Me, Myself and Math, a six-part series by Steven Strogatz, looks at us through the lens of math.

Sharpen your pencils, dust off your abacus and join me once again for a few weeks of mind-bending pleasure. No, I’m not speaking about politics.

We’ll travel to a place where problems have answers and truth exists.

The haven of mathematics.

My previous series offered a panoramic view of the field. This time, in “Me, Myself and Math,” we’ll focus on how the subject I love — math — relates to the subject we all love — ourselves.

From the DNA that encodes us, to the fingerprints that characterize us, to our place in the universe and our friend counts on Facebook, we are mathematical marvels. In the coming weeks we’ll see what math can reveal about us and our world, and at the same time, how the wonders of us have inspired advances in math. No specialized knowledge or background will be required, just curiosity and a sense of fun.

Let’s begin with what our bodies can teach us. We all know that toddlers learn to count with their fingers and sometimes their toes. Those appendages are called “digits,” and it’s no accident that the same word refers to the 10 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in the decimal system. Our bodies are our first arithmetic teachers.

But what is less widely known is that our bodies are also trying to teach us higher math, if only we’d let them. Look at a baby’s first hairdo:

Courtesy of Sheila Larson

The cowlick at the center of that cute little swirl is, in mathematical parlance, a “singularity,” a point of confusion where the baby’s hair can’t seem to decide which way to grow. On the back of the cowlick the hair falls to the left; on the front it grows to the right; and on the sides it falls forward and backward. What makes a cowlick “singular” is that a variable (the hair’s direction) changes abruptly and discontinuously there.

Nature abounds with singularities. At the eye of a hurricane the wind doesn’t blow at all, yet the wind nearby blows in any direction and every direction. Something equally paradoxical happens at the North Pole. If you ask, in the style of Lewis Carroll, what time it is at the North Pole, the only sensible answer sounds like a joke: it’s all times. All the time zones converge at the North Pole, so by stepping away from that singularity along different lines of longitude you can put yourself into any time zone you like.

Singularities reflect nature’s attempt to resolve mismatches, to enforce continuity against all odds. When disagreements become inevitable (between hair directions, or wind directions, or time zones), singularities confine those mismatches to the smallest space possible: a single point.

One of the most remarkable features of singularities is their persistence. They have a kind of permanence to them. As you grow, your skull and scalp get bigger but you never lose that cowlick.

Topology is the branch of higher mathematics that deals with such durable features. It’s often defined as the study of those properties of shapes that stay the same after continuous deformations. To visualize what this means, imagine drawing a sketch of a baby’s hair pattern on a thin sheet of Spandex. Now deform the sheet — stretch it, bend it, twist it — but don’t rip it or glue any parts of it together. Your exertions distort the geometrical structure of the sketch (some hairs move farther apart and the angles between them change) but its topological structure stays the same (cowlicks persist, as do the crossing or non-crossing of any two given hairs).

Or take a close look at your fingers and palms. See all those neatly patterned fingerprint ridges? On small patches of skin they run nearly parallel to one another. That’s nature enforcing continuity again. But when different sets of ridges collide head-on, it’s hard to keep everyone happy. Each ridge wants to stay parallel to its neighbors, but also wants to merge with the newcomers. The collisions create unavoidable discontinuities — singularities — that are not only of interest to palm readers and the FBI. They’re your skin’s way of offering you a lesson in topology.

When you look at your fingerprints, you’ll notice just a few types of singularities. The two most fundamental are the triradius

and the loop.

All other singularities on fingerprints can be built up from these two. For instance, the singularity known as a whorl

can be regarded as the fusion of two loops

that have been squashed together so that their two inner endpoints coincide.

In 1965 Lionel Penrose, a British medical geneticist, pointed out that fingerprints and palm prints obey a universal rule: no matter what your personal pattern looks like, everybody with five fingers always has four more triradii than loops. (His bookkeeping treated a whorl as two loops, for the reason explained above.)

The irony here is amazing. The most distinctive feature we sport — the geometry of our fingerprints and palm prints — is also the least distinctive: the same topological rule holds for all of us.

Penrose generalized his rule to include people born with other than five fingers, as sometimes occurs as a result of genetic abnormalities. If D denotes the number of digits on each hand, Penrose’s rule is that the number of triradii, T, exceeds the number of loops, L, by one fewer than the number of digits:

T – L = D – 1.

In 1979, Penrose’s son Roger, a mathematical physicist, published a beautiful paper dedicated to the memory of his father, in which he derived his father’s rule from topology. Let me outline his proof for you now.

The first step is to assign a number to each type of singularity. This number is traditionally called, with no pun about fingers intended, the singularity’s “index.” To see what it measures, imagine a tiny gymnast, Jim, who circumnavigates the singularity by going around it once counterclockwise. (He’s a device to help us probe how the ridges change direction around the singularity.) At every point along his path, Jim carefully orients himself in the direction of the fingerprint ridge there, causing him to rotate.

Here’s what happens when he moves around a triradius.

At position A, Jim stands straight up, to align with the ridge there. At position B, he tilts slightly clockwise. After one full circuit around the triradius, he ends up standing on his head, flipped upside down from how he started, having performed a single 180° flip clockwise, opposite to the counterclockwise sense of his journey.

In general, the index of a singularity measures the net number of 180° flips Jim executes during one counterclockwise circuit; its sign indicates whether they were counterclockwise (+) or clockwise (–) flips. Therefore the index for this kind of singularity is defined to equal –1.

Circumnavigating a loop singularity, however, causes Jim to flip 180° in a counterclockwise sense — the same sense as his circumnavigation. Hence the index of a loop is +1.

The next step in the argument invokes a key fact about indices: if Jim goes around a circuit that encloses two or more singularities, the sum of their indices dictates how many 180° flips he’ll make. And conversely, the net number of 180° flips he makes on a circuit equals the sum of the indices of the singularities inside. This pair of statements is known as the index theorem. It implies Penrose’s rule, as follows.

Let Jim circumnavigate the boundary of a flattened handprint:

Empirically it’s known that the ridges on palms and fingers behave as shown here — the ridge directions tend to lie parallel to the ends of the fingertips and the wrist, and to stick out from the sides of the fingers and the sides of the palm. So as Jim goes all the way around the flattened hand counterclockwise, the only places where he flips are the gaps between the fingers. There are four of these gaps for a person with five fingers. At each of them Jim executes a single 180° flip clockwise. Hence the net number of 180° flips Jim makes equals –4, and by the index theorem, this must equal the sum of the indices inside. Since triradii count as –1 and loops count as +1, the number of triradii must exceed the number of loops by 4, which is precisely Penrose’s rule, T – L = 4, for a five-fingered hand.

Following Roger Penrose’s gesture, I’d like to dedicate this piece to the memory of my own mentor, Art Winfree, one of the world’s great mathematical biologists. Much of his work concerned the singularities of biological clocks in our bodies and brains.

Using topological reasoning like that shown above, Art predicted that rhythms ranging from heartbeats to sleep cycles would have their own North Poles, states where the phase of the rhythm would become singular and the cycle could cease. His ideas were confirmed experimentally and are now regarded as important clues by doctors and biomedical researchers working to unravel the mysteries of cardiac arrhythmias.

Although the culprits remain at large, there are promising leads. For as Art loved to say, quoting Sherlock Holmes, “singularity is almost invariably a clue.”

NOTES

1. A lively and accessible introduction to topology, including index theory as well as much more, is given in D. Richeson, “Euler’s Gem” (Princeton University Press, 2008).

2. Fingerprint and palm print patterns were first analyzed topologically by Lionel Penrose, based on his clinical observations, in L. S. Penrose, “Dermatoglyphic topology,” Nature, Vol. 205 (1965), pp. 544–546. This paper states what has now come to be known as Penrose’s rule.

For Roger Penrose’s derivation of his father’s rule from index theory, see R. Penrose, “The topology of ridge systems,” Annals of Human Genetics, Vol. 42 (1979), pp. 435–444. My treatment of these topics, as well as the sketches of the loop, whorl and triradius singularities shown above, were directly inspired by this elegant paper.

For an independent analysis that arrives at the same conclusions and extends the results to cultures of fibroblasts (cells found in connective tissue), see T. Elsdsale and F. Wasoff, “Fibroblast cultures and dermatoglyphics: The topology of two planar patterns,” Wilhelm Roux’s Archives of Developmental Biology, Vol. 180 (1976), pp. 121–147.

3. For introductions to the history of fingerprinting and its use in criminal investigations, see S. A. Cole, “Suspect Identities” (Harvard University Press, 2002) and C. Beavan, “Fingerprints” (Hyperion, 2002).

4. When you look at your own hands, you’ll probably see your fingerprints without any trouble, but you might have to use a bright light and magnifying glass to see the ridges running down the sides of your fingers and continuing to the palm. On older people or manual laborers like bricklayers who handle rough, heavy materials frequently, these ridges might be barely visible, like worn treads on a tire. They’re more conspicuous on children’s hands.

5. Art Winfree’s work on phase singularities in biological rhythms, and his wide-ranging synthesis of research by others, can be found in his magnum opus, “The Geometry of Biological Time,” 2nd edition (Springer, 2001). Many people find this book hard to read because of its idiosyncrasy, but it’s brilliant and worth the effort. For Winfree’s earlier and perhaps more accessible accounts of how singularities might be implicated in cardiac arrhythmias, see A. T. Winfree, “Sudden cardiac death: a problem in topology?” Scientific American, Vol. 248, No. 5 (1983), pp. 144–161, and “When Time Breaks Down” (Princeton University Press, 1987).

6. “Singularity is almost invariably a clue”: The quotation is from “The Boscombe Valley Mystery” in A. Conan Doyle, “The Adventures of Sherlock Holmes.”

Thanks to Margaret Nelson for preparing the illustrations; Sheila Larson for sharing her photograph of her nephew Diego; Paul Ginsparg, Jon Kleinberg, Tim Novikoff, Andy Ruina and Carole Schiffman for their comments and suggestions; and Roger Penrose for his permission to adapt the images from his 1979 paper, drawn originally by A. J. Lee.