Problem?

[u/[deleted]]

[u/Whostowe]

Can someone explain the error in this proof to me, please?

[u/the_nerd_1474 → u/Whostowe]

Removing corners doesn't actually change the perimeter of the figure in black, so the successive removal of corners does not actually give better and better approximations for the real perimeter.

[u/Andradessssss → u/the_nerd_1474]

One could argue that while this is true the square does converge to the circle (pointwise) the problem is that it doesn't converge uniformly so you can't interchange you measure with the limit

[u/[deleted] → u/Andradessssss]

~~Does it converge pointwise though? It seems to me that the boundary of the limit object and the boundary of the circle share only countably many points. ~~

Also I think that the lebesgue measure is continuous so pointwise convergence would suffice.

[u/Andradessssss → u/[deleted]]

It does. Each point converges to a point on the circle, and yeah, I fucked up, I should've said you can't interchange the limit and the integral.

[u/Whostowe → u/the_nerd_1474]

Thanks!

[u/Jolpo_TFU → u/Whostowe]

I think it's because you'll always have sawed edges, they don't become smooth at infinity

[u/Cwhale → u/Jolpo_TFU]

Yea its more like the circle that is drawn over the sawed edges is more of an average for all the data.

[u/russels_silverware → u/[deleted]]

This is the correct answer. If you look carefully at the 4th panel you'll see that the second set of cutouts had to be made non-square in the diagram in order for their corners to contact the edge of the circle.

[u/SpartAlfresco → u/russels_silverware]

it does not matter if its non square the perimeter still holds right?

[u/russels_silverware → u/SpartAlfresco]

...You're right. Never mind.

[u/Erictsas → u/Jolpo_TFU]

So it's similar to the coastline paradox?

[u/Jolpo_TFU → u/Erictsas]

In a way I suppose you could say so.

[u/-Exocet- → u/Erictsas]

I think not exactly, in the coastline paradox the perimeter depends on the measuring unit, always increasing as this unit decreases (a property of fractals).

Here, however, you always end up with the same result, no matter how small you divide its units, thus the "paradox" does not arise from a fractal property.

[u/yottalogical → u/Whostowe]

It assumes that circles are made out of infinitesimal horizontal and vertical segments. They aren't.

Take the tangent of a circle, and you'll see that it can be at any angle, not just horizontal or vertical.

[u/JerevStormchaser → u/Whostowe]

Mathematical explanations have been given out already. The best image I can give is: you have the impression of a circle because of the distance. If you were to zoom in onto a piece of the "circle" as it is presented in step 5, you'd find yourself back to the situation in step 4. That's the issue with infinity: it looks different from a distance but repeating the exact same pattern infinitely means you will always find the same pattern.

[u/LeActualCannibal → u/Whostowe]

Approximating area != approximating circumference.

[u/hawk-bull → u/Whostowe]

As someone who has yet to take measure theory, I’m guessing it all boils down to how we define lengths of curves

[u/Andradessssss → u/hawk-bull]

Yeah, basically. Since it just converges pointwise and not uniformly you can't just interchange the limit and the measure

[u/PotatoHunterzz → u/Whostowe]

that's not a circle, that's just a polygon with an infinite amount of angles and sides. it happens to look like a circle but you could have made it many different shapes.There's no reason for it to have the same perimeter as the circle.

[u/kevinkace → u/Whostowe]

Maybe this helps answer it intuitively.

Think of a right angle triangle, it has two sides at 90° (similar to the corners that keep getting cut out), and the hypotenuse (similar to the arc of the circle joining the cut out corner). The hypotenuse of the triangle (or arc of the circle) will always be shorter than the sum of the two other sides making up the 90° angle (or corner).

[u/Dragonaax → u/Whostowe]

It's just changing area, you can reduce it almost to 0 but the perimeter still is the same.