting: to stay on top of what goes on in class. I myself must be able to prove everything. In class I must stay on my toes every second. … I am convinced that the Moore method is the best way to teach there is—but if you try it, don’t be surprised if it takes a lot out of you.”

References

Paul R. Halmos (1975), The Problem of Learning to Teach: The Teaching of Problem Solving, American Mathematical Monthly, Volume 82, pp.466-470.

Paul R. Halmos (1985), I Want to Be a Mathematician, Chapter 12 (How to Teach), New York: Springer-Verlag, pp.253-265.

William S. Mahavier (1998), What is the Moore Method?, The Legacy of R. L. Moore Project, Austin, Texas: The University of Texas archives.

Devlin’s Angle is updated at the beginning of each month.

JULY-AUGUST 1999

Supermarket Math

Many people, perhaps most people, claim to be unable to do math. What they generally mean by “math” in this context is basic arithmetic. They usually base their self-assessment on their poor performance in school arithmetic tests. But does a low score on a school arithmetic test necessarily indicate a poor head for figures? Not according to several researchers who have looked into the matter.

One such scholar is Jean Lave, an anthropologist in the Department of Education at the University of California at Berkeley. Between 1973 and 1978, when she was on the faculty at the University of California at Irvine, Lave spent time among a group of apprentice tailors in Liberia, studying the way the young tailors learned and used the arithmetic they required for their work. Her goal was to compare the way people learned in school—where the learning takes place out of any context of use and without any immediate use for what is learned—with the way they learned particular skills on the job, when they really needed what they learned. (In her own terminology, she wanted to compare “formal” learning with “informal” learning.) She chose arithmetic not because she had any particular interest in mathematics, but because it was easy to test people’s ability and compare the results.

One particular hypothesis Lave wanted to test was that arithmetical skills learned on the job will be learned much better than skills learned at school. But when she presented the young apprentices with (practical) problems designed to test their arithmetical skills, she found there was little difference between the two. The tailors had as much difficulty solving the problems designed to test the arithmetic they learned and used at work as they did with the problems designed to test what they had learned at school. No matter whether the students learned a particular arithmetic skill at school or at work, they seemed unable to apply that skill in a different, real-life context. Lave published her findings in her 1988 book Cognition in Practice: Mind, Mathematics and Culture in Everyday Life. I re-read this book recently, and this column is based on what she said there.

Can we use what we learn at school?

Behind Lave’s work lay a fundamental question: Is it really possible to acquire skills—mathematical or otherwise—in a de-contextualized fashion in a classroom setting, then make use of those skills in any real-life situation that may require them? The assumption that this is indeed possible is so fundamental to our system of schooling, and so firmly ingrained in our way of thinking about learning, that is it easy for us to not realize that it is (just) an assumption, not an established fact.

Of course, there are plenty of circumstances where people do apply in their work things they learned in a classroom. For example, scientists and doctors make regular use of knowledge they acquire as students in the classroom. But in these cases, either the job has a recognizable “scholastic” nature, or the training is specifically geared to preparation for work, or both. Lave’s question was getting at something more general: Is it possible to acquire general skills out of context in a classroom, and then use them in any real-life situation where they are in principal applicable. Arithmetic, as traditionally taught in schools, is a prime example of a collection of general skills that are taught in classrooms all over the world, and which should, in principal, be applicable in a wide variety of real-life situations.

Lave’s initial study led her to the tentative conclusion that when people acquired skills out of context in the classroom, they could not, in general, apply those skills in real-life situations for which they were theoretically appropriate. But that did not prevent them for developing their own, often highly effective, ways of doing things. One example that Lave quotes (Lave 1988, p.65) comes from a series of verbal purchase transactions taped by researchers in a Brazil market. The subject is a twelve-year old boy manning a stall selling coconuts.

Customer: How much is one coconut?

Boy: 35.

Customer: I’d like ten. How much is that?

Boy: (Pause) Three will be 105; with three more, that will be 210. (Pause) I need four more. That is . . . (pause) 315 . . . I think it is 350.

As a mathematician, my initial reaction on reading the above passage was, “How silly! The quickest way to arrive at the answer is to use the rule that to multiply by 10 you simply add a zero—so 35 becomes 350.” Indeed, that is by far the quickest method, provided you know the rule. If I did not know any better, I might have gone on to conclude: “That young boy obviously doesn’t know his multiplication table, not even the easiest case!” But having read Lave (and other authors who make essentially the same point), instead I asked myself, how does the situation appear if I put myself in the position of the young market trader? Presumably, he often finds himself selling coconuts in groups of two or three. So he needs to be able to compute the cost of two or three coconuts; that is, he needs to know the values 2 x 35 = 70 and 3 x 35 = 105. Faced with the highly unusual request for ten coconuts — the shopper was in fact a researcher who set out to see how the young traders could cope with transactions involving such arithmetical problems — the young boy actually adopts an approach that, in its own way, is mathematically very sophisticated. First, he splits the 10 into groups he can handle, namely 3 + 3 + 3 + 1. Arithmetically, he is now faced with the determining the sum 105 + 105 + 105 + 35. He does this is stages. With a little effort, he first calculates 105 + 105 = 210. Then he performs 210 + 105 = 315. Finally, he works out 315 + 35 = 350. Altogether quite an impressive performance for a twelve-year old of supposedly poor education.

Losing weight and finding the best buy

Her curiosity aroused by her study of the Liberian tailor apprentices, Lave began an in-depth study in the United States: The Adult Math Project. The AMP was designed to see to what extent ordinary adult Americans used basic arithmetical skills in their everyday lives, and what methods they used to carry out their calculations. The two main settings Lave chose for her investigations were the supermarket and the kitchen: How did shoppers identify the best-buy among a range of competing products, and how did dieting cooks calculate the amounts of ingredients required to prepare a particular meal?

To give you a taste of the kinds of things Lave observed in the AMP, a male dieter preparing a meal was faced with having to measure out 3/4 of the 2/3 of a cup of cottage cheese stipulated in the recipe he was using. Before reading on, how would you go about this?

Here is what the subject did. (Lave 1988, p.165.) He measured out 2/3 of a cup of cheese using his measuring device, and spread it out on a chopping board in the shape of a circle. He then divided the circle into four quarters, removed one quarter and returned it to the container, leaving on the board the desired 3/4 of 2/3 a cup. Perfectly correct.

What is my reaction as a mathematician? That there is a much “easier” way: When you multiply the fraction 3/4 by 2/3 the 3 cancels and you end up with 2/4, which simplifies to 1/2; so all the subject needed was 1/2 a cup of cheese, which he could have measured out directly. Simple. But our man did not see this solution. Nevertheless he clearly knew what the concept “three-quarters” means, and was able to use that knowledge to solve the problem in his own way.

The AMP studied thirty-five subjects in Orange County, in Southern California. The subjects varied considerably in terms of education and family income, and included some people of poor educational background and low income, for whom buying groceries economically was very important. Twenty-five subjects were in the supermarket study, ten in the dieting cooks study.

Since the idea was to examine the way ordinary people actually used mathematics in their everyday lives, the researchers could not simply test them with questions such as “If you were faced with three kinds of frozen french-fries with the following weights and prices, how would you decide which is the most economical?” As Lave and her colleagues showed, the answer that people give to such a question has very little to do with what they would actually do in a real shopping situation. To put it bluntly, “What if?” questions don’t work.

Instead, the researchers chose to follow the subjects around and observe them, taking copious notes, occasionally asking the subjects to explain their reasoning out loud as they went about their shopping or food preparation, and sometimes asking for explanations just after the transaction had been completed. Of course, this procedure is highly contrived. The very presence of the observers changes the experience of shopping or of preparing a meal, as does the request that the subjects describe what they are doing. Thus, to some extent the study is not really one of people “in their normal, everyday activities.” But it’s probably as close as you can get. Moreover, anthropologists have developed ways of going about such work so as to minimize any influences their presence has on their subjects’ behavior.

Each of the researchers spent a total of about forty hours with each of her or his subjects, including time spent interviewing subjects to determine their backgrounds (education, occupation, etc.). Though most of the subjects were women, there were some men in the group. However, the researchers noticed no difference in the mathematical performances of the men and the women in either the supermarket study or the dieting cooks study, so gender did not seem to be a significant factor.

Out of a total of around 800 individual purchases that the subjects made in the course of the study, just over 200 involved some arithmetic—which the researchers defined to be “an occasion on which a shopper associated two or more numbers with one or more arithmetical operations: addition, subtraction, multiplication, or division.” (Lave 1988, p.53.) The shoppers varied enormously in the frequency with which they used mathematics. One shopper used none whatsoever, while three of the subjects performed calculations in making over half their purchases. On average, 16% of purchases involved arithmetic.

One interesting observation that came out of the study was that the attitudes the subjects reported having had towards mathematics when they had been at school had no effect on their arithmetical performance in the supermarket or the kitchen.

Another fascinating result is that, in comparing competing products to decide which was the best buy, shoppers made relatively little use of the unit price printed on the label—an item of information included on the label by law specifically to enable shoppers to compare prices. The most likely explanation is that the unit price is essentially abstract, arithmetical information. Unless the product is something that the shopper either buys or uses in, say, single ounce units, then the price per ounce has no concrete significance for that shopper. Thus, even though direct comparison of unit prices is the most direct way to determine value-for-money, shoppers often ignore it. And, needless to say, they are even less likely to calculate it.

A common approach was to calculate ratios between prices and quantities in a way that made direct comparison possible. This could be done if the quantities were in a simple ratio to one another, say 2:1 or 3:1. For example, if product A cost $5 for 5 oz and product B cost $9 for 10 oz, the comparison was easy. A typical shopper would reason like this: “Product A would cost $10 for 10 oz, and product B is $9 for 10 oz, hence product B is the cheaper buy.”

Notice that the kind of argument just described does not involve the explicit calculation of unit prices, although mathematically it is entirely equivalent. One difference is that the unit price provides a single figure associated with each item, thereby allowing the comparison of any two similar items in the store. But shoppers generally only resorted to arithmetic when making a comparison between two (or sometimes three) items, and then they made whatever transformations would bring the price information for those two items into a form that made comparison possible.

Another advantage of working with the actual amounts that might be bought is that the price comparison is often just one part of a more complex decision making process, in which the shopper’s storage capacity, size of family, likely rate of usage, and the estimated storage period before a particular item might spoil all play a part. As the AMP researchers observed time and again, what shoppers did was to juggle all of these variables in order to reach a decision, thinking about the purchase options first one way, then another, then another. The price-comparison arithmetic was certainly a part of this process, but it was by no means the only part. Despite the complexity of this process, shoppers did not have to expend any great effort. Indeed, they were not aware that they were “thinking” much at all; they were “just shopping.”

Another method that many shoppers used to decide between two options was to calculate the price differential, a procedure that requires just two subtractions. For example, faced with a choice between a 5 oz packet costing $3.29 and a 6 oz packet priced at $3.59, the shopper would argue, “If I take the larger packet, it will cost me 30 cents for an extra ounce. Is it worth it?”

Among the arithmetical techniques that the researchers observed shoppers performing were estimation, rounding (say to the nearest dollar or the nearest dollar and a half), and left-to-right calculation (as opposed to the right-to-left calculation taught in school). What seemed to be absent, however, were most of the techniques the shoppers had been taught in school. Lave and her colleagues set out to investigate where the school math had gone.

Putting shoppers to the test

In order to compare the shoppers’ arithmetical performance in the supermarket with their ability to do “school math,” the researchers designed a test to determine the latter. Again, the results were fascinating. Despite the significant efforts the researchers made to persuade the subjects that this was not like a school test, rather that its purpose was purely to ascertain what arithmetical ability they had retained since school, with nothing at stake, the subjects approached it as if it were indeed a school test. For instance, when the researchers asked if they may observe the subjects taking the test, the subjects responded with remarks such as “Okay, teacher.” They made comments about not cheating. They asked if they were allowed to rewrite problems. And they spoke self-depreciatingly about not having studied math for a long time. In other words, the subjects approached the math test in “math test mode,” with all the stresses and emotions that entailed.

In fairness to the subjects, it has to be said that, despite the surrounding circumstances, the “math test” did have all the elements of a typical school arithmetic test: questions involving whole numbers, both positive and negative, fractions, decimals, addition, subtraction, multiplication, and division. All the problems were, however, designed to test the kind of arithmetical skill that the researchers had observed the subjects using (in context) in the supermarket. For instance, having observed that shoppers frequently compared prices of competing products by comparing price-to-quantity ratios, the researchers included some problems to see how the subjects fared with abstract versions of such problems. For example, faced with an item costing $4 for a 3 oz packet and a larger packet costing $7 for 6 oz, many shoppers would—in effect—compare the ratios 4/3 and 7/6 to see which was the larger. So the researchers would include on the test the question: “Circle the larger of 4/3 and 7/6.” One obvious difference between such a question in the formal test and its equivalent in real life was that the subjects took the test question as requiring a precise calculation, whereas they were much more likely to use estimation in real life, though often with considerable accuracy.

Perhaps the greatest surprise in the study was the huge disparity between how the subjects performed in the real life situations and their results on the test. Although the arithmetic test was designed to test the very mathematics that the subjects used in the supermarket, assuming they performed the calculations using the method they learned in school, the shoppers performance was rated at an average 98% in the supermarket as against a mere 59% average on the test. In other words, the shoppers in the supermarket were probably not using the arithmetical skills they learned in school. Rather, they were solving the problem another way.

This last conclusion is supported by the fact that performance on the test was higher the longer subjects had studied math at school and the more recently they had finished school, whereas neither length of schooling nor the time since schooling had any measurable effect on how well they did in the supermarket. Thus, school math classes seem to teach people how to perform on school math tests, but not how to solve real-life problems that involve math.

Since the subjects were highly successful when it came to performing arithmetical tasks in real-life situations—regardless of their schooling history—one obvious question is, how were they doing it?

Of course, part of the difference in performance might have been due to the difference between actually being in the store as against “taking a test.” As we observed a moment ago, the subjects could not help viewing the arithmetic test as a “school quiz,” with all the psychological stress that might entail. But that did not seem to be the major factor. Rather, what seemed to make the biggest difference was the kind of test the subjects were asked to take and the manner in which the questions were presented. This was shown by a further test the researchers put the subjects through: a shopping simulation, where, in their homes, the subjects were presented with simulated best-buy shopping problems, based on the very best-buy problems the researchers had observed the subjects resolve in the supermarket. In some of these simulations, the subjects were presented with actual cans, bottles, jars, and packets of various items taken from the supermarket and asked to decide which to buy among competing brands; in others they were presented with the price and quantity information printed on cards. In this simulation, which was evidently a kind of “test” situation, but with the questions clearly of a shopping nature as opposed to being school-like “math questions,” the subjects scored an average of 93%. (The fact that the simulation was done in the subject’s home, carried out by the researcher who had accompanied the subject on the shopping trip, also seems to have been a significant factor.)

To put this in terms of a specific example, a subject would perform extremely well (in the 93% success rate category) in the home shopping simulation, when presented with a card that said 3 oz of product A cost $4 and another card that said 6 oz of product B cost $7 and asked which was the best buy; but in the context of being presented with a list of arithmetic problems, the same subject would do far worse (in the 59% success rate category) when asked to circle the larger of 4/3 and 7/6. And yet, the arithmetic problem underlying the two questions is exactly the same!

The conclusion seems to be not that people can’t do math; rather they can’t do school math. When faced with a real-life task that requires elementary arithmetic, most people do just fine—indeed, 98% success is virtually error free. And yet, school math—at least the more elementary parts—is supposed to provide us with just the arithmetic skills we need in our everyday lives. Indeed, the methods taught in school are supposed to be the simplest and the best—that’s why they are taught!

Incidentally, though a number of the AMP shoppers did carry a calculator with them, only on one occasion during the entire project did one shopper take it out and use it in order to carry out a price comparison. And no one ever used a pencil and paper to carry out a calculation. When the calculation became too hard to do in their heads, they simply resorted to other criteria on which to base their decision.

Does Lave’s research have implications for the way we should teach basic arithmetic in our schools? You bet it does. But that’s another story, one that deserves—and requires—far more space than I have left in this issue of Devlin’s Angle. I’ll come back to it at a later date.

Reference

Lave J (1988) Cognition in Practice: Mind, Mathematics and Culture in Everyday Life, Cambridge, UK: Cambridge University Press.

SEPTEMBER 1999

What Can Mathematics Do For the Businessperson?

Question: How much mathematics do you need to be a successful business CEO?

Answer: None.

That’s not a conclusion or a hypothesis. It’s an observation. Many—perhaps most—of the CEOs of top companies have virtually no mathematical knowledge. If their company needs to have someone with mathematical expertise, such a person will be hired.

What can be extremely valuable in business, however, is the ability to dig beneath the surface of a problem to see what the real underlying issues are, and the capacity for clear and logical thought. And learning mathematics provides an excellent way to develop that capacity.

Just as the 1997 DOE Riley report showed that completion of a formal mathematics course in high school results in better performance at college whatever major the student chooses to pursue, so too completion of one or two college mathematics courses seems likely to confer benefits to practically anyone, regardless of their particular career. For the individual who goes into business, doubtless the main benefit of doing some college mathematics is the capacity for clear, precise thinking that is the bedrock of mathematics.

My own research having brought me into regular, close contact with a number of people in business over the past fifteen years, I have observed that nowhere do you find greater fuzzy thinking than when it comes to dealing with information.

To take just one example, the words data, information, and knowledge are often used interchangeably, and opinions differ as to whether knowledge management is about managing technology, managing people, or both.

Does this matter? You bet it can. How many times have you heard about a company that introduced a new computer system to improve its information management, only to discover that, far from making things better and more efficient, the new system led to scores of new problems that had never arisen with the old way of doing things. The new system is capable of providing vastly more information than was ever available before, but it is hard if not impossible to extract that information, or it’s the wrong kind, or presented in the wrong form, at the wrong time, or delivered to the wrong person. Or there is simply too much of it for anyone to be able to use. What used to be a simple request over the phone becomes a lengthy battle with a seemingly uncooperative computer system, taking hours or even days, and eventually drawing in a whole host of people.

Why does this happen? The answer is that, despite all that we hear about living in the Information Age, what we are really living in is an age of information technology, or more precisely a collection of information technologies. We do not yet have an established science of information. As a result, we do not yet have the ability to properly design or manage the information flow that our technologies make possible.

The engineers who produce our buildings, bridges, automobiles, aircraft, household appliances, and communications technologies base their work on the solid foundation of decades and even hundreds of years of scientific progress in physics and other sciences. But the people who design and manage our information systems (which may comprise people or machines or a combination of the two) have to work with much shakier foundations.

Since 1987, I’ve been attached to Stanford University’s Center for the Study of Language and Information, involved in a research project that sets out to analyze information as a theoretical concept. Though we are nowhere near having a “science of information” akin to, say, physics—indeed, it’s not clear there can be such a science—the research work has led to some clarification of the basic issues involved.

For instance, here is a useful way of thinking about the difference between data, information, and knowledge:

Data is what newspapers, reports, and “computer information systems” provide us with. For example, a list of stock prices on the financial page of a newspaper is data.

When people acquire data, and fit it into an overall framework of previously acquired information, that data becomes information. Thus, when I read the list of stock prices in the newspaper, I obtain information about various companies. What allows me to acquire information from the data in the newspaper is my prior knowledge of what such figures mean and how the stock market operates.

When a person internalizes information to the degree that he or she can make use of it, we call it knowledge. For example, if I know how to buy and sell stocks and am familiar with some of the companies whose stock values are listed in the newspaper, the information I obtain by reading the figures can provide me with knowledge on which to trade stocks.

This simple analysis is not rocket science. Hey, it’s not even science. Nor is it mathematics. What it is an application of clear thinking designed to establish exactly what a number of key terms mean. Mathematicians and scientists do this kind of thing as a matter of routine.

But, I’ll tell you something: Many other people don’t think that way. Even some very smart people. Indeed, many people find this way of thinking very alien.

After witnessing many times the enormous confusion that reigns in the world of information/knowledge management, and examined some of the problems that result from the installation of a new “information system”, about a year ago I was motivated to set down on paper what I thought were some basic, common sense ideas about information. Those common sense ideas were based on some of the work that had been carried out at CSLI. But I deliberately cut out any of the deep philosophical issues or the mathematical modeling, and presented the basic ideas using simple, everyday examples, taken from the real business world.

The result—the book InfoSense—was just published by W. H. Freeman. If you are a regular MAA Online reader, you’ll probably hate it—at least, you will if you are expecting a mathematics book. If so, then don’t think of buying it. It wasn’t written as a mathematics book, nor was it written for people used to thinking mathematically.

Nevertheless, you might find it informative to glance at the book. Why? Well, when you do look through the pages (it’s quite short), remember this one thing. Though many people have told me they like it, and find it very helpful, I have also been informed that it’s “too technical” and “very heavy going in places”. I am sure that no regular reader of MAA Online would make such a remark. Indeed, I doubt that anyone who had completed a high school geometry course could react that way.

And that’s my point.

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.

OCTOBER 1999

Those Amazing Flying Mathematicians

October is the month when those of us living in the United States see migrating birds moving south for the winter. How do they know which direction to fly? There are several possibilities. Most of them seem to require mathematical computations that most humans would find challenging. Assuming that the average bird is not the mathematical equivalent of a college math major, how do the birds do it?

To put the question another way, why is it that a pilot of a Boeing 747 needs a small battery of maps, computers, radar, radio beacons, and navigation signals from GPS satellites—all heavily dependent on masses of sophisticated mathematics—to do what a small bird can do with seeming ease, namely, fly from point A to point B.

In fact, scientists still have a long way to go before they understand completely how birds navigate. But some parts of the picture have started to fall into place.

The evidence seems to suggest that birds use a combination of different methods. Let’s look at the various possibilities in turn.

1. Visual Clues: Many animals learn to recognize their surroundings to determine their route. They remember the shape of mountain ridges, coastlines, or other topographic features on their route, where the rivers and streams lie, and any prominent objects that point to their destination. For example, a digger wasp always memorizes the land marks around its burrow. Birds may use this method to locate their nest, but it seems unlikely that it will support flights over long distances. And it clearly cannot be used for navigating over large bodies of water or for flying at night, both of which many species of birds do every year.

Of, course, navigating by recognizing the terrain does not seem to require much by way of mathematics. The same cannot be said of any of the other navigational methods we’ll look at next.

2. Solar Navigation: Many birds—and other creatures such as the honey bee—use the sun to navigate. This requires knowing where the sun is located in the sky at each time of the day at the time of the migration. For a human navigator, plotting course from the position of the sun in the sky requires mastery of trigonometry. Can birds and honey bees do trigonometry?

3. Magnetic Fields: One of the many methods used by homing pigeons to find their way home is to follow the magnetic field lines of the earth. The birds have a magnetic compass in their heads. This has been demonstrated by attaching small magnets to the heads of homing pigeons. The magnets deflect the Earth’s magnetic field around the birds, and cause them to fly off course in the same degree of deflection. Again, when humans navigate by means of a compass, they use trigonometry. Is this how the birds do it?

4. Star Navigation: At least one species of bird—Indigo Bunting—use the stars to navigate at night. It appears that they learn to recognize the pattern of stars in the night sky when they are still in the nest. For instance, a few years ago, a study found that nestling Indigo Buntings in the northern hemisphere watch as the stars in the night sky wheel around Polari—the north star, located above Earth’s north pole. Polaris lies due north for those in the northern hemisphere. Being able to identify Polaris in the night sky could help birds find their way north.

To test this hypothesis, the researchers showed the birds a natural sky pattern inside a planetarium. They seemed to fly in a direction consistent with being able to detect the motion of the stars. They knew in their own way which direction was north.

When the experimenters changed the set up so that Betelgeuse was now the pole star which the stars rotated around, the birds flew in a direction consistent with Betelgeuse being the pole star. They no longer went where they should have relative to Polaris. So, they weren’t using the locations of specific star patterns. It was just that they were noticing which star the others rotated around. In other words, it wasn’t the star patterns, but how they moved that counted.

To further substantiate the claim, it has been recognized that some birds become disoriented on cloudy nights, when they can’t see the stars. (It should be noted, however, that despite what you might read in books and articles, the Indigo Bunting is the only species of bird which has been demonstrated to use celestial navigation.)

Navigation by the stars is, of course, one of the ways human mariners of times past found their way around the globe. As with solar navigation and magnetic field navigation, the human version involves trigonometric calculations. How do birds solve the equivalent problems?

5. Polarized light. One additional navigational possibility is that birds discern polarization patterns in sunlight. As the sun’s rays pass though our atmosphere, tiny molecules of air allow light waves traveling in certain directions to pass through, but they absorb others. The resulting polarized light forms an image like a large bow-tie—located overhead at sunset—pointing north and south. You can see the bow-tie created by polarized sunlight if you go out at sunset and look upwards. You should see the bow-tie straight above you, pointing north and south.

It has been suggested that some birds can detect the polarization, and use it like a large compass in the sky. Most likely, birds and bees don’t see the bow-tie effect that humans do. Rather, they probably see the actual gradations in polarization between the sun’s nearly unpolarized light to the almost 100% polarized light 90 degrees away from the sun.

Honeybees appear to use the same technique to find their way, even on cloudy days, when the sun can’t be seen. All they need is a small patch of blue sky to see the sun’s rays through, and the polarization effect shows them the way.

However, polarized light is almost certainly not the only component of avian navigation. It merely allows a bird to calibrate its compass, at is were. Determining the direction in which to fly requires something else — something that in the human case requires mathematics.

The mathematical ant

Of course, it’s not just migrating birds that have to navigate. Traveling around on the ground can often present a significant challenge. Many creatures—for example dogs and several kinds of ants—navigate by chemistry, finding their way to their destination by following scents and chemical trails laid down by themselves or by other members of the species. And then there are the salmon, the whales, and other sea creatures who regularly navigate their way over the oceans, using what seems to be a combination of the sun and the stars and the earth’s magnetic field.

A particularly intriguing example of overland navigation is provided by the Tunisian desert ant. This tiny creature can wander across the desert sands for a distance of up to fifty meters until it stumbles across the remains of a dead insect, whereupon it bites off a piece and takes it directly back to its nest—a hole no more than one millimeter in diameter.

How does it find its way back? By the ant-equivalent of the same process the Apollo astronauts used to plot their course to the Moon: dead reckoning. A linguistic derivation from “deductive reckoning” (strictly, it’s “ded. reckoning”), the idea of dead reckoning is to calculate your position relative to your starting point from a knowledge of your speed and your direction of travel.

The evidence that this particular creature navigates by dead reckoning comes from some painstaking research carried out by R. Wehner and M. V. Srinivasan in 1980-81. They discovered that if a Tunisian desert ant is moved after it finds food, it will head off in exactly the direction it should have to find its nest if it had not been moved, and will stop and start a bewildered search for its nest when it has covered the precise distance that should have brought it back home. In other words, it remembers the precise direction in which it should head to return home, and how far in that direction it should travel.

For the mathematician, the puzzling aspect of the behavior of the Tunisian desert ant is, how on earth does it figure out its path? For humans, dead reckoning requires arithmetic, trigonometry, a good sense of speed and time, and a good memory (or else good record keeping). When human mariners and lunar astronauts navigated by dead reckoning they used charts, tables, various measuring instruments, and a considerable amount of mathematics. The tiny desert ant has none of these.

Does animal math add up?

People who have struggled with, and failed to master, high school mathematics frequently marvel at how a lowly creature such as a bird or an ant can perform a mathematical feat that used to defeat them in the classroom.

But, of course, animals don’t do their thing “using mathematics” the way we do. Rather, natural selection, acting over millions of years, has equipped them with a range of physical and mental abilities that enable them to survive in their own evolutionary niche. A bird that navigates by the sun or the stars “solves” a problem in trigonometry only in the same way that a river flowing down hill “solves” a differential equation of fluid dynamics or the Solar System, by its very planetary motions, “solves” a particular instance of a many-body problem in gravitational dynamics (both feats that remain well beyond the capabilities of present-day mathematicians, incidentally). That is to say, it is only when the bird’s activity is interpreted in human terms that the creature can be said to “solve a mathematics problem.” The bird itself simply does what comes natural to it.

As a result of our own evolutionary path, we human beings (at least, those of us who do not live in Kansas) have found ways to be able to extend our own range of instinctive, unconscious behaviors so that we can mimic some of the activities of our fellow creatures. Using mathematics, science, and technology, we too can navigate our way around the globe (and more recently to other planets). But it is important to remember that there is a huge difference between a physical system (say, a river or a bird’s brain) performing an action by virtue of its structure and the description, simulation, or mimicking of that activity using mathematics.

In fact, the mathematics to describe even seemingly simple, everyday activities of humans, animals, and physical and biological systems can be extremely complex. This is in large part why the original goals of artificial intelligence and robotics remain a long way from being achieved.

To my mind, there are two ways to look at this situation, both of which fill me with awe. First, starting with the mathematics, and knowing the complexity of the mathematics required to describe an activity such as bird navigation, I am filled with awe at the power of natural selection and its ability to give rise to the rich variety of successful lifeforms with which we share this planet, some of which can perform feats that we humans can do only with considerable effort and ingenuity, if indeed we can do them at all.

Second, when I take the evolution of life on earth as my starting point, I am filled with awe that, within the last four hundred years or so, we humans have been able to develop theories, and from them technologies, that have enabled us to perform, in our own way, some of the activities that natural selection has—over millions of years—equipped other species to perform.

To many people, mathematics is merely a collection of techniques for calculating. But when you look around at many of the things we do with mathematics, you realize that it is a powerful mental framework that enables human beings to extend their capabilities well beyond those for which our evolution directly equipped us. According to the old joke, “If God had meant us to fly, he’d have given us wings.” A more accurate version would be, “Obviously God wanted us to fly; that’s why He gave us a brain that was capable of developing mathematics, which we could use to invent and build airplanes and develop methods and technologies to navigate when we are in the air.”

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.

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.

NOTE: 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.

DECEMBER 1999

About time

Note: This special Millennium edition of Devlin’s Angle is much longer than normal. You might want to print it out to read. That will also give you a version absolutely impervious to any Y2K bug.

The dawn of a new year provides us with a reminder that we live much of our lives by the clock and the calendar. The message is even stronger when the new year is the last of a millennium, carrying the number 2,000. But what exactly is time? There are three answers: one in physics and philosophy (time as a physical phenomenon), another in psychology (our sense of passing time), the third in mathematics and engineering (the time that we measure and use to regulate our lives). Devlin’s Angle will, of course, concentrate on the last of these three notions. How did we come to measure time in the first place? What exactly is it that our timepieces measure? (This is where mathematics comes in.) And what scientific principles do we use to construct ever more accurate clocks? (More mathematics here.)

Time past

The measurement of time began with the invention of sundials in ancient Egypt some time prior to 1500 B.C. However, the time the Egyptians measured was not the same as the time today’s clocks measure. For the Egyptians, and indeed for a further three millennia, the basic unit of time was the period of daylight. The Egyptians broke the period from sunrise to sunset into twelve equal parts, giving us the forerunner of today’s hours. As a result, the Egyptian hour was not a constant length of time, as is the case today; rather, as one-twelfth of the daylight period, it varied with length of the day, and hence with the seasons. It also varied from place to place on the surface of the Earth. And of course, time as a measurable concept effectively ceased during the hours of darkness.

The need for a way to measure time independently of the sun eventually gave rise to various devices, most notably sandglasses, waterclocks, and candles. The first two of these utilized the flow of some substance to measure time, the latter the steady fall in the height of the candle. All three provided a metaphor for time as something that flows continuously, and thus began to shape the way we think of time.

Though their accuracy was never great, these devices not only provided a way to measure time without the need for the sun to be visible in the sky, they also provided the basis for a concept of time that did not depend upon the length of the day. But it was to be many centuries before advantage was taken of that possibility. Instead, each of these time-measuring devices carried elaborate systems of markings designed to give the time based on the sundial. Fragments of one thirteenth century waterclock found in France gave instructions on how to set the clock for every single day of the year! Because the hours of darkness are the antithesis of the daylight hours, the scale for the nighttime hours was simply the daytime scale for the day exactly half a year earlier. For example, the scale for the nighttime on July 1 was the daytime scale for January 1.

In addition to their lack of accuracy, sandglasses, waterclocks and candles were also limited in the total length of time they could measure before having to be reset. As a result, they were largely used for measuring the duration of some activity, such as a speech made by an orator, cooking time, or the length of a legal consultation.

For most of history, ordinary people did not have regular and easy access to any kind of time measuring device whatsoever, other than to glance at the sky on a sunny day and see where the sun was. For them, time as we understand it today did not really exist. The one group in medieval times whose day was ruled by time in a way not unlike people today were the Benedictine monks, with their ecclesiastically regulated prayer times, the eight Canonical Hours: lauds (just before daybreak), prime (just after daybreak), terce (third hour), sext (sixth hour), nones (ninth hour), vespers (eleventh hour), compline (after sunset), and matins (during the night). The signal that announced each canonical hour and regulated the monks’ day was a ringing bell. This gives us our word “clock,” which comes from the medieval Latin word for bell, clocca.

Regardless of whether they were regulated by a sundial, a waterclock, a candle, or the stars, the bells that were used to signal each new canonical hour were rung according to a schedule based ultimately on the period of sunlight at that location and at that time of year. Because they were not spaced equally apart, the canonical hours provided a concept of time that, in addition to changing throughout the year and from location to location, did not flow evenly as modern time does.

Buying time

During the Middle Ages, the idea of a regulated time started to spread out from the monasteries along with the associated religious observances. At the end of the fourteenth century, the best selling book in Europe was the Book of Hours, a collection of devotional readings that a well-to-do lay person in his home could read or recite at the appropriate canonical hour.

Today, most people keep themselves and their families alive by selling their time—explicitly in the case of workers, professionals, or consultants “paid by the hour,” less explicitly but no less real for salaried employees. Moreover, present day economies are largely sustained by the lending and borrowing of money, for which the lender charges interest—a charge for the time the borrower has use of the money. The situation was different in the Middle Ages. Though allowed by the Romans, usury—the charging of interest on a loan—was banned from early Christian society until well into the twelfth century, the argument being that time belongs to God and therefore cannot be bought or sold. (Officially, usury is still banned by Islam.)

The growing dependence on international trade from the thirteenth century onward required the support of a money market, and as a result usury gradually crept into the Christian societies of Europe. With the growing acceptance of time as a commodity that could be bought and sold, humanity started along the path of developing a sense of time as something separate from the familiar cycle of night and day and the changing of the seasons. As time became rationalized, it also grew more secular, part of the daily activities of commerce, industry, and daily life.

The invention of time machines

It was into a world of “natural time,” based on the sun’s march across the sky, and varying with the seasons, that the first mechanical timepieces—time machine—were introduced in thirteenth century Europe. At odds with the conception of time as something that flows, with the first clocks came the idea of measuring time by splitting it into equal, discrete chunks and counting those chunks.

Most of us think of the time produced by our clocks as time itself. Yet the only thing natural about the time produced by clocks is that it is originally based on a complete revolution of the earth (or more precisely, the average of such revolutions). The division of that period into 24 equal hour—generally treated as two successive periods of 12 hours each (AM and PM), the division of each hour into 60 minutes, and the further division of each minute into seconds are all conventions—human inventions.

In fact, there’s a fundamental circularity in the way we measure time. The time that is measured by a clock is itself produced by that clock. The clock’s time is independent of the flow of the seasons or the cycle of day and night, and is independent of the clock’s location on earth. Today we don’t give this matter any thought—time is what the clock tells us. But in the early days of clocks that was not the case. Indeed, so different was the time determined by the clock that the practice developed of indicating when a time given was produced by the clock by adding the phrase “of the clock”—later abbreviated to the “o’clock” we use today.

With the invention of the clock, the basic unit of time ceased to be the day and was replaced by the hour. With clocks, people could correlate their activities to a far greater degree than ever before. And the ability to measure time in a mathematical fashion helped prepare the way for the scientific revolution that was to follow three hundred years later.

All clocks depend on the laws of physics, which provide potentially reliable timepieces in the form of oscillators. Any object that oscillates will have a preferred period of oscillation, and by finding a way to capitalize on that regular period, a reliable clock may be constructed.

Early oscillating mechanisms were called escapements. The first escapement, the “verge-and-foliot,” comprised a freely swinging horizontal bar (the foliot) attached to a centrally located vertical shaft (the verge). The mechanism was driven by gravity. A heavy weight hung from a cord wrapped round a horizontal spindle. As the weight slowly descended, the cord turned the spindle. A toothed crown-wheel on the spindle made the escapement oscillate, the escapement regulated the rate at which the spindle turned, and the rotation of the spindle measured the passage of time by moving a hand around a marked clock face. (The rate of oscillation, and hence the speed of the clock, was adjusted by moving symmetrically-placed small weights along the foliot bar.)

Some time in the fifteenth century, clockmakers started to use tightly coiled blades of metal—springs—to power their timepieces, instead of gravity. Following Galileo’s famous 1583 observation that the period of oscillation of a swinging pendulum seemed to depend only on the dimensions of the pendulum, not on the size of the arc, the verge-and-foliot escapement was modified—and improved—so that the swing of a pendulum arm regulated the motion. The pendulum clock was itself improved when the verge-and-foliot mechanism for controlling the rate of rotation of the crown wheel was replaced by the anchor escapement, where a caliper-like “anchor” performed the task previously carried out by the verge-and-foliot.

Despite the various improvements, most early clocks were highly unreliable. This was of little consequence, however, since they could be checked and adjusted regularly by reference to the sun. Thus, despite the technology and the mechanical nature of the time it produced, time was still ultimately dependent on the sun.

But by the middle of the seventeenth century, pendulum clocks with an anchor escapement were being manufactured that were accurate to within ten seconds per day. This was far more precise than reading the time from a sundial. Not only was it not easy to read the time accurately from a sundial, the speed of the sun across the sky varied slightly from one day top the next. Indeed, with the availability of precise time machines, it became possible to measure the variation in the sun’s speed. It was then, at the start of the scientific revolution, that people effectively started to live by mechanical time. Though, for the vast majority of the population, the sun would continue to provide the principal means of telling the (approximate) time, the definitive time was that provided by the clocks. From then on, clocks were used to set and calibrate sundials, rather than the other way round as had previously been the case.

Where am I?

The introduction of accurate clocks not only provided an accurate way to measure and tell the time, it also enabled sailors to make us of the variation of time with longitude to determine their position when at sea.

In the fifteenth century, when explorers such as Christopher Columbus and Amerigo Vespucci first started to sail into the great oceans, they faced a major hurdle: How could they keep track of their position? For earlier generations of sailors, such as the Mediterranean and Northern European traders, there was no such problem—they always kept close to a shoreline. From early times, charts called portolans (harbor guides) were available to provide details required by the coastline-hugging sailor—depth of the water, location of treacherous rocks, special landmarks, et cetera. But how do you keep track of your position when you have left the shoreline behind?

Part of the answer was provided by Greek geographers of the third century B.C., who used astronomical calculations to draw three reference lines on their world maps—the three lines of latitude known nowadays as the equator and the tropics of Cancer and Capricorn. Eratosthenes subsequently added further east-west lines of latitude, positioned to run through familiar landmarks. A century later, Hipparchus made the system more mathematical regular, by making the lines equally spaced and truly parallel, not determined by the lay of the land or by places that people found important. He also added a system of north-south lines of longitude, running from pole to pole, and divided the 360 degrees of both latitude and longitude into smaller segments, with each degree divided into 60 minutes and each minute into 60 seconds. (Both the 360 degrees of the circle and the 60-fold division of the degree and the minute come from the fourth century B.C. Babylonian sexagesimal system of counting, adopted because of the ease of subdividing the whole numbers 60 and 360.)

In the second century B.C., the Greek astronomer Claudius Ptolemy wrote eight books on geography, in which he described how to draw maps with lines of latitude and longitude for reference. Ptolemy’s manuscript was accompanied by twenty-seven world maps, drawn according to his ideas. (It is not known if Ptolemy himself drew those maps.) Ptolemy made two major errors: his estimate for the circumference of the world was only three-quarters of the true figure, and he extended Asia and India much too far to the east. (It was the combination of these two errors that made Columbus — who had a copy of one of Ptolemy’s maps — think he could sail west to the Indies, and thereby led to the discovery of America by the Europeans in the fifteenth century.)

To make use of the grid lines drawn on a map, a navigator had to have a way to determine the ship’s latitude and longitude. Latitude was not much of a problem. All the sailor had to do was measure the altitude of the sun at noon. This varies with latitude (and with the time of the year), and a simple geometric calculation allows the latitude to be computed from the noon altitude on any given day of the year. Already in the Middle Ages, astronomical tables showed the altitude of the noon sun throughout the year at different latitudes, and a sighting instrument such as a quadrant could be used to measure the altitude. Even in those days, determination of latitude could be made to within half a degree.

But how do you determine longitude? The first practical answer was in terms of speed. If an explorer knew the speed he was traveling, he could compute the distance covered each day, and in that way keep track of his longitude. Columbus had no instrument to measure his speed, so he simply observed bubbles and debris floating past his ship and used those observations to make an estimate of the speed. A better solution was to use time. Even the Greeks had observed that longitude could be regarded as a function of time. Since the earth makes one complete revolution every twenty-four hours, in each single hour it rotates through fifteen degrees of longitude. This means that every degree of longitude corresponds to four minutes of time. If a navigator knew the time at his starting point, and also knew the local time, then by comparing the two times he could compute his current longitude relative to the initial longitude. By carrying a clock on board, all a sailor needed to do to determine his longitude was read the clock’s time at (local) noon (i.e., when the sun was at its highest point) and convert the clock’s discrepancy from noon to give the ship’s longitude relative to the starting longitude. Every four minutes of difference would indicate 1 degree of longitude to the east or the west. To make this process work, of course, the sailor had to have a reliable clock; moreover, a clock that remained reliable when carried out to sea on a ship.

From the sixteenth century onwards, the need for an accurate clock to determine longitude became so important to growing world trade, that a number of monetary rewards were offered for the first person to produce such a device. In 1714, England’s Queen Anne offered £20,000 (several million pounds in today’s currency) for the first person to find a way to determine longitude to within half a degree. Many attempts were made to solve the problem and win the various prizes. In 1759, a Yorkshireman called John Harrison tested a 5.2 inch diameter clock on a trip from Britain to Jamaica and back. The clock lost only five seconds on the outward journey, corresponding to a longitude error of only one and a quarter nautical miles. Harrison won Queen Anne’s prize, and the world finally had a way to determine longitude: by the accurate measurement of time.

Times change

In the case of ocean travel, the development of reliable timepieces brought time and space together, and enabled the traveler to make use of time in order to determine location. For land travel, however, the arrival of accurate clocks created a conflict between time and location.

The first inklings of the problem occurred in Europe in the eighteenth century, with the introduction of mail coach services. Designed to convey passengers as well as mail, the coaches kept to a strict schedule, and as with today’s express delivery services, each company needed to maintain a good reputation for reliability and punctuality. The problem with keeping to a strict timetable was that the actual “time of day” varied from town to town. Even in a small country like England, towns to the west of London could be up to twenty minutes behind the capital. Much like today’s jet-setting international business executives, coachmen were forever having to adjust their watches to give the correct local time.

The problem became much worse with the arrival of the railway network in the nineteenth century. The greater speeds, together with the need to change from one line to another—possibly from one railway company to another—in the course of a single journey made the plethora of different local times a confusing annoyance. In England, the railroads decided that they would run their operations according to London time, as determined by the Royal Observatory at Greenwich, and by 1848 practically all British railroad companies operated according to what would eventually become known as Greenwich Mean Time (GMT). For a while, many local towns continued to keep their own time, determined by local observations of the sun, but gradually the benefits of having a single time began to outweigh tradition and local pride. By 1855, almost all public clocks throughout Great Britain showed GMT.

The method used to synchronize all the clocks brought with it another acknowledgment that time could be a commodity to be sold. The Greenwich Observatory maintained an electrical Standard Clock that defined GMT. Each day, the Observatory took stellar readings to correct the Standard Clock. (Measuring the positions of certain stars at night was a much more accurate way to measure the earth’s rotation than trying to identify the moment when the sun was at its midday highest point.) After the invention of the electrical telegraph in 1839, telegraph lines were laid alongside all the major railway tracks. In 1852, the Astronomer Royal, George Arey, instituted a system whereby time signals from the Standard Clock were transmitted along the telegraph lines to electrical clocks at railway stations, government offices, and post and telegraph offices throughout the country. For a fee private subscribers could also be hooked up to receive the time signal. (Clockmakers and clock repairers were major customers of this service.) They were, quite literally, buying time.

In this way, the entire British Isles came to conform to a single system of time, determined by the stars, and distributed along telegraph wires.

The confusion caused by the differences in local times generated in Britain by the introduction of the railroad system was nothing compared to the United States, where, because of the much greater distances involved, the differences could be far greater than a few minutes. As the U.S. railway system grew between 1840 and 1850, most railroad companies operated according to the time of their home city. The result was that, at the height of the temporal confusion that developed, there were around eighty different railway timetables in use around the country.

To try to bring order to the chaos, regional time zones started to develop. For example, by the early 1850s, all New England railroads kept to the same time, determined by the Harvard College Observatory. Likewise, there were standardized time zones around New York, Philadelphia, and Chicago.

The next step toward uniformity occurred in 1869, when a far-sighted individual called Charles Dowd put forward a plan to divide the entire nation into four uniform time zones, each fifteen degrees of longitude wide, and hence each one an hour apart from its neighbor. The time zones were not designed to be adopted by local residents. Rather, they provided a systematic basis for coordinating railroad schedules, and Dowd published timetables that gave the conversions between each local time region and the zonal railroad time. Eventually, however, people started to suggest making railroad time the only time, with the entire nation having just the four zonal times.

The proposal to abolish all the city-based local times caused enormous controversy, with many civic leaders seeing it as a matter of local pride to maintain their own time system. It was not until 1883 that the majority of the country were prepared to make the move to adopt railroad time. The final step was orchestrated by a railwayman called William Allen, who lobbied long and hard for what he saw as an obviously advantageous move. At 12 noon on November 18, 1883, Allen saw his dream come to fruition. At that precise moment, the vast majority of the nation switched to the new time.

The switch was achieved by getting all the different observatories, which regulated the time in their regions, to agree to send the new time signal at precisely the same moment. By then, most towns and cities provided a standard time signal, which in turn was based on a time signal received from an observatory. (A common means of providing the local residents with a daily time signal was by means of a time-ball, a ball that slid down a vertical pole to provide a countdown to noon. Today, New York City uses such a device to provide a ritualistic time signal for the start of the new year at midnight on New Year’s Eve.) By coordinating all the time signals, in one fell swoop the entire nation was switched from local time to one of the four zonal railroad times (apart from a small number of renegade regions that vainly held out for a year or so longer).

In 1918 the four-zone time system was legalized. After two thousand years, a completely abstract, man-made, uniform, mathematical notion of “time” was starting to work its way into — and condition — our view of the world. But there were still some further developments to take place. First, although, by the late nineteenth century, many countries had adopted uniform time systems, there was hardly any coordination between different nations. In particular, there was the fundamental issue of where to locate the base line for measuring longitude. Unlike latitude, where the earth’s axis of rotation determines two poles and a corresponding equatorial base line, there is no preferred baseline for measuring longitude. England used the line of longitude through Greenwich—where the Royal Observatory was located—as the zero meridian, and by 1883, Sweden, the United States, and Canada had also adopted the Greenwich Meridian as the baseline.

With the growth of international commerce, discussions started to take place to try to establish a uniform worldwide system for measuring longitude. In 1884, an International Meridian Conference was held in Washington, D.C. to try to resolve the issue. On October 13, the twenty-five participating counties put the matter to the vote. Twenty-two of them voted for Greenwich (San Domingo voted against, France and Brazil abstained). Greenwich was chosen for two reasons. First, the meridian had to lie on a major observatory. Second, because so much international shipping at the time was British, Greenwich was already the most widely used meridian in sea transport, having been adopted by around two-thirds of the world’s shipping companies.

The establishment of a worldwide system to measure longitude brought with it a notion of worldwide time. Since there are twenty-four hours in a day and 360 degrees in a circle, each fifteen degrees of longitude represented one hour. Thus, by wrapping a 360 longitudinal grid around the earth, humankind automatically divided the planet into twenty-four time zones, each one hour different from its two neighbors.

Just as the adoption of uniform time in Britain had been brought about by the development of coach travel and the railways and the adoption of uniform time zones in the United States was in response to the growth of rail travel, so too the main impetus for having a uniform worldwide system of measuring time was Marconi’s invention of wireless telegraphy in 1899. With instantaneous communications between counties around the world, and between land and ships at sea, it became imperative to have a uniform system of world time.

Surprisingly, this momentous development in human civilization, a major step toward today’s “global village,” went almost unnoticed at the time. Admittedly, not all countries made the switch to using the Greenwich Meridian straight away. Many countries did not adopt Greenwich until well into the twentieth century, with the last one, Liberia, not making the change until 1972. National pride was the principal inhibiting factor. But ultimately, there was nothing that could stand in the way of what was, quite literally, the march of time.

Living by the clock

Today, we live much of our lives “by the clock.” We are awakened by an alarm clock, we listen to the radio at a particular time, we travel to and from work at a certain time of day, we attend meetings that start and finish at predetermined times, we eat our meals according to the clock, not simply when we feel hungry, and the clock tells us when to go to a movie, to a concert, to the theater, or to watch our favorite television program. Indeed, not only are most of our daily activities regulated by the clock, they are often ruled down to the precise minute. This way of living is very recent. Not only does it depend on the uniform system of worldwide time measurement, it also requires that each one of us carries on our person a reliable means to keep track of time. The development of first the pocket watch and then the wrist watch also changed the way we view, and live, our lives. Completion of the revolution in human life brought about by the evolution of our concept of time was as much a technological step as an intellectual one. To live according to the regular beat of man-made time, we have to carry time around with us. More accurately, since our present-day watches do not (yet) communicate with each other or with any centralized “time station,” we carry around with us a device that manufactures a personal time, that is built to be in synchronization with official time to within a few seconds.

The accuracy (and cheapness) of today’s watches and clocks comes from an observation made by the Frenchman Pierre Curie in 1880. Curie noticed that when pressure is applied to certain crystals—quartz crystals, for example—they vibrate at a certain, highly constant frequency. Subsequent investigations showed that subjecting crystals to an alternating electric current also caused them to vibrate. The first use of this phenomenon was in the design of radios, to provide a broadcast wave of constant frequency. Then, in 1928, W. A. Marrison of Bell Laboratories built the first quartz-crystal clock, replacing the pendulum and the various other mechanical oscillating devices of previous timepieces by the constant vibrations of the quartz crystal. The quartz clock was so accurate and reliable, that already by 1939 it had replaced the mechanically-regulated clocks at the Observatory in Greenwich.

Though the resulting accuracy was not discernible to human consciousness, the arrival of the quartz clock to measure time changed the nature of time yet again. Since quartz crystals can vibrate at millions of times a second, the underlying basic unit of time provided by our timepieces changed from the second—the unit provided by mechanical oscillating devices—to units up to a million times smaller. The meant that our timepieces had developed to the point where time finally broke free of the natural phenomenon with which our very notion of time had originated: the earth’s daily rotation. With devices capable of measuring up to a millionth of a second, it was possible to measure the small discrepancy in the earth’s rotation from day to day. It no longer made sense to define the second as 1/86,400th of a mean solar day (86,400 = 24 x 60 x 60). Instead, we now base our time on the daily movement of distant quasars. Administered in Paris, this international, astronomical time is called Coordinated Universal Time (UTC).

The time of our lives

Astronomical observations provide a stable basis for our modern time. But even quartz-crystal clocks are not sufficiently accurate to provide the precision of measurement required for many present-day technologies. For one thing, no two crystals are exactly alike, and differences in size and shape affect the frequencies at which the crystals oscillate. Also, over time, the oscillating frequency of a given crystal tends to drift, as its internal structure changes slightly. Far greater accuracy is provided by the atomic clock, the first of which was constructed by the English physicists L. Essen and J. Parry in 1955. It makes use of the fact that when suitably energized, the outer electron of a caesium atom flips its magnetic direction relative to the nucleus, in the process emitting or absorbing a quantum of energy in the form of radiation with a constant frequency of 9,192,631,770 cycles per second. The idea behind the atomic clock (or the caesium clock) is to bombard cesium with microwaves of close to 9,192,631,770 cycles per second. The microwaves cause an energy oscillation of exactly 9,192,631,770 cycles per second in the caesium atoms, and that in turn regulates the microwaves, holding them to exactly that frequency—a simple feedback loop that provides the ultimate, perfect timekeeper. By using the very basis of matter, we can define the second to be 9,192,631,770 ticks of the caesium clock. The official definition, adopted in 1967, is that the second is “9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.”

Although units of time less than about a tenth of a second are not discernible to human consciousness, present-day life depends heavily on the extremely accurate measurement of time provided by quartz clocks and atomic clocks. For example, consider how dependent we are today on broadcast electromagnetic waves for various kinds of communication. Suppose an FM radio station is assigned the broadcast frequency 100 MHz. (That’s 100 million cycles per second.) If the station’s second is just 1/1,000th different from the true second, its broadcast signal will be off by 100,000 Hz (i.e., 100,000 cycles per second). Without highly accurate timekeeping, our communications network would be chaos. Another example is provided by our current desktop computers, which derive their speed from a highly accurate internal clock capable of measuring (or, if you prefer, creating) extremely short periods of time, currently approaching the 500 MHz range.

A third example of the use of the high accuracy of atomic clocks is provided by the ground-based LORAN-C navigation system and the satellite-based Global Positioning System (GPS), both modern-day versions of the use of time to determine position. GPS, for instance, depends on a network of twenty-four satellites that orbit the earth at an altitude of 11,000 miles. Each satellite continually beams down a signal giving its position and the mean-time determined by the four atomic clocks it carries on board. By picking up and comparing the time signals from four satellites, a ground receiver—which may be small enough to be hand held—can compute its latitude and longitude with an accuracy of up to 60 feet and its altitude accurate up to 100 feet. The clocks on the satellites have to be extremely accurate for two reasons. First, the satellite uses its clocks to determine its own position at any instant. Second, the determination of the position of the ground receiver depends on the tiny intervals of time it takes an electromagnetic signal to travel from each of the satellites to the receiver. Since the signal travels at 186,000 miles per second, a timing error of one-billionth of a second will produce a position error of about one foot. The on-board clocks are accurate to one second in 30,000 years. (Ground-based atomic clocks can be accurate up to one second in 1,400,000 years.).

In the United States, the U.S. Naval Observatory (USNO) in Washington, D.C. is charged with the responsibility for measuring and disseminating time. American time is determined by the USNO Master Clock, which is based on a system of many independently operating caesium atomic clocks and a dozen hydrogen maser clocks. Their web site at http://tycho.usno.navy.mil/ provides a rich source on information on modern timekeeping.

Though largely hidden from our view, the fine-grained notion of time in use today, based on the movement of massive objects far away in the universe and measured by the tiny quantum energy states of the atom, is quite literally the time of our lives. It affects the very fabric of our daily lives and the way we view ourselves and the world we live in. We live by the clock, and in many ways we are slaves to the clock. Yet in terms of utility, time is something that only exists because we have the means—both the conceptual framework and the associated technology—to measure it with reasonable accuracy.

Future time: The clock of the long now

Anyone who has visited Stonehenge—which among possible other purposes was undoubtedly a timekeeping device—will have felt the sense of awe at the technological skills of our ancestors many thousands of years ago. With many of today’s timepieces barely lasting from one Christmas to the next, will the present generation leave a similar legacy to the future?

Several years ago, the computer scientist Danny Hillis (the designer of the massively parallel computer called the Connection Machine) asked whether modern technology would allow us to build a mechanical clock that would keep running and yield accurate time for at least 10,000 years. Such a device would be a twentieth century legacy to the future, a present-day analogue of Stonehenge.

Hillis first wrote about his idea in 1993: “When I was a child, people used to talk about what would happen by the year 2000. Now, thirty years later, they still talk about what will happen by the year 2000. The future has been shrinking by one year per year for my entire life. I think it is time for us to start a long-term project that gets people thinking past the mental barrier of the Millennium. I would like to propose a large (think Stonehenge) mechanical clock, powered by seasonal temperature changes. It ticks once a year, bongs once a century, and the cuckoo comes out every millennium.”

Such a device could do for time what photographs of Earth from space have done for thinking about the environment, Hillis suggests. It could become an icon that reframes the way people think.

With assistance from Whole Earth founder Stewart Brand, the musician and technophile Brian Eno, and others, Hillis established a foundation to support the design, construction, and maintenance of his clock. Eno called it “the clock of the long now,” which Brand took as the title for a book on the project, published last year. The Long Now Foundation was officially established in 1996—or rather 01996 in Long Now Time, since a 10,000 year clock will have to count years with five digits to avoid a future Y10K problem.

The prototype of the 10,000 year clock Hillis is working on stands eight feet tall, and is constructed of Monel alloy, Invar alloy, tungsten carbide, metallic glass, and synthetic sapphire. (The eventual one may be larger.) The prototype is due to debut on January 1st 2000. It measures time by what Hillis calls a serial-bit adder, a highly accurate binary digital-mechanical system he invented. Its 32 bits of accuracy gives it a precision equal to one day in 20,000 years. It self-corrects by intermittently locking on to the sun. The mechanical power to “wind it up” could be provided by the alternate solar-heating and nighttime-cooling from the daily cycle of day and night, or from the annual warming and cooling of the seasons. An intriguing supplementary—or even alternative—source of power Hillis has suggested is to establish an annual “winding of the clock” as a worldwide cultural event.

For a society that lives its life by the clock and by technology, Hillis’s Clock of the Long Now would surely be a fitting memorial to mark the close of the twentieth century. Without doubt, it’s about time.

For further information about Hillis’s clock, check out the web site of the Long Now Foundation at http://www.longnow.org/.

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, was published by W. H. Freeman in August.