Devlin's Angle

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 birds -- Indigo Buntings -- 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 Polaris -- 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 ( 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.