As you walk through the butterfly room of the Texas Discovery Gardens, take time to slow down and notice what’s going on in the space around you. What do you notice about the flight of the butterflies? I spent some time watching the different flight paths used by different butterflies. What shapes of paths in the air do you see?
If you’re like me, you probably noticed that when a butterfly is flapping its wings, the path is usually very erratic, darting this way and that. But does a butterfly have to fly erratically? No, if you watch for a while, you can find times when they flay in more regular paths. So why would they fly so erratically?
That question leads us to the topic of extrapolation. If you knew that someone had drawn a shape and erased part of it and what was left looked like this:
how might you try to complete the shape? You would probably draw something like this:
On the other hand, if what was left looked like this:
you’d really have no idea how to complete the shape. It could turn out to be any of thousands of different paths.
How does this relate to the butterfly’s flight? Well, who might want to figure out where the butterfly’s path will take it? Who would be interested in where the butterfly would end up? Maybe something that wanted to eat it? Butterflies are not big or strong, and they are easy to spot amidst the plants. So they need other defenses, and the erratic geometry of their flight is one: they make it very difficult for predators to extrapolate their paths and intersect with them to catch them. So mathematics can be useful, even to a butterfly!
If you look at their non-erratic flight, you can also see another mathematically interesting shape. You’ll notice that sometimes they glide in straight lines, but that they also glide in curved paths. In these curved paths, they tend to go around in a circle when viewed from above, while also gradually descending. This kind of curve, that curves like a circle while descending (or ascending), has a special name in mathematics. It’s called a helix and it’s a shape we can see in many places around us: the handrail of a circular staircase, the thread of a screw, the rail of some roller coasters, and more. It’s also a kind of flight pattern that occurs spontaneously over and over: in nature, like the path of a spinning seed falling from a tree, or in man-made flying objects, as this illustration from the Federal Aviation Administration’s handbook for glider pilot’s shows.