I don't get why this troll physics is wrong.

Yuqing

As I've mentioned, for each iteration of removing squares (which I will call cycles), the area will differ. (i.e. the second and third squares removed will be smaller than the first square removed). In fact it gets more complicated than that. For each removal cycle you will remove 2 squares near the edge of the quadrant and you will remove squares interior to the quadrant. These squares will also have different areas in fact. This effect will build up and for later cycles there may be several different sizes in a single cycle (the two most interior squares share an area, then pairwise as you progress outwards of the quadrant). You cannot simply take every square to be the same size.

guss

My equation takes all of that into account, except for the fact that the squares change size. If someone could derive the right equation I'd love to see it.

Yuqing

But the fact that the squares change size is precisely what makes it difficult.

guss

I know.

You can think of the squares being removed as being in sort of rings of size. The smaller ones are closer to the edge, the next smallest are one in, and so on. The amount of rings in a certain layer can be given by 2(n-2k)-1 where n the n value discussed before and k is a constant that decreases incrementally.

Thinking about it like that was easier for me, but I still can't get it.

olivermsun

It seems as if a proof of the "nonconvergence" of the jagged lengths to the circumference could be constructed by looking at the ratio of "Jaggy Lengths" to the circumference C. The jaggies between two touches on the circle are always two legs of a right triangle, while the hypotenuse is the chord length (which should converge to the arc length). I realize this "triangle" fact was pointed out a few times earlier in the thread, but my point is mainly that you don't need to explicitly evaluate the Jaggy Lengths to prove the inequality.

FlexGunship

Uh, not to oversimplify a discussion that is certainly interesting, it would be enough to say that the boundaries of the blue-area will never converge to be tangential to the circle except on the four cardinal points.

Without solving for the area of the blue-shaded region, you can at least conclude that the blue shaded region will never converge with the surface of the circle (no matter how many times the process is repeated).

atomthick

The problem with this proof for pi = 4 is that no matter how small the little squares get, the ratio of the length of the arc and the sum of the square edges that encompases it is always the same pi/4 no matter how small the square becomes. And from the triangle inequality we get that pi/4 < 1 therefore pi can not be equal to 4!

If the ratio was to converge to 1 then the proof could have been correct.

Ryumast3r

To put the above posts into "time to teach a 6th grader why pi = 3.14 and not 4" speak:

It doesn't equal 4 because, no matter how many times you cut the squares smaller and smaller, there will always be area of the squares that is not touching the circle.

Here's a little picture I drew in paint to help visualize:



In the above picture, there is a section of it that is red. You'll notice that this area exists on the other pictures as well (but it's blue, I believe). This area, no matter how many times you cut away at the square, still exists to an extent. At some point, it's a very small extent, but the upper-right corner of the square will never touch the circle, there will always exist 2 more upper-right corners for every single upper-right corner that you cut away.

Eldar

I agree with you completely, but the diameter is specified as being 1 and not 4, the perimeter is 4 :P I'm sure you realised this but just had a quick lapse and mistyped.

Unrest

I don't think that counts as a reason. The area keeps getting smaller, and it approaches something, presumably it approaches the area of the circle.

The pi=4 comes from the perimeter, not the area. The perimeter seems to remain 4 no matter how many squares (or rectangles, which is what the original picture shows) you cut away.

But how's this for a simple proof:

pi < 4 (already been proved)
therefore pi doesn't equal 4

Ryumast3r

See below as well, What I meant is that there is area unaccounted for (though it becomes infinitesimally small), which also means that there is a section of the perimeter that does not touch the circle.

Put more simply what I was trying to say: Every time you cut a corner, 2 more appear that do not touch the circle, ergo the perimeters of both objects will never touch at every point.

Yeah, that's what I meant. I had a test and... well... you know what happens after tests. :P

Femme_physics

Great explanation, actually. *thumbs up*

However, I will definitely use it to mess with ppl! :)


Edit: Wait, I'm confused. If this fails to prove that an area is really the sum of infinitely small rectangles, why should I trust taking the integral to give me a correct approximation?

D H

The jagged curve *does* converge to the circle in almost every¹ sense of the word "converge". The area *does* converge to that of a circle. The distance between any point on the jagged curve and the circle *does* converge to zero.

Think of the upper (or lower) half of the jagged curve / circle as a function of x. While the jagged semi-curve converges uniformly to the semi-circle, the derivatives of those curves are miles apart. In fact, the jagged semi-curve is nowhere differentiable in the limit N → ∞. Just because two functions converge to one another does not mean that their derivatives, or their lengths, do so.


¹Addendum: Well not quite every sense. The jagged curve does not converge smoothly to the circle. The jagged curve is nowhere differentiable in the limit N → ∞ while the circle (upper semicircle) is infinitely differentiable almost everywhere.

hillzagold

It's the idea that with every step, the difference between areas of the jagged figure and the circle is constantly shrinking, until the difference is zero, and also that the jagged figure and the circle are the exact same shape. Because the jagged shape always has a perimeter of 4, the circumference of the circle is also 4, and not 3.14.

Without getting into differentials and tangents and other topics more advanced than the thought that went into the trolling, I prefer the simpler answer that the jagged shape never does become a circle. In math it's called an infinitesimal, where 1/[tex]\infty[/tex] (or any other mind-bogglingly small number) is not zero. The difference between the jagged shape and the circle keeps shrinking, and you can choose to disagree with me and believe whether or not the difference becomes zero. I'm no math professor, so I won't claim to be more than pretty sure about this.

disregardthat

Approximation by polygons will also never become the circle, but still the perimeter of the polygons will approach the circle. The main problem here is that one will not be certain that length is preserved under the limit operation unless the (almost everywhere) derivatives of the sequence curves (such as the jagged curves in our example) has a (almost everywhere) continuous (maybe simply integrable, not sure) limit. If this condition is satisfied, and it is for polygons, we will have length preservation. And this is why we can trust the approximation by polygons, as someone commented on earlier.

Unrest

I think you're mixing up area with perimeter by using the vague word "shape". You can't say "the difference between these two shapes is less than the difference between those other two shapes". Difference implies subtraction, and subtraction isn't defined on "shapes". This confusion is what the original picture plays on. The area of the jagged shape does approach the area of the circle, but the length of the perimeter doesn't even begin to.

Here's an even more extreme example of how our intuition about the relationship between area and perimeter length doesn't work: Koch snowflake

guss

How do you know it is true that the shape never becomes a circle?