Devlin's Angle

December 2004

The Amazing Ahmed

Let me introduce you to a remarkable individual called Ahmed.

Ahmed, who was the subject of a research paper published in 1981 by scholars R. Wehner and M. V. Srinivasan, lives in the Tunisian desert. He has had no formal education, and everything he knows he has picked up by experience.

Each day, Ahmed leaves his desert home and travels large distances in search of food. In his hunt, he heads first in one direction, then another, then another. He keeps going until he is successful, whereupon he does something very remarkable. Instead of retracing his steps - which in any case might have been obliterated by the wind blowing across the sands - he faces directly towards his home and sets off in a straight line, not stopping until he gets there, seemingly knowing in advance, to within a few paces, how far he has to go.

Because of language and cultural problems, Ahmed has been unable to tell researchers how he performs this remarkable feat of navigation, nor how he acquired this ability. But the only known method is to use a technique known as "dead reckoning." Developed by the ancient mariners of long ago, the method was called "deductive reckoning" by British sailors, who abbreviated the name to "ded. reckoning," a term that in due course acquired an incorrect spelling as "dead reckoning." In dead reckoning, the traveler always moves in straight lines, with occasional sharp turns, keeping constant track of the direction in which he is heading, and keeping track too of his speed and the time that has elapsed since the last change of direction (or since setting off). From knowledge of the speed and the time of travel, the traveler can calculate the exact distance covered in any straight segment of the journey. And by knowing the starting point and the exact direction of travel, it is then possible to calculate the exact position at the end of each segment.

Dead reckoning requires the accurate use of arithmetic and trigonometry, reliable ways to measure speed, time and direction, and good record keeping. When seamen navigated by dead reckoning they used charts, tables, various measuring instruments, and a considerable amount of mathematics. (The main impetus to develop accurate clocks came from the needs of sailors who used dead reckoning to navigate vast tracts of featureless ocean.) And dead reckoning wasn't just important in days of long ago. Until the recent arrival of GPS navigation, sailors and airline pilots used the method to navigate the globe, and in the 1960s and 70s, NASA's Apollo astronauts used dead reckoning to find their way to the Moon and back.

Yet Ahmed has none of the aids that mariners and lunar astronauts made use of. How does he do it? Clearly, this particular Tunisian is a remarkable individual.

Remarkable indeed. For Ahmed measures little more than half a centimeter in length. He is not a person but an ant. A Tunisian desert ant to be precise. Every day, this tiny creature wanders 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?

Many kinds of ants find their way to their destination by following scents and chemical trails laid down by themselves or by other members of the colony. Not so the Tunisian desert ant. Observations carried out by Wehner and Srinivasan, those two researchers I mentioned a short while ago, leave little room for doubt. The only way Ahmed can perform this daily feat is by using dead reckoning.

The two investigators found that, if they moved one of these desert ants immediately after it had found its food, it would head off in exactly the direction it should have taken to find its nest if it had not been moved, and moreover, when it had covered the precise distance that should have brought it back home, it would stop and start a bewildered search for its nest. In other words, it knew the precise direction in which it should head in order to return home, and exactly how far in that direction it should travel, even though that straight-line path was nothing like the random-looking zigzag it had followed in its search for food.

A recent study by S. Wohlgemuth, B. Ronacher, and R. Wehner (Odometry in desert ants: coping with the third dimension, Journal of Experimental Biology, to appear) has shown that the desert ant measures distance by counting steps. It "knows" the length of an individual step, so it can calculate the distance traveled in any straight-line direction by multiplying that distance by the total number of steps.

Of course, no one is suggesting that this tiny creature is carrying out multiplications the way a human would, or that it finds its way by going through exactly the same mental processes that Neil Armstrong did on his way to the Moon in Apollo 11. Like all human navigators, the Apollo astronauts had to go to school to learn how to operate the relevant equipment and how to perform the necessary computations. The Tunisian desert ant simply does what comes natural to it - it follows its instincts, instincts that are the result of hundreds of thousands of years of evolution.

In terms of today's computer technology, evolution has provided Ahmed with a brain that amounts to a highly sophisticated, highly specific computer, honed over many generations to perform precisely the measurements and computations necessary to navigate by dead reckoning. Ahmed no more has to think about any of those measurements or computations than we have to think about the measurements and computations required to control our muscles in order to walk or run or jump. In fact, in Ahmed's case, it is not at all clear that he is capable of anything we would normally call conscious mental activity at all.

But just because something comes easy or natural or without conscious awareness does not mean it is trivial. After all, almost fifty years of intensive research in computer science and engineering has failed to produce a robot that can walk as well as a small child can manage a few days after taking its first faltering steps. Instead, what all that research has shown is how complicated are the mathematics and the engineering required to achieve that feat. Few adults ever master that mathematics - as consciously performed mathematics - let alone a small child that runs with perfect bodily control for the candy aisle in the supermarket. Rather, the ability to carry out the required computations for walking comes, as it were, hard-wired in the human brain.

So too with the Tunisian desert ant. It's tiny brain might have a very limited repertoire. It may well be incapable of learning anything new, or of reflecting consciously on its own existence. But one thing it can do extremely well - indeed far better than the unaided human brain (as far as we know) - is carry out the particular mathematical computation we humans call dead reckoning. That ability does not make the desert ant a "mathematician," of course. But that one computation is enough to ensure Ahmed's survival.

How unique is the Tunisian desert ant in having some fairly sophisticated built-in mathematical ability? The surprising answer is, not at all unique. Nature, through the mechanism of evolution by natural selection, has produced a large number of similar "natural born mathematicians." Not mathematicians in the sense of having the ability to solve a range of mathematical problems, as we humans can do - at least some of us, and then after some considerable training. But mathematicians in the sense of being able to solve one particular mathematical problem; the one problem that is, for the creature concerned, a matter of life or death.

The innate mathematical abilities found in many of our fellow creatures is the subject of my latest book, THE MATH INSTINCT: Why You're a Mathematical Genius (along with Lobsters, Birds, Cats, and Dogs), to be published next spring by Thunder's Mouth Press.


Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition.