As a mathematician, I can attest that my field is about ideas above anything else. Ideas that inform our existence, that permeate our universe and beyond, that can surprise and enthrall. Perhaps the most intriguing of these is the way infinity is harnessed to deal with the finite, in everything from fractals to calculus. Just think about the infinite range of decimal numbers—a wonder product offered by mathematics to satisfy any measurement needed, down to an arbitrary number of digits.
Despite what most people suppose, many profound mathematical ideas don’t require advanced skills to appreciate. One can develop a fairly good understanding of the power and elegance of calculus, say, without actually being able to use it to solve scientific or engineering problems. Think of it this way: You can appreciate art without acquiring the ability to paint, or enjoy a symphony without being able to read music. Math also deserves to be enjoyed for its own sake.
Sadly, few avenues exist in our society to expose us to mathematical beauty.
So what math ideas can be appreciated without calculation or formulas? One candidate is the origin of numbers. Think of it as a magic trick: harnessing emptiness to create the number zero, then demonstrating how from any whole number, one can create its successor. One from zero, two from one, three from two—a chain reaction of numbers erupting into existence. I still remember when I first experienced this Big Bang of numbers. The walls of my Bombay classroom seemed to blow away, as nascent cardinal numbers streaked through space.
For a more contemplative example, gaze at a sequence of regular polygons: a hexagon, an octagon, a decagon, and so on. I can almost imagine a yoga instructor asking a class to meditate on what would happen if the number of sides kept increasing indefinitely. Eventually, the sides shrink so much that the kinks start flattening out and the perimeter begins to appear curved. And then you see it: What will emerge is a circle, while at the same time the polygon can never actually become one. The realization is exhilarating—it lights up pleasure centers in your brain. This underlying concept of a limit is one upon which all of calculus is built.
The more deeply you engage with such ideas, the more rewarding the experience is. For instance, enjoying the eye candy of fractal images—those black, amoebalike splotches surrounded by bands of psychedelic colors—hardly qualifies as making
a math connection. But suppose you knew that such an image depicts a mathematical rule that plucks every point from its spot and moves it. Imagine this rule applied over and over again, so that every point hops from location to location. The “amoeba” comprises those well-behaved points that remain hopping around within this black region, while the colored points are more adventurous, loping off toward infinity. Not only does the picture acquire more richness and meaning with this knowledge, it suddenly churns with drama, with activity.
Would you be intrigued enough to find out more—for instance, what the different shades of color signified? Would the Big Bang example make you wonder where negative numbers came from? Could the thrill of recognizing the circle as a limit of polygons lure you into visualizing the sphere as a stack of its circular cross sections, as Archimedes did over 2,000 years ago?If the answer is yes, then math appreciation may provide more than just casual enjoyment: It could also help change negative attitudes toward the subject. Students have a better chance of succeeding in a subject perceived as playful and stimulating, rather than one with a disastrous PR image.
Perhaps the most essential message to get across is that with math, you can reach not just for the sky or the stars or the edges of the universe but for timeless constellations of ideas that lie beyond.