The geometry gap is much like the “ingenuity gap,” a concept raised in the book of the same name by Thomas Homer-Dixon, director of the University of Toronto’s Trudeau Centre for Peace and Conflict Studies. The Ingenuity Gap chronicles examples of people and societies unable to solve problems of their own creation. For Whiteley, the sedentary mathematical parts of our brains certainly result in an inability to solve scientific problems.
As an applied geometer, he works with biochemists researching the shapes of proteins and viruses, the latter of which are almost always icosahedral (twenty-sided) or dodecahedral (twelve-sided). The geometric molecular structure and the way it moves determine how proteins and viruses behave in the body and how they “dock” with drugs designed to alleviate or eliminate diseases such as cystic fibrosis and Alzheimer’s. In order to perform effectively, a drug’s molecular shape must match with viruses and proteins—fitting like a “lock and key,” or with an “adaptive fit,” like a flexible baseball glove catching a fastball. Collaborating with scientists, especially students coming into the fields of biology, chemistry, and engineering, Whiteley has observed that they don’t have the mathematical background and geometric visualization skills needed to see possible solutions. “Several times in my own work I have had to trick applied mathematicians into sitting down and looking at a problem in terms of a simpler geometry,” he says.
The dearth of the visual in mathematics, and the alienation it causes, is something Douglas Hofstadter, director of the Center for Research on Concepts and Cognition at Indiana University, experienced as a student. Hofstadter grew up “deeply in love with mathematics” and its “abracadabraic” qualities. But the prickly algebraic jungle of double-subscripted symbols, he says, “brought me to my knees...I was forced by the terrifying heights of abstraction to abandon math at Berkeley. It was a shattering experience to find myself repelled by what I had thought I loved.” He obtained a Ph.D. in physics instead, discovering the now-famous fractal shape that became known as the “Hofstadter Butterfly” and then writing his 1980 Pulitzer Prize-winning book, Gödel, Escher, Bach: An Eternal Golden Braid. It was only many years later, when he was a professor and could dictate his own terms, that he returned to math.
Hofstadter had always been fascinated with how the mind works. He set out to teach himself Euclidean geometry as an experiment, a way of observing the process of intellectual reasoning and discovery. He used the humble triangle as his guinea pig. “I was so naive at the outset of my many-year binge with geometry that I was caught off guard by the fact that the triangle had more than one centre,” says Hofstatder. “I had heard the terms ‘orthocentre,’ or ‘centroid,’ and ‘circumcentre.’ I sort of knew these things existed, but to tell the truth, when I found out that any triangle has several centres, I was really thrown. It seemed like a miracle. In point of fact, any triangle has an infinite number of centres, but they are not all equally interesting.”
“One time,” Hofstadter continues, “I made an analogy to the human body. Suppose I were to ask various people, ‘What is the most important part of your body?’ One person might reply, ‘It’s my brain.’ Somebody else might say, ‘No, it’s my stomach;’ somebody else might insist ‘my heart,’ ‘my sexual organs,’ or ‘my belly button’....I became inflamed and impassioned in trying to invent new geometrical notions. And I saw that the process of analogy-making lay at the core of all my discoveries. I started noticing beautiful patterns not only in triangles and among triangle centres, but there were exquisite patterns at the level of the ideas themselves.”
Following his triangle experiment, Hofstadter returned to math with the intention of reclaiming it. “I came back to group theory and Galois theory some thirty-five years later and I said, ‘By god, I’m going to make this hateful stuff lovable.’ ” He started a course called Group Theory and Galois Theory Visualized, though the running joke is that his approach is more like Pizza Theory. The visual metaphor Hofstadter uses to explain difficult group theory theorems such as Zassenhaus’ lemma is simple circles with slices and concentric circles within, which look a lot like pizzas. (It didn’t hurt either that he always brought pizzas to class.)
Hofstadter’s overriding conclusion from these intellectual experiments is that analogy-making plays a crucial role in doing quality mathematics—analogies are what allow the illustration and conceptualization and understanding of otherwise seemingly unfathomable ideas.
Of course, one could say math phobia has its societal upside. Like computer phobia or ignorance about the inner workings of an automobile engine, it has spawned expert classes—for example, accountants, who will happily do math for you. But if you disavow the paralyzing assumption that math is merely juggling numbers and puzzlingly hard, it is not so difficult to take charge of one’s repressed mathematical alter ego and have some fun.
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