My Favorite Paradox

Sometimes people make fun of my for my tendency not to wave to them when I’m walking about campus. And while it’s true that most of the time I’m daydreaming about just what I would do if at that moment I suddenly had a jetpack, occasionally it’s something interesting. Here’s a puzzle which used to rattle around my brain whenever I was walking to/from school as a little kid, which I still think of occasionally and in fact, have only just in the last year or two come fully understand.

Not surprisingly, this paradox pertains directly to the act of walking about campus itself. Being a bit of an impatient fellow (or perhaps more accurately, one who’s always just dangerously close to being late for something), I always try to find the fastest path between my where I’m going and where I’m coming from. It has the advantage of both getting there as quickly as possible and also giving me something to think about along the way.

Of course, as a little kid, the first trick I learned was that, whenever you have the choice, you should cut across the diagonal of the path instead of walking along the sides. It’s common sense, of course, and also just simple geometry: one side of a triangle is always shorter than the sum of the other two. So I still use this trick when I walk to class in the morning nowadays, by taking the path through the little patch of forrest instead of walking along the outside:

But sometimes, it’s not possible to actually cut straight across the diagonal;  for example, before I get to the patch of trees,  I have to make my way across a parking lot. Like the trees, I could just walk around the outside of the entire parking lot. But as a kid, I used to think, “Wouldn’t it be better to sort of weave my way through the parking spaces, approximating that ideal diagonal line?

If you don’t see why third-grade me might have thought that, it might help to imagine a case where there are slightly more obstacles in my way. Let’s say, for example, that it’s a weekend and so there aren’t any cars in the parking lot. But it’s Stony Brook, so it’s either raining or it has recently rained, in which case I will encounter the following mess of puddles when I head in to pick up the stack of notes I foolishly left in my office:

You can see now, I hope, that the more you increase the number of stairsteps in the path, the more it just begins to look like the diagonal of the triangle. So what to make of this? It it still a decent way to save time?

As a little kid, I foolishly thought so. It seemed to me that, if a true diagonal was the shortest path, and the square on the outside was the longest, then as you increase the number of stairsteps and gradually turn into a diagonal, the path must be getting shorter and shorter.

Well that’s not the case, as I think it took me a sold year or two to realize, if I remember correctly. But it shouldn’t take you that long, because (a) you probably realized it already, and (b) you have the benefit of my magic stick-figure drawings to help you out. Let’s take the stairstep pattern from the puddle picture above. If we remove the puddle, and then take just the horizontal bits of the stairstep, we can imagine sliding them all straight downward:

And of course you’ll see that the amount of horizontal distance transversed along the stairstep is exactly the same as if you go around the perimeter. Moreover, you can play the same trick sliding the vertical bits all to the right, and you’ll see that the total distance is exactly the same. The staristep is just a re-arrangement of the horizontal and vertical segments that make up the perimeter path. And since I could play this game with any stairstep path, regardless of now many steps are involved, the fact is that even though the stairstep seems to begin to look more and more like the diagonal, it is always the same length as the path around the outside. The true diagonal, by contrast, is always the shorter path. What’s more, it is often substantially shorter; for example, if the length of each “outside” path is, say, 100 yards, then every stairstep path (regardless of the number of steps!) requires 200 yards worth of walking to traverse, where as a true diagonal takes only √(200) = 141 yards of travel, or a full 30 percent less.

So how can this be? You might be inclined to mumble something about how you can never actually walk a stairstep with 1000 steps in real life, because the steps would be smaller than the size of your feet, and thus you’d actually end up walking a diagonal, or something like that. That’s what I did for many years, actually, every time I needed to explain the paradox to myself. But that doesn’t really get at the heart of the matter; after all, you can always imagine a smaller little critter (an ant, say) and ask why his stairstep path is so much longer even though it looks like a true diagonal to us*. Even if you got down to the atomic scale, it doesn’t remove the heart of the paradox, which is basically the question of how nature knows to make a true diagonal suddenly so much shorter than a stairstep with a nearly infinite number of steps** if the length isn’t gradually decreasing as the number of steps goes up. After all, to our untrained eyes it looks like as you add steps, you gradually turn into a straight line.

No, even though I thought for the longest time that it had to do with the fact that there was no such thing as infinitessimal motion in the real world  (and was therefore related to Zeno’s paradoxes), I discovered in the last couple years that it really has to do with an odd fact of basic mathematical analysis applied to differential geometry. The secret is this: mathematically speaking, the length of a path has nothing to do with it’s “shape.” You can see this by imagining that one day (after a particularly dizzying ride on a homemade merry-go-round, perhaps) I decide to take a foolish curlycue-path to school:

of course , can make this path as long as I like just by adding more and more pointless loop-de-loops. And yet the overall “shape” still looks just like the diagonal. Unfortunately, neither mathematics nor mother nature herself have this (faulty) intuition that you and I have, that two paths that look alike must necessarily be alike in other aspects as well. As I add more and more stairsteps (or curlycues) the paths may begin to look more and more like a diagonal shape, but “shape” is a so-called “global” property–it depends on all of path at once. “Distance,” in a strict sense, depends on the properties of each infinitesimally small segment of the path, which are then all added up. And the two aspects don’t really need to know anything about each other in order to get along.***

So why do we have this rather misguided sense of how paths should behave? Because they often behave like this. If I walk to school along a sort of wobbly line like so:

 

 

 

 

 

and then imagine slowly flattening out the wobbles, then both things will happen: the path will begin to resemble the diagonal in shape, and also in length.

 

Without getting too technical, this works because the path is nice and smooth, and the way I’m changing it each time is nice and smooth (unlike adding stairsteps, which is a rather pointy procedure). Because most everything in life follows smooth paths that change in smooth ways, we can’t help but think that this is the only way that things can happen. Indeed, it’s been bred into us as part of our basic neurological structure for comprehending how the world works.

But mother nature knows things that we don’t because she gets to know all the details. And one of those details is that by changing a path in a rough, pointy way you can change its shape without changing its length in the same fashion.

Touché, mother nature. Touché.

 

UPDATE: I should say that I am not in any way claiming to have “discovered” this paradox. Although I do take pride in having thought of it independently, it’s well-known among mathematicians as the “Diagonal Paradox.” See here.

__

*No fair also saying that I should just cut diagonally across the spaces between the cars, only moving horizontally or vertically when I absolutely have to. Of course that would be the next best thing to a true diagonal, but it ruins the fun of the paradox!

**or indeed, even a countably infinite number of steps!

***The mathematical way to say all this, of course, is that the profiles of a sequence of curves can converge uniformly to the profile of some limiting curve, but that this does not imply that the lengths converge to the length of the limiting curve (they may not converge at all). Curve lengths are NOT automatically passed under limits–only for smooth, well-behaved curves. The stairstep, of course, is not differentiable everywhere, and in fact becomes nowhere differentiable in the limit as it becomes the diagonal. So it’s simply wrong to expect the lengths to match up. They are two different curves–at least, as seen under the L2 norm. Interestingly, the two paths DO have the same length under the L1 norm, and it’s worth asking why it is we feel that the L2 norm is the “real-world” distance norm.

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About Colin West
Colin West is a graduate student in quantum information theory, working at the Yang Institute for Theoretical Physics at Stony Brook University. Originally from Colorado (where he attended college), his interests outside of physics include politics, paper-folding, puzzles, playing-cards, and apparently, plosives.

One Response to My Favorite Paradox

  1. derek says:

    i often ponder this because of the shape of the paths available to walk on my campus ( check out the aerial view here: http://maps.google.com/maps?f=q&source=s_q&hl=en&q=butler+university&sll=39.83745,-86.170921&sspn=0.004828,0.009645&ie=UTF8&t=h&rq=1&ev=zo&split=1&radius=0.31&hq=butler+university&hnear=&ll=39.83745,-86.170921&spn=0.004828,0.009645&z=17)

    i often had to walk from the figure 8 shaped building, or alpha shaped building (half light, half dark) to the buildings around the area labled “west mall”

    my roomates and i had a nearly 3 hour discussion about it sophomore year that ended in frustration and tears for some.

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