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I’ve never taken calculus (the last math class I took was an algebra class in high school in the late 1980s). But I read a description of calculus once that used the example of trying to figure out the length of a coastline and described how one could use smaller and smaller sections to estimate the length and this is what calculus is all about. But, this article doesn’t mention calculus at all, so now I’m wondering if my impression of what calculus is used for is all wrong.

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Your impression is exactly right. The difference lies in something called smoothness. Loosely speaking, a smooth curve is one where if you zoom in far enough at any point, the curve starts to look like a straight line. You might have to zoom in a very long way, but as long as you eventually end up with something that looks straight, it counts as smooth, even if the overall shape is very wiggly. Loosely speaking, smooth curves behave well when you use calculus on them.

The coastline paradox is a situation where that doesn't happen. No matter how far you zoom in on a part of the coastline, you always find more detail -- bays and peninsulas, cliffs and inlets, pits and protrusions, tiny bumps, and so on. And the more of that detail you try to include in your measurement, the more your measured length will increase. In mathematics, we say that the length diverges. If you were measuring a smooth curve, the length would converge on a single value -- incorporating more detail would change the length, but only by tinier and tinier amounts.

Now of course a real coastline is a very complicated thing, and there are lots of practical obstacles to measuring one very precisely. So at some point we stop talking about real coastlines and start talking about a mathematical ideal, a kind of perfect roughness that's the opposite of the perfect smoothness we use for calculus. Such a perfectly rough curve is called a fractal.

There are also other things that can break calculus -- pointy curves, discontinuities, and infinities can all cause trouble.

Great explanatory writing, thanks.

> In mathematics, we say that the length diverges. If you were measuring a smooth curve, the length would converge on a single value -- incorporating more detail would change the length, but only by tinier and tinier amounts.

Can you say something bout the exponential curve in this context? IIRC whatever level we zoom to we see the same curve, neither converging nor diverging. Is there something special we should take note of here?

I think you might be remembering something else -- an exponential curve is proportional to its own rate of change. Exponential curves are smooth in the way I described -- if you zoom in far enough, they look straight.

The part about being proportional to its own rate of change does make exponential curves very important in calculus (and thus in science in general). Sine waves also have this property (although in a more complicated way). This is why you see exponential decay and sinusoidal oscillation so much in physics.

The exponential curve does look like a line when you zoom in very close to a point. For example, f(x) = e^x looks linear with slope e^x near x. You can see that, for small epsilon, e^(x+epsilon) - e^x is approximately epsilone^x, with an error term of the order of epsilon^2 e^x

Calculus deals with things that are sufficiently smooth on a small scale, and thus converges. For example, approximating a circle with smoother and smoother polygons. The length of those polygons will converge to the circumference of the circle.

The coastline paradox is the classic example of a fractal; the dimension of a fractal is actually greater than that of the space it lives within!

The word converge always confused me here when learning English. It's not like the edges of a circular polygon "converge" even in a hypothetically perfect circle. They more....blend together, or lose their edge. Either way maybe I've never properly understood the term.

What converges is the resulting value of the calculation. As you add more lines the approximate result (of the circumference) gets closer and closer to the true answer. It's not necessarily referring to the geometric shape of the polygon. (Though you can also see it as, if you take any given point on the circumference of the polygon, as you add more segments, that point will get closer and closer to the true edge of a circle.)

Unless when it doesn’t, see the problem of “squaring the circle”. Somewhat related to the coastline paradox as well.

I think what you're referring to is the false proof suggesting that pi=4. That is not what squaring the circle is, and does not in fact become smoother and smoother but rougher and rougher

The Wikipedia article states it's not really fractal.

It's not really a fractal in the sense that a fractal is a mathematical object, and a coastline lives in the real world. As your ruler gets smaller, it also becomes impossible to define what even is the coastline anymore - is it the wetted sandline? On a sufficiently small scale, where is the interface between land and sea?

Thus at sufficiently small scale it does break down, but the fractal nature over a very broad range of scales is clear; I feel like the Wikipedia article is tripped into pedantry at that point :)

On a sufficiently small scale, I guess you would have to start worrying about quantum effects. So, you would still have trouble measuring the coastline maybe not a paradox anymore but still quite a pain.

> But I read a description of calculus once that used the example of trying to figure out the length of a coastline and described how one could use smaller and smaller sections to estimate the length and this is what calculus is all about.

According to https://en.wikipedia.org/wiki/Outline_of_calculus:

> Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.

The coastline paradox is definitely an example of something that interests calculus people, and to tackle the problem they most likely will use a bunch of concepts and theorems of calculus.

However calculus, as a field is not this paradox.

Calculus and why it doesn’t help here are mentioned in the “mathematical aspects” section of the article

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