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

[VS63]

As the steps get smaller and smaller, each one can be defined as a right angled triangle, and therefore calculated out using Pythagoris Theorum. At some point the triangles will become so small that the length of the hypotenuse will almost exactly equal the length of the arc of the perimeter of the circle at that point. In fact, being a straight line, the hypotenuse will actually be minutely shorter than the arc.

It can readily be seen that at the 45degree position the corresponding triangle will have a horizontal and vertical size of 1 unit, and a total length of 2 units, whatever size that unit might be. Using Pythagoris, it can also be seen that the Hypotenuse of that triangle will have a length of sqrt(2) or approx 1.4142 units, which is obviously less than 2. In all cases of any triangles defining the perimeter of the circle, the hypotenuse will be less than the sum of the horizontal and vertical sides. Once all the hypotenuse' are added together, their sum should approximately equal PI, although as noted earlier, because the hypotenuse' are all contained within the arc of the circle, they will actually sum to slightly less than PI.

 

[Samuelb88]

The reason why the circumference of the square doesn't approach \pi as we change the shape of the square to that of the circle is that the edges doesn't touch the circle tangentially, as mentioned in earlier posts.

Let us zoom in sufficiently close to a section of the circle such that the section of the circle looks like a straight line. No matter how many times we repeat the process of changing the shape of the square to that of the circle, we will always be able to zoom in sufficiently close such that:

http://imageshack.us/photo/my-images/847/zoom.jpg/

Now if the edges touched the circle tangentially and we zoomed in sufficiently close to a given section, it would look like the section of the circle and the tangential intersection of the approximating curve would be "on top" of each other.

*Credit of the explanation should be given to my classmate, whose name I have forgotten.

** My picture isn't showing up on my screen, so for those who cannot see it as well, the picture is that of a straight inclined line and two lines, a vertical line that intersects the inclined line at 45 degrees and a horizontal line that intersects the inclined line at 45 degrees. It is just a simple picture of the circle and on of the fringes of the square whose perimeter has been changed to fit that of the circle zoomed in sufficiently small.

 

[0xDEADBEEF]

Wow mathematics knowledge here is terrible. There is a whole field of mathematics devoted to this. It is called the theory of measures. And this problem is not trivial at all.

1) The rectangle curve sequence does converge to the circle (for every epsilon there is a delta...)
2) The length of the rectangle curve is always 4
3) The "Manhattan metric" has little to do with it, as I can produce sequences of curves that have different lengths than the limiting length in that metric too.
4) "Tangentialness" and differentiability have no direct meaning for the result either (I could build a wiggling sine wave approach to the circle and get the same result)

This means that arc length must be defined in a non intuitive way in two dimensions.
The reason for this is that into dimensions there is always enough space for a two dimensional curve to have infinitely many wiggles of non zero length. This is studied further in fractal dimensions. A common way is the integral over the tangent vectors of the curve using the Lebesgue measure, but the Lebesgue measure is fairly abstract.

But I guess this doesn't really help. The main message is, that you have to know exactly what you are doing when dealing with infinities. If you really want some brain damage look up the Banach Tarski paradox. It has been shown that you can make one sphere into two identical ones by taking it apart and putting it back together.

 

[Dadface]

Why is such a simple problem being made to look so complicated? It has already been pointed out many times here that no matter how many right angled sections are used the total perimeter of the surrounding shape remains constant (see for example Dave's posts above)This result is simple and intuitive and all that is needed to see it is to spend a few minutes with pencil and paper.

 

[granpa]

well it shows that the length of the hypotenuse doesn't simply follow
from the length of the other 2 sides.
Instead it requires a special additional axiom.

 

[Dadface]

That may be so (I'm not sure what hypotenuse you are referring to) but the problem as originally set does not require any reference to hypotenuses.

 

[disregardthat]

4) "Tangentialness" and differentiability have no direct meaning for the result either (I could build a wiggling sine wave approach to the circle and get the same result)

Uniform convergence (almost everywhere) of the differentiated curves have everything to do with this. This is the essential property that fails which makes lim(length) =/= length(lim). This has been stated several times in this thread. That the non-differentiability in the corners doesn't matter has also been stted.

 

[Dadface]

Having quickly scanned through the latest posts on this thread it appears to me that an emphasis has been placed on area calculation which is a different problem to that originally posted which implied that pi=4 comes from perimeter calculation.

 

[DarthPickley]

here is an analagous situation: take the function given by the limit as a --> 0+ of the function f(x) = a*sin(x/a).
Clearly, this converges to the function F(x) = 0.

However, suppose that this is like physics, so you are going to talk about the energy of the wave (no quantum stuff here though! It's not a photon or any duality with a particle)
So, you say that the energy is proportional to frequency and to amplitude, so E = h*f*A, which in this case gives E = h*(1/a)*(a) = h. And suppose that the constant h = 1. then Energy E=1 is constant for all a>0.

However, the energy in the simple function that the function converges to as a approaches 0, which is F(x) = 0, will just be 0. F'(x) is always 0 as well.

So, just because the points in the curve approach 0, other properties of the curve, such as its energy (aka the maximum slope), are not also converged to in the model.

Here is another example, this time more relevant, because it uses perimeter of a circle:
you have the polar curve defined by r = 1 + sin(a*Θ)/a . As a --> ∞, r --> 1, and the curve converges to a circle.
However, it should be easy to see that for any finite, positive integer a, taking the path defined by r = 1 + sin(a*Θ)/a will be longer (and more windy) than taking the path r = 1 (a circle). (also it will be longer by a factor of about 1.2)
But the limit of these perimeters is not equal to the perimeter of the circle.

For something like that to work, I think it has to have the lines that are approximating it to converge towards being the same location AND direction as the curve is.
One way you could do this is you could take the original method of having lines that only go horizontal and vertical, and then just take the convex hull of that curve (basically wrap a string tightly around the rectangularized curve of perimeter 4, and find the length of that string)

 

[RationalPi]

Take a 1 cm diameter ring and a 3 meter length of fishing line.

Wrap fishing line tightly around the ring such that the ring is completely covered in fishing line.

The circumference of the pipe is equal to 3 meters.

Problem?

 

[DaveC426913]

Problem?

yeah. On several counts.

1] How does multiple wraps of fishing line result in the circumference? You do know what a circumference is, yes?
2] Where's the pipe come from? (bad copy editing I'd guess)