Cutting corners (or: "math, ugh!")

Factual Questions
1

So. I’m standing on one corner of a square (on the proverbial “point A” we’ve heard so much about). I walk along one side, then turn left and walk the next side, arriving at the opposite corner (say, oh, “point B”). I’ve walked a total of 2 units. Fine.

Now, say instead I am a crow, and I fly from A to B. Any moron (that’s where I come in) knows that the distance as the crow flies from A to B is Sqrt(2). Fine.

Now imagine (I promise this will end soon) that instead of walking all the way along both edges, I walk 1/2 way, then make a 90 degree left, walk a bit, make a right, walk a bit, make another left, etc. until I hit the end, always making 90 degree turns. It seems like this way I would end up walking 2 units also, I just broke them up into a different order of left and right turns.

OK. Now, finally, imagine you know calculus and I don’t (not far from the truth as it turns out). :slight_smile:

No, seriously, now, finally, imagine that I make more and more “left-90, small step, right-90, small step” maneuvers, with ever-increasing fineness. Sort of like the old thing about walking 1/2 way to the end, but we’ve done that before, this one is different. :slight_smile:

So explain to me why, as the lengths of my left- and right- turning segments get shorter and shorter, they don’t approach the limit of sqrt(2), they stay 2, and then at some point – boom! – it’s sqrt(2).

2

I’ll attempt an explanation – though I think you’ve hit on an essential part of calculus that is both as simple and yet apparently difficult to understand as most of mathematics.

While it may look from one point of view like your path is approaching the diagonal, it really isn’t. You’ve just changed the scale, which doesn’t change the ratios of the distances.

At the infinitesimal level, the orthogonal steps (left, right, etc.) will still be the legs of a triangle, and the (infinitesimal) diagonal will still be shorter than their sum. Imagine if you were a giant, and the whole big square that we started with appeared to be a tiny step of an enormous square. The diagonal of this square is of course shorter than the sides (as already agreed), we’re just looking at it from farther away. That can’t change what’s really going on.

The point you’ve realized is that calculus is not just approximations, but a means of simplifying to get an exact answer. Numerical methods, on the other hand …

3

Why? I don’t know how to answer a “why” question with a mathematical proof. Why do six pennies exactly surround another penny with their edges all tangent? I dunno, that’s just the way the world is put together.

Maybe what you’re looking for is this: The distance stays at 2 and then boom shanges to sqrt(2) because the function that describes how the distance changes has a discontinuity at x=2. That is,

lim f(x) .ne. f(2)
x->2

4

A very interesting question. I guess the reason limits don’t apply here is that you’re doing two different things. In one case you’re walking north one unit and east one unit, albeit in an increasing number of smaller and smaller chunks. The sum of your north paths is still one unit, and the same is true of your east paths. In the other case you’re walking northeast sqrt(2) units. You’re not walking north or east at all. However your path would start to appear closer and closer to the diagonal, which is what makes the question so interesting. It seems like limits should apply, but in fact they don’t.

City Gent, what discontinuity do you mean? What is x and what is f(x)?

5

You’ve asked the question backwards. Why SHOULD the sum of the lengths of your segments approach SQRT(2)?

There is no reason to believe that the sum of the lengths of your various sets of stair steps should approach SQRT(2). In fact, it is obvious that they don’t. Each set has a length of 2. The limit of the sequence is trivially seen to be 2, not SQRT(2).

The flaw that prevents this from being a paradox is in your statement “… they stay 2, and then at some point – boom! – it’s sqrt(2).”

But, there is no point at which the length of the stairsteps is SQRT(2). There is no boom. No matter how small you make the steps, the sum of their lengths is always 2, never anything smaller.

It’s tempting to suppose that the problem is those sharp angles in the step or tooth function. But your example could have used semicircles. Consider a semicircle spanning a line segment of length 2/pi. The semicircle has a arc length of 1. Now consider two semicircles, half the size of the original, each spanning half the segment. Each has a arc length of 1/2; total length 1. Keep halfing the size and doubling the number of semicircles. The total length will always be 1, but the tiny semicircles will appear to approximate the length of the segment, 2/pi. But they don’t, and there’s no reason to believe they should.

The moral here is, when dealing with the infinite or the infintesimal, never trust your intuition. It will almost always be wrong.

6

The small step segments are in a different coordinate system than the diagonal. You can’t compare the integration of the steps with the diagonal distance, because the coordinate system of the diagonal distance is rotated 45 degrees from the step coord system which is straight up and down (x,y).

7

As a consolation prize, there is a similar problem that does, in fact, go to the expected sqrt(2). In your version, at each corner, you turn 90[sup]o[/sup] right, then 90[sup]o[/sup] left, and so on. Now suppose that, at each corner, you turn 89[sup]o[/sup], or 88, or 87… Now, whenever you decrease the angle, the path approaches a straight line, and the distance does, in fact, approach the straight-line distance.

8

Thanks all!

And Chronos, thanks for the consolation prize. As usual, I’ll take the rest on a gift certificate.

9

Heck, I’ll add another perspective, which may help (or may make things more confusing… but that’s fun, too).

The key is that by making smaller stair-steps, you are cutting down on the actual amount of inches that the length of each stair-step exceeds the length of its diagonal. [if your stair steps are one unit long, their total length is 2, while the diagonal is 1.4, so the stair is 0.6 units longer. If they’re one-half unit long, the numbers are half of that, so 0.3units longer]

However, you’re also increasing the number of stair steps so when you add up the [shorter] extra length over the larger number of steps, you still end up having the same total extra length, no matter how small you make each step.

The reason it intuitively seems as though the smaller-step route is shorter is because if you were walking the stair-shaped path, as the segments got down to the length of your legs, you would start cutting across diagonally naturally. And when the segments got down to the size of your foot, you couldn’t walk along the stair-shaped path, even if you tried. So in real life, your route would be shorter as the stair-steps got smaller, but only because you wouldn’t be following the stair-shaped path exactly any more.

Next week, we can discuss walking along a fractal path!