The arithmetic of our bodies

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. So says Steven Strogatz in this rather interesting piece:

"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."

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. 

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. 

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... 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.

Using topological reasoning like that shown above, Art Winfree, one of the world’s great mathematical biologists, 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.

Of course, Leonardo was there first. Leonardo envisaged the great picture chart of the human body and believed the workings of the human body to be an analogy for the workings of the universe. According to Leonardo if you open your legs enough that your head is lowered by one-fourteenth of your height and raise your hands enough that your extended fingers touch the line of the top of your head, know that the centre of the extended limbs will be the navel, and the space between the legs will be an equilateral triangle.

He predicted the following arithmetic of the body:

• the length of the outspread arms is equal to the height of a man
• from the hairline to the bottom of the chin is one-tenth of the height of a man
• the maximum width of the shoulders is a quarter of the height of a man
• the distance from the elbow to the tip of the hand and from the breasts to the top of the head is a quarter of the height of a man
• the length of the hand is one-tenth of the height of a man
• the foot is one-seventh of the height of a man