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

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OmCheeto

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That does not mean the limiting function has an infinite number of jags. That is faulty logic. It just means that the number of jags are increasing without bound for the functions in the sequence. The limiting function need not resemble these.

Consider a similar variant along these lines: You iteratively pick some arbitrary number from the stack of rational numbers. Each pick leaves a new disjoint interval in the rational number line. As you continue, the number of disjoint intervals increase. Still, the "limit" of this process will exhaust the rational numbers since they are countable, leaving no such intervals.
Exhaust?

Please.

I may be daft, but I'm not stupid.

Where is the wizard?
 

disregardthat

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Exhaust?

Please.

I may be daft, but I'm not stupid.

Where is the wizard?
Don't you know the word?
Definitions of exhaust on the Web:
consume: use up (resources or materials); "this car consumes a lot of gas"; "We exhausted our savings"; "They run through 20 bottles of wine a week"
run down: deplete; "exhaust one's savings"; "We quickly played out our strength"
use up the whole supply of; "We have exhausted the food supplies".

Please explain what genuine questions you have if any.
 

disregardthat

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No, its not that simple. The limit of the steps is a curve that converges uniformly to the circle but is nowhere differentiable. In short, it is not the circle.
If you are right then that is if you insist on the induced parametrization of the circle. I may have been sloppy and written path when I meant curve and vice versa. Sometimes it matter. Only the range (curve) of this limit function is relevant though, and that is the circle. Why would you say "it is not the circle" when you say it converges uniformly towards the circle? The circle does not depend on any parametrization and does not have a preferred (differentiable) one.

As far as I know the length of a curve is defined independently of any particular parametrization, and is only applicable if a differentiable parametrization exists. Hence the length of the limit curve is not necessarily calculated using the induced parametrization. If I have understood you right.
 
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OmCheeto

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Don't you know the word?
Definitions of exhaust on the Web:
consume: use up (resources or materials); "this car consumes a lot of gas"; "We exhausted our savings"; "They run through 20 bottles of wine a week"
run down: deplete; "exhaust one's savings"; "We quickly played out our strength"
use up the whole supply of; "We have exhausted the food supplies".

Please explain what genuine questions you have if any.
Since there are an infinite number of rational numbers, how long do we have to wait before they are exhausted? Or is that the next question? "Prove that there are an infinite number of rational numbers."

I must say, is General Math always this lively?:smile:
 

disregardthat

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Since there are an infinite number of rational numbers, how long do we have to wait before they are exhausted? Or is that the next question? "Prove that there are an infinite number of rational numbers."

I must say, is General Math always this lively?:smile:
Note that is said the limit of this process. A limit is never "reached". I don't see what you are getting at.
 
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The correct way to approximate pi would be to take the diagonals of the "bumps", which are illustrated in the attached image (View attachment Pi aproximation.bmp). Now you add up all the diagonals' lengths and then you get some number under pi. :smile:
 
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Maybe I'm not seeing it, but I picture repeating this process "infinitely" just yields a diamond or sideways square, never a circle.
 

lurflurf

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The main point (what makes circumference difficult at times) is that circumference is a local property. That is a small (in some metric) deviation can cause a large difference in circumference. Our jagged circle thing is almost a circle in some sense, but it has a very different circumference.
 

coolul007

Gold Member
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An interesting book containing "proofs" that are wrong is:

The Mathematical Recreations of Lewis Carroll: Pillow Problems and a Tangled Tale

He is a master at them.
 
Limits don't always commute with other operations.

In other words, if s_i is a sequence, it is it not necessarily true that f(lim s_i) = lim f(s_i). In this case in particular, the limit of the perimeters isn't necessarily the perimeter of the limit.

Why is this? It just is.

Let's put it this way. You expect there should be a law that says the perimeter of the limit is the limit of the perimeters. But you can't just assume it's true. You have to PROVE it's true.

The problem is that you can't prove it to be true, because it's false. Why is it false? Because we can find a counter example. What counter example is that? The circle-square limits.

Another trivial example is to take the sign function: sgn(x) = 1 if x is positive, -1 if x is negative, or 0 if x is 0.

Now, take the sequence s_i = (-1)^i (1/i). So the sequence starts off -1, 1/2, -1/3, 1/4.

The limit of s_i is 0, because the numbers get closer and closer to zero, as close as we want if we go far enough. Thus, sgn(lim s_i) = 0.

BUT, the sign of each element in the sequence alternates. sgn(s_1) = -1, sgn(s_2) = 1, sgn(s_3) = -1. In fact, the sequence sgn(s_i) doesn't even converge! It has no limit. Thus, sgn(lim s_i) cannot equal lim sgn(s_i).
 
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Maybe I'm not seeing it, but I picture repeating this process "infinitely" just yields a diamond or sideways square, never a circle.
It does if you cut out squares, then you can get another trick about the lengths of sides of a square or right-tiangle, like alphachapmtl said.

But here the diagram shows you have to use rectangles so the new internal corners touch the circle.
 
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EDIT: THIS IS WRONG. Working on the correct one now.

There is always going to be area between where the "ridges" are and where the outside of the circle lies. That area is given by:

[tex]A = 1 - \pi\ - \frac{4n+4}{n(1 - 2\sqrt{2})}[/tex]

Where n is like this (and A is the blue area):
QH6Uk.png


The limit of the above function as n -> infinity is .0461.

P.S. My constant of [tex] 1 - \pi\ [/tex] may be off. If someone wants to check/correct me on that equation that would be great, I'm a little tired at the moment. If not, I can go over this tomorrow.
 
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Deveno

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it is not true that lim(lengths) = length(limits).

the (euclidean) length of a curve is not well-defined on the set of all possible curves. the notion of length, can be counter-intuitive, it depends on 2 things: what you are measuring, and how you measure it. the various fractal curves give examples of how length can be "worse than it looks" (the koch snowflake, for example. its length doesn't appear to be infinite).

if you were to produce a specific parametrization (piece-wise) for the "boxed" approximation, you would find that in the limit, it is not differentiable. the differentiability of the parametrizations are a key hypothesis in proving this independence.
 

disregardthat

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if you were to produce a specific parametrization (piece-wise) for the "boxed" approximation, you would find that in the limit, it is not differentiable. the differentiability of the parametrizations are a key hypothesis in proving this independence.
Not only that, the derivatives of the parametrizations must themselves converge uniformly to a continuous function. At least this is a sufficient condition, and counter-examples to "limit(length) = length(limit)" where this is not satisfied (while your condition is) exists. E.g: One could easily make the jagged lines around the circle smooth by substituting the tip with a quarter of a circle or something similar, but the lengths would still not converge to the correct value.

And it is not true, as suggested by someone eariler, that the jagged curves around the circle does not converge to the circle. The curves does converge uniformly to the circle, the problem is just that the lengths does not.
 
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Ugh, finally got it. Maybe. Could someone try to do this out themselves?

n is the number of points in each quadrant where there is a point touching the circle. A is the area of the blue. See below image for sample. Basically, there is always going to be area surrounding the circle that is blue.
VgJkD.png


Equation:
[PLAIN]http://www4b.wolframalpha.com/Calculate/MSP/MSP67019f5i3id7f23f46800006a6bfeede1b8g0ih?MSPStoreType=image/gif&s=30&w=215&h=61 [Broken]

lim x->infinity = 1-(pi/4) = original area of blue when n=0. But I don't understand how that could be the case. I think the above equation must be wrong. I'll have another look tomorrow. I suppose it could make sense when there are thousands of points touching the circle, but I don't think so.

Graph:
[PLAIN]http://www4b.wolframalpha.com/Calculate/MSP/MSP96719f5hfca6ccfc80700004fie2ha6eb8fca2b?MSPStoreType=image/gif&s=10&w=300&h=166&cdf=Coordinates&cdf=Tooltips [Broken]
http://www.wolframalpha.com/input/?i=y+=+(4-pi)/4+-+(x+(x+1)+(sqrt(x^2+1)-x)^2)/(2+(x^2+1))+from+x=0+to+x=5
 
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I'm quite interested in seeing a derivation of that formula if it's correct. From my knowledge this is quite a difficult problem. In each removal, the areas of the squares removed are non-uniform. For example, for the third "cycle" (n=7 according to your notation), the area of the squares removed at the edges of each corner are larger than the squares removed at the interiors of the corner and this effect builds up as n increases. Did you manage to account for that?

edit: Upon reading your post again, it may seem like I've misunderstood. That formula seems to suggest that the area of the blue surrounding does not decrease which is certainly not the case. I believe that the area of blue surrounding does converge towards the circle and its just the perimeter that does not.
 
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Alright, I'll post the derivation I did. I'm almost 100% sure there's a mistake in here, so please try to help.

Let's say that n is the number of times a white "corner" touches the black circle in one of it's quadrants. See previous image above for a guide. For simplicity's sake, let's only focus on one quadrant for now. We can multiply it by 4 to get the total area lost later.

Now, to get the area lost (in each quadrant) by having n number of touches (in each quadrant).

I THINK we can break the whiteness up into even squares. The amount of squares along the top row is simply n, and the amount of squares along the outside column is also n. This means (I THINK) we can get the total number of squares for touches n by using the triangular number formula (but I could be wrong because the circle being a curved surface might change this).

[tex]\frac{n^2+n}{2}[/tex]

So that is the total number of squares lost in each quadrant. Now, let's get the area of each of those squares. Let's focus now on the corner-to-circle touch on the edge of the quadrant. (in quadrant 1, it would be the farthest right touch).

First, we should simply get the angle on the circle where this touch is. cos(theta) gives us the distance from that touch to where the top edge of the original square was (if n=0 and there were no touches). sin(theta) gives us the distance from that touch to the right edge of the original square was (in quadrant 1, but that's all we have to worry about for now). That means, to get theta, all we have to do is:

[tex]n(sin(\theta)) = cos(\theta)[/tex]

Now, let's get the distance from where theta lies on the outside of the circle to the edge of where the original square would be. This will give us the length of each side of each square. This equation is:

[tex]L = \frac{1}{2} - \frac{cos(\theta)}{2}[/tex]

After some simplifying and combining the previous two equations, we get

[tex]L = \frac{1}{2} - \frac{n}{2\sqrt{n^2-1}}[/tex]

The original area with just the original square and circle is just 1- pi/4. Let's square the L in the equation above to get the area of each square, then multiply it by the triangular number formula (giving us the total area subtracted from each quadrant), then multiply that by 4 to get the resulting area subtracted from the entire figure. After all this, we get this:
WolframAlpha equation

I'm almost certain my math is correct, it's just the concept. I'm not sure if we can count every square as having the same area, and I'm not sure if we can simply use the triangular number equation due to the nature of the circle being curved.

If someone could take a look at this that would be great, because I'm stuck, and not to mention kind of sick of working on this problem myself.

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

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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

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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).
 
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.
 
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