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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

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.

nonrandomstring 3y ago
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?
- luca3v 3y ago
  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
- AdamH12113 3y ago
  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.

nonrandomstring 3y ago

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?

luca3v 3y ago
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
AdamH12113 3y ago
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.