The Gods Have Descended

By Marc Chamberland

Marc Chamberland

Some of the most deeply felt moments in life concern our connection to beauty— heart-stirring music, a baby’s laugh, a magnificent sunset. This same wonder applies to scenarios that would more often be described as technical rather than beautiful: the stunning precision of a dance troupe, a clever chess maneuver, a climber’s brilliant combination of moves on a rock face. What makes this techni- cal beauty so appealing? Perhaps it’s the uncom- mon mastery of a skill or the element of surprise. Beyond our analysis, however, these experiences capture our imagination and inspire our creative spirits.

But why do we seldom hear such stories connected to math? Defenders of mathematics can argue fervently about the mesmerizing beauty of their discipline, but it seems that their epiphanies are hidden and rarely celebrated. The mathematical community—indeed, the general public—could benefit from our tales of math- ematical allure. So, it’s time that I offer one of my own stories.

A Mysterious Series

In my first semester as an undergraduate at the University of Waterloo, Canada, I took Advanced Honours Algebra from Peter Hoffman. His tall, wiry frame, bulbous eyes, and 1970s shaggy hair came alive as he danced across the lecture platform. The University of Waterloo in Canada is a magnet for mathematical aspirants, so this 60-strong class was packed with some very bright students. One day Hoffman scribbled the following formula on the board:

“We’ve all seen this formula before, right?” he queried.

The only infinite series I had previously encountered was also the most accessible one: the geometric series. Although Hoffman’s equation is a standard result taught to calculus students, it was new to me. And it left me in awe. How could adding infinitely many polynomials—a mess in my mind—equal such a concise and elementary transcendental function? And where did the factorials come from? Aren’t those related to counting problems? My next question left me even more perplexed: How could somebody prove that this equation was true? To claim that such a formula was legitimate suggested madness, but to have a proof seemed divine.

I found all this so astonishing that a biblical phrase came to mind. The response of the people to seeing the apostle Paul perform a miracle in Lystra was my thought at seeing the new formula: “The gods have descended among us in the form of men.”

Many people, when overwhelmed by a stunning or surprising occurrence, experience a momentary shut- down of their chattering minds. It’s as if their brains need all available resources to process the experience. My reaction was a spontaneous response, an uncon- scious attempt to make sense of this inexplicably beautiful formula. Today, I routinely teach the mortal underpinnings of Hoffman’s formula, but it has never lost its wonder.

Unfortunately, mathematics is often taught as a col- lection of symbol-manipulating rules that are neither inspiring nor obviously applicable. Any good teacher knows that she will win over more hearts—and ac- companying good will—if she can show the wonder of her subject. Mathematics has much to teach students concerning beauty, usefulness, and connections to other disciplines.

Even if my students do not go on to do something groundbreaking with the math they learn, I hope that most of them will grow in their respect for, and even be charmed by, mathematical ideas. And if they are ever so awestruck, so captivated, so overwhelmed that their response to a new idea is something like “the gods have descended,” then I’ve succeeded in showing them that soul-stirring beauty can be found in mathematics.

Marc Chamberland is a professor of mathematics at Grinnell College and creator of the YouTube channel Tipping Point Math.