Struck by Genius: How a Brain Injury Made Me a Mathematical Marvel (11 page)

Part of the trouble is that no one has a way to accurately measure the circumference or area of a perfect circle. Instead, mathematicians have to approximate. One way of calculating the value of pi dates back to the Greek mathematician Archimedes. Around 250
BC
, he tried to find the area of a circle by placing one polygon inside a circle and another polygon outside the circle. He calculated the perimeter of the two polygons and theorized that the value of pi lay between those two numbers. Then he kept increasing the number of sides of the polygons—working his way up to ninety-six sides—so the areas of the two polygons got closer to equaling each other. Using this method, he calculated that the value of pi was between 3
10/71
and 3
1/7
. The ancient Greeks didn’t use decimals, but his fractions were the equivalent of about 3.1408 and 3.1429, respectively—not too far off the figure we use today.

I had never been taught about Archimedes or this visual approach, but now that I had arrived at the realization independently, I wanted to run through the streets announcing my profound discovery. I wanted to tell everyone this was a great secret revealed. Suddenly an idea I’d known in school only as “that 3.14 number” took on a relevance it had never had for me in a textbook or lecture.

I raced home and began my research further into pi that very day. As I read academic papers and popular-science articles online, I started to feel that pi literally defined everything—not just the ratios in a circle, but all of creation. It pertained to so many naturally occurring spheres, from pebbles to planets. And in more complex mathematics, like calculus, it helped define slope. I thought about the spirals of my seashell and draining water and coffee swirls. My own pupils were circles. Where would we be without the invention of the wheel? Circles were everywhere I looked and they felt fundamental to existence.

I began trying to describe what I saw to the rare visitor or person who telephoned. I said things like “Have you ever seen those boats where you push the lever forward from stop to fast? Push that lever all the way up and think of it as an obtuse, greater-than-ninety-degree angle, say a hundred-and-seventy-nine-degree angle, a really large one. Move that lever back down to a right angle, then slowly bring it down to an acute, less-than-ninety-degree angle until you collapse it in on itself. Think that every click along that lever makes a certain triangle. And every triangle is defined by pi at a certain value.” While this visual helped me a great deal and seemed very on point, my audience remained confused.

I wished I could give everyone the eureka moment I had had that day with the car—a circle subdivided by glistening, illuminated triangles. But when I tried to describe my inspiration, people told me that it would have been just an arc of light or a halo or a reflection to them. I couldn’t believe it was so mundane to them when for me it was a peak experience. I searched for the words to best represent this. I would wave my hands in the air, tracing what I’d seen with a pointed finger. All I got in return were blank stares. I realized that things would never be the same for me—all my life I would see down deep into the structure of things while everyone I knew was still skating on the surface. It was as though I’d been fitted with some sort of microscopic, x-ray-vision contact lenses. I searched for the words to capture its beauty and, more than that, the truth I believed its structure represented, but I was stammering.

Finally, I picked up a pencil and tried to sketch it.

I had never been able to draw in the past, but I was now pretty adept. The pencil didn’t feel like a foreign object in my hand but an extension of me and my mind. I felt compelled to draw and did almost nothing else. I found I was better able to represent things on paper than I had been. The joke in my family until now had been that when we played Pictionary, my doodles were always the worst! For one round of the game in which I had to represent the god Zeus, my scratch marks were little more than a carrot shape for a mountain and a zigzag for a lightning bolt above it.

I was amazed by my sudden facility with a pencil. It was as though someone else were clutching my fist and guiding my hand. This was another ability I’d never had before, and I had to set the pencil down for a moment to take it all in. What really made it come together, however, was when, in a rare conversation during my continued self-imposed isolation, a friend suggested that I add a ruler and a compass to my toolbox. I began to draw forms very close to the beauty I’d witnessed.

When I first tried to render the vision, I drew a perfect circle with a smooth perimeter, which was
not
what I had seen. The circle I saw was not perfectly curved—the circumference of it was jagged. It was more of a polygon with countless sides and filled with triangles, only approximating a perfect circle. So, as I drafted circle after circle at my desk, I began filling them with triangles. As I added more triangles I realized I was filling in more and more area at the circle’s edge.

“You can only fill in more and more triangles,” I said to my mom during one late-night phone call about this quest.

“Hence, pi goes to infinity,” she responded.

Her statement made everything click, and I realized how this insight was reflected in the circles I saw all around me and had been attempting to draw. What if the base of these triangles became smaller and smaller? First I filled a circle with 60 triangles, then 180:

Then 360:

All the way up to 720, when the width of the pencil’s lead wouldn’t allow for any more lines:

One day my daughter, Megan, was halfway through an episode of
Pokémon
when she hollered a question to me: “Dad, how does the TV work?” I explained that each image is made up of hundreds of little rectangular pixels, and when the pixels change color, they change the larger picture too. Just then a commercial for Overstock.com, with its giant
O
logo, came on, and Megan said, “That’s
impossible,
Dad. How do you make a circle out of rectangles?”

It was like a bomb went off in my mind. In a matter of minutes, I was no longer just a receiver of geometric imagery or a researcher; I was a theorist.

Ever since the mugging, curved objects in my line of sight had lost their smooth edges. They looked jagged and inexplicably discrete from their surroundings, and while I had spent years puzzling over the distortion, its significance had eluded me, and my drawings hadn’t cracked the code. That afternoon I found the right words.

“Circles don’t exist,”
I told her.

The deceptively simple observation hijacked my thought process. I described the basics of the familiar concept to Megan: “When you see a circle on the television, the edges appear to be curved only because the pixels are so tiny relative to the scale of your perception. The smaller the pixels, the smoother the edge becomes, but it never becomes perfectly smooth because the pixels can be made smaller and smaller, on to infinity.” I didn’t want to get too technical with Megan, but this realization reminded me of Mandelbrot’s fractal geometry, which emphasized the roughness in the world around us. In my head, I was quickly moving beyond the fundamentals.

Pi was becoming my mathematical soul mate. I began to obsess over it the way I did the cleanliness of my hands. Other mathematicians had discovered the utility of pi long before I stumbled on it. It is believed that ancient Egyptians knew about it and used it in the construction of the pyramids, as evidenced by those structures’ proportions. Pi has even been found in the Mandelbrot set. Its value lay in mathematics—not just in geometry but also in calculus applications and computing algorithms. The record for calculating pi (as of this writing)—which has been an obsession of people around the world ever since pi was first discovered—was achieved in March 2013 by Ed Karrels of Santa Clara University, in California. He computed the number to eight quadrillion places using a supercomputer. Though the sophisticated calculations of pi humbled me, I realized all of us were seeking the same thing.

However, I thought even the most sophisticated attempts to calculate pi were not taking it down to the quantum level—not one that I was able to find from my humble home computer station, anyway. So I began contemplating a hypothetical correction using one of the concepts for which Max Planck was awarded the Nobel Prize: the Planck length, an infinitesimally small unit of measurement equal to 1.62 × 10
⁻³⁵
 meters. Just to clarify, that’s millions, billions, trillions, and even quadrillions times smaller than anything that can be seen with the naked eye. The Planck length is the scale of length measurement where the usual rules of gravity break down and quantum mechanics comes into play. It’s also the smallest possible building block of space in the universe that can be observed (or exist relative to us). If the triangles I placed in my circle were each on the scale of a Planck length—
would that not be a more perfect pi?
I later posted my theory about this on a physics forum online because I was so sure I was the first to discover it.

I didn’t know much about academic procedures and I was far from being able to write a journal article on my thoughts and submit it for peer review, but this little step in taking my thoughts public was a real change for me. I found that society interested me again—if only in the context of having a conversation with people about math. I had some great responses, which encouraged me, and some real trolls, who insulted me and made me realize online forums aren’t the best place to seek feedback. But it was still a big step for me to post my thoughts.

After that brief and mixed-results excursion into public discourse, I doubled up on my reading and ignored the pull of the outside world. Historically speaking, I was in good company—Archimedes was allegedly so engrossed in pi that he failed to notice when Roman soldiers captured his home city of Syracuse. Before he was beheaded, he yelled, “Don’t disturb my circles!” I understood how he felt.

The term
pi,
which is the sixteenth letter of the Greek alphabet, was first used to describe the irrational number in a 1706 paper by the mathematician William Jones. The concept of pi has existed for four thousand years, and it can be used as a litmus test to indicate where humans were technologically at any given time. As Petr Beckmann wrote in
A History of π,
“The history of π is a quaint little mirror of the history of man.” I read that Simon Newcomb, the astronomer and mathematician who accurately measured the speed of light in the nineteenth century, had been quoted as saying humankind’s fascination with pi vastly transcends practical need. “Ten decimals are sufficient to give the circumference of the earth to the fraction of an inch,” he said, “and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful telescope.” But as humankind advanced, so did the utility of pi. Since ancient times, it has been important in construction and architecture. Now, everything from manufacturing machine parts to navigation and global positioning makes use of the constant in engineering formulas. Its infinite trail stretches out before us like a road map to the future, driving us forward. If pi really is tied to the Planck length, I imagine this finding could prove even more useful, for everything from space travel to supercomputing to things we have not yet imagined. If explorers from another world one day found our lifeless planet and excavated the ruins, I fantasized they would conclude, “Throughout their history, humans sought the end of pi.”