At its core, mathematics is the study of patterns—a realization that changed Eddie Woo’s life: suddenly, the “dry” subject he dreaded in high school was transformed into a never-ending adventure. Now an award-winning math teacher, Woo explores the patterns all around us in It’s a Numberful World: in the “branches” of blood vessels and lightning, in the growth of a savings account and a sunflower, even in his morning cup of tea.
In the following excerpt, Woo considers just what makes a rainbow round—exploring the math and the beauty of this natural phenomenon. Read on to learn more, and don’t forget to check out It’s a Numberful World, available today!
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“Daddy, look out the window!”
I’m trying to concentrate on driving on this wet afternoon, and I’m squinting hard even through my sunglasses because the sun is so low on the horizon and the road is glistening with moisture. The school pickup is a stressful operation at the best of times, but my daughter’s voice catches my attention from the back seat, and I lift my gaze to look at her in the rearview mirror. Her elbow is on the door’s armrest and she’s supporting her chin with her hand as she stares through the rain-speckled window. I can see by the look in her eyes that something has transfixed her. So I turn and look, and there it is: the brightest rainbow I’ve seen in years. I look at it for longer than I should, given the fact that I’m still moving slowly through traffic, but like my daughter I find it hard not to stare. The iridescent green, the glowing red, the unearthly indigo…. Though I’ve seen rainbows like this hundreds
of times before, there’s something about today’s display that makes it particularly arresting.
“Why’s it round, Daddy?”
“Mmmm?” I reply, in that way parents do when they are just too distracted to give a decent response. My eyes lock back onto the road ahead of me and the traffic that’s now stopped around us. My brain finally catches up, but I reflexively repeat the question to stall for time anyway. “Round?”
She’s still looking out the window, but I can see her nod in my peripheral vision. “Yeah, why’s it round?”
There are so many things I love about my children. One of my favorite qualities is their perpetual wonder. Due to their age, or rather lack thereof, they have eyes to see things in the world— genuinely beautiful and amazing things—that I have become bored with, that my brain has trained itself to ignore.
Case in point: Why is a rainbow a bow? What gives it its elegantly round shape?
It turns out that the elegant roundness of a rainbow comes from a surprising source: the elegant roundness of each and every raindrop that makes up the rainbow.
I say surprising because most people, ironically, do not picture raindrops as round. On the contrary, a cursory search on the internet for “raindrop” will turn up millions of images that are distinctly pointy at the top. However, a search for “raindrop photos” reveals a more realistic picture: While sometimes slightly stretched or squashed, raindrops are nonetheless much closer to spherical in shape than these caricatures would have us believe.
But we’re getting ahead of ourselves. Let’s rewind a little to think about what’s happening when we see a rainbow in the sky. As we know from experience, rainbows don’t always form after rain; the sun needs to be shining brightly enough soon after the rain ends for a bright rainbow to appear, which is why they are often seen around sun-showers. If your entire sky is blanketed in thick clouds, then you’re out of luck. Rain is necessary, but not sufficient—you really need the light.
Fans of Pink Floyd and Isaac Newton alike will know that light does a curious thing when it passes through something like a prism. Due to a phenomenon we call refraction, white light—like the kind emitted by our sun—splits up into its components, which we call a rainbow.
Raindrops behave a bit like the piece of the glass on the album cover, refracting sunlight and dispersing it into a spectrum. But if that was all that was going on, we’d expect there to be rainbow light everywhere after a shower ends. Why is the rainbow bound up in a tight band that seems to belong in a perfect circle? And besides that, why does the rainbow curve away from the sun rather than around it?
That’s because the geometry of a circle makes the rays of light coming from the sun behave in a very predictable—and literally dazzling—way when they interact with a sphere of water. They not only disperse into their component colors, but they also reflect inside the raindrop in such a way that they all shine back perfectly in one particular direction, revealing the whole spectrum of colors as they do so.
Although there are millions of raindrops in the air, only certain raindrops are in the right position to reflect light toward exactly where you are standing—and every single one of those droplets lies on the surface of a gigantic cone with your eye at the tip. When you look at a cone from the perspective of the tip, though, you don’t see the full cone—you just see the cone’s cross section, which is a circle. You might then ask why we only see semicircular rainbows; the reason is that the horizon tends to obscure the bottom half of the circle. The full version can indeed be seen when viewed from the air, such as out the window of a plane if your timing is lucky enough!
For me, this is what mathematics is. The world around us is filled with patterns, structures, shapes, and relationships that beg to be not just marveled at but understood.
Humanity developed mathematics as a language we can use to interpret the world, but realities such as the rainbow have taught us that mathematics is no mere invention. It has always been woven into the fabric of the universe all around us, if we’re willing to open our eyes and see it.
I don’t remember how I answered my daughter that afternoon while we sat in traffic and admired the sky. But now I can tell her— and you—that rainbows are round because a choir of raindrops has conspired together to put on a light show so breathtaking and ethereal that if it didn’t just appear in the sky above our heads, we probably wouldn’t believe it.
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Excerpt from It’s a Numberful World: How Math Is Hiding Everywhere. Copyright © 2018, 2019 by Eddie Woo.
