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Calculus deals with things that are sufficiently smooth on a small scale, and thus converges. For example, approximating a circle with smoother and smoother pol

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!

Solvency 3y ago
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.
- yuriks 3y ago
  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.)
infradig 3y ago
The Wikipedia article states it's not really fractal.
- deleted 3y ago
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- Remnant44 3y ago
  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 :)
  - bee_rider 3y ago
    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.
  - deleted 3y ago
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fellerts 3y ago
Unless when it doesn’t, see the problem of “squaring the circle”. Somewhat related to the coastline paradox as well.
- ravi-delia 3y ago
  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

Solvency 3y ago

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.

yuriks 3y ago
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.)

infradig 3y ago

The Wikipedia article states it's not really fractal.

deleted 3y ago
[deleted]
Remnant44 3y ago
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 :)
- bee_rider 3y ago
  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.
- deleted 3y ago
  [deleted]

fellerts 3y ago

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

ravi-delia 3y ago
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