This weekend I was astonished to learn that I’ve been operating my entire life under a fundamental misunderstanding of a very basic mathematical concept.
Here’s what happened:
My family and I are biking to the Davis Farmer’s Market on Saturday morning when my younger son (age 12) objects to our route. Rather than going due east on our street and then taking a right turn to head due south to our destination, I’m taking us on a series of rights and lefts—zigzagging, in other words. “This takes longer than just going straight out and making one turn,” he says.
“I don’t know that it’s any faster,” I concede, “but it’s certainly shorter.”
“What?” he says. “No, mom. It’s the same distance.”
No, no, no, I say, happy that I can still impart some math knowledge to my almost 7th grader. If our house is at point A of rectangle ABCD, I explain, and the Farmer’s Market is opposite, at point C, then the shortest route between them is the diagonal line segment AC, not line segment AB followed by line segment BC. (See Figure 1) Our zigzaggy route approximates that diagonal and is therefore shorter.
“No, mom,” he says. “Imagine this on a graph. If our rectangle is a square five spaces wide and five spaces tall, then bicycling across one side and down another is ten, and zigzagging back and forth through the middle is also ten because you still have to go out one, then down one, then out one, down one, et cetera. See?”
By this time we’ve nearly arrived, and I am getting rather uncomfortable because I can easily call up this graph in my head and the boy’s logic seems unassailable. My husband and older son (age 15) have joined in the conversation at this point, and they agree that a zigzag across a rectangle does not, in fact, approximate a diagonal line across that space.
I appeal to Pythagoras. I use the word hypotenuse. I argue that in our example square, 5 √2 is shorter than 10. Yes, they say, but zigzags are not hypotenuses. “But if the zigzags were small enough,” I finally say, “it would eventually look like a straight line. At some point the zigzaggy route would become shorter, no?”
I get off my bike and lock it. I think about the myriad unnecessary turns I’ve made throughout my life, convinced that I was cutting my traveling distances: in Verona, New Jersey, where I spent most of my childhood; across the campus of my alma mater, Princeton; wandering around Tokyo and Yokohama (where streets are generally not laid out in grids and I was mostly just lost); driving from home to childcare to workplace and back in San Francisco; and now, biking through Davis, California with my family.
“Wow,” I say. “I think it’s possible that every decision I’ve ever made has been based on a fundamental misunderstanding about the world.”
My husband makes a sad face. “Every decision?”
“Yeah, mom,” my older son says, “Dad wants to know if you mean every decision.”
I’m thinking about the routes I’ve worked out around town: the one that goes right-left-right-left-right from home to the Safeway, or the left-right-left-right-left-right-left to the UC Davis library where I do a lot of my research. And how generally, metaphorically, I seem to take the longer, more complicated path. Like all the effort I spent not writing in my twenties and thirties. “A lot of decisions,” I say.
“I always thought you liked to zigzag because it’s just more interesting that way,” my husband says.
“Well, it is more interesting.”
“Exactly,” he says.
This is one of those moments when I am reminded of why I’m spending my life with this man: he often sees the best in me, even when it turns out I am acting out of ignorance.
After loading up on summer fruit and veggies, we take the more direct route home: straight north up one street, then due west to our house. On the way back, my older son says he’s worked out a proof for why a zigzag, even when the turns are tiny, even when the tininess approaches infinity, is still longer than a straight line. I don’t actually understand the proof, but I’m happy he’s worked it out. I like it when my kids are smarter than I am. Given the revelations of the morning, I should probably just say I like it that my kids are smarter than I am.
And I suspect I’ll continue to zig and zag my way through life. It’s more interesting that way, which is a good enough reason. But maybe in some non-Euclidean universe out there, it’s also a shortcut.
I think you invented calculus
Me, I would just go the route that gave me the fewest left turns. I hate left turns.
I loved this post, Naomi. I’m a zig-zagger — not with the idea of taking short cuts, but more from satisfying curiousity (wonder what’s down this street….) But I found all those arguments above simply dazzling. And your kids sound dazzling, too!
Thanks! And I’m beginning to wonder if the penchant for zigzagging correlates at all with gender….
Why not just use a stop watch, and see which ever one is faster than the other?
Love this, Naomi. I’m also a zig zagger and (sigh) fear that I always will be. Here’s to not sticking to the straight and narrow…
For what it’s worth, Naomi, I’ll give you that you are wrong about the math problem but you may still be right about biking to the veggie stand. It all depends on whether you go straight down the streets and make right angle turns or you cut corners and angle you way down the streets. You see, if you angle your way down each of two long streets you don”t save a lot compared to the length of the street (a small percentage). If you angle your way down each of ten short streets, however, then the angles become greater and the distance savings goes up as well.
Think of solving the problem first by putting all the horizontal streets together and then all the vertical streets. The first adds up to AB in your diagram, and the second to BC. Now add width to each street. To make it extreme, assume the width of all horizontal streets is AB/10 and the width of all vertical streets is BC/10, where street lengths are all measured from the middle of an intersection to the middle of the next intersection. Also assume that you always travel from one extreme corner to another, making a diagonal across the street. (Don’t do this in real life 🙂 Finally, assume the zigzag route is made up of 5 equal zigzags.
In this case, you never travel horizontally or vertically. Rather, you always travel at an angle. Furthermore, it’s always the same angle. So you can get the total but adding up the segments in the same way.
The AB – BC route is now root((AB-AB/10)^2 + (AB/10)^2) + root((BC-BC/10)^2 + (BC/10)^2)). If AB = BC = 10, this makes the AB-BC route = 2*(root(82)), which is very close to 18.
The zigzag route is now 5*(root((AB/5-AB/10)^2 + (AB/10)^2) + 5*(root(BC/5-BC/10)^2 + (BC/10)^2). If AB = BC = 10, this makes the zigzag route = 10*(root(2)), which is closer to 14.14.
Of course, this ends up being the length of the diagonal AC since you end up traveling that exact path given the numbers assumed. However, even with narrower streets and or fewer zigzags you will still save more distance following the zigzag path. For example, with only 2 zigzags, the zigzag route ends up as 2*(root((AB/2-AB/10)^2 + (AB/10)^2) + 2*(root(BC/2-BC/10)^2 + (BC/10)^2), which assuming AB=BC=10 gives 4*(root(17)), which is close to 16.
All of which is to say that your intuition may have been right even if your reasoning was flawed.
I love this, Chris. It puts a whole new “angle” on the problem! Watch out, Davis drivers & pedestrians. 🙂
Does your bike have an odometer?
No odometer. It’s just a running-errands-around-town bike. It hadn’t been a got-something-to-prove bike till Saturday!
As he probably already told you, Eliot is using something called taxi-cab geometry, and it is very interesting in the world of geometry and in the real world. May you enjoy whatever routes you take.
We did not know this term, Kathy, but were delighted to find, when we looked it up at Wikipedia, a graphic that is almost exactly like the one Julian made for me for this blog!
Naomi Williams, you are nothing short of being one of the most lovable human beings on the planet. Period.
Please, continue to use the word hypotenuse at will.
Back at you, Rae!