Anti-proof by demonstration:

construct physical circle, diameter = x.
construct physical square over circle, length of side = x.

Wrap string around physical square exactly once. Cut to length.
Wrap string around physical circle exactly once. Cut to length.

Compare strings.

Quote: Original post by JoeCooper
So I guess what I'm trying to say is that it doesn't matter what the flaw in it is; the important thing is that it only looks like there isn't a flaw because through our limited perception it looks circleish.

That makes more sense to me if I never took calculus. I spent 3 years learning that little rectangles = curved lines.

Quote: Original post by samoth
You wouldn't want to doubt the guys who built the Great Pyramid... [snip] ...and an awful lot of slave hands.

Apparently the pyramids were not built by slaves at all. Have a look at the Who Built the Pyramids? section here.

Quote: Original post by ChurchSkiz

Quote: Original post by JoeCooper
So I guess what I'm trying to say is that it doesn't matter what the flaw in it is; the important thing is that it only looks like there isn't a flaw because through our limited perception it looks circleish.

That makes more sense to me if I never took calculus. I spent 3 years learning that little rectangles = curved lines.

no you didn't. You learned that little rectangles approximate areas involving curved lines. At least I hope that's what you learned.

Quote: Original post by szecs

Remove more corners, perimeter is still 4.

That's the trick. The relation between the side and diagonal of a square isn't rational.
The act of removing the corners implies reducing the perimeter of the resulting shape.

Could anyone draw the formula that represents "removing the corners" from the square so we can appreciate the resulting error?

Quote: Original post by Way Walker
But these criticisms also apply to circumscribing regular polygons

That was the exact previous thing I went into. There isn't a correct method. Just approximations of varying usefulness. Pi = 4 isn't normally going to be a useful figure.

As for the shape, we can plain see that its area does approach that of a circle (implying the familiar Pi value) even if the perimeter is artificially inflated.

With Pi=4, you can't use its diameter to find its area. As long as we agree that Pi is supposed to relate the diameter and area, that's a problem.

But if you work it the other direction, you actually can find Pi from its area.

Quote: Also, because "I said so" isn't a satisfying mathematical explanation.

Excuse me, but that's not what I said.

[Edited by - JoeCooper on December 2, 2010 1:35:29 AM]

Quote: Original post by JoeCooper

Quote: Original post by Way Walker
But these criticisms also apply to circumscribing regular polygons

That was the exact previous thing I went into.

Quote: Also, because "I said so" isn't a satisfying mathematical explanation.

Excuse me?

I said that if you try to calculate it through the area, you get a radically different figure, and while using the regular polygon method also isn't perfect, the difference is dramatically smaller to the point of being useful.

I didn't mean to single out your comment since it was a comment on the whole discussion (including the linked thread) and why I included another post there as well. A lot of the explanations of why it doesn't work are no more insightful than "I said so". For example, in the linked discussion the explanations are, "you cannot interchange limits and lengths," and, "Suppose that X(n) is a sequence of objects that have a meaningful limit X. If all of hte X(n) have a property P, then [...] most people will accept that the limit must have P without thinking about." The first isn't entirely true because you can if you have the right limiting sequence (e.g. regular polygons in this case) and the second gives no reason as to why it doesn't work in this case while there are still "numerous theorems in maths that follow this pattern." Basically, it doesn't work because they said so.

Students (and others) new to a particular area of math sometimes get the right answer by doing something wrong or even completely irrelevant. Their answer is quantifiably correct (there being literally 0 difference between their answer and the correct answer), but there's not necessarily any reason it should be or that it will be anywhere near correct in other cases. Why do regular polygons produce a better result? Or is it just chance, like a coder who makes a working program by randomly copy-and-pasting code from the internet? Or is it just that it was on the internet so it must be true?

I think it's related to the fact that the limiting set really is a circle so it will be pi and the derivative of the limiting set exists everywhere, but the limit of the derivative is undefined everywhere. Maybe the full explanation requires a deeper knowledge of fractals than one can give in a single post to a technical but still general audience?

And you're excused. [smile]

The "because I said so" bit has me thinking.

There are a lot of assumptions going into this.

The whole exercise in question (in the OP's pic) is founded on the assumptio that the circle has less perimeter than the square, but we can do this reducing trick so that its shape approaches that of a circle.

But we can see immediately from the first step that the perimeter is artificially maintained - and I'd posit that this is no different than drawing horns onto the side of the square - therefore it doesn't meet it's own goal; its perimeter is not reduced.

We can also see that we're actually adding a lot of angles.

So as far as the parimeter is concerned, the reduction trick is not actually being done on the property in question.

Does that work?

Having fun?

Actually it's an interesting problem, because as pointed out:

Quote: However, the construction in the OP still seems to fulfill the usual definition of a circle: all the points in a plane that are a given distance away from a given point. That definition is apparently incomplete.

It has to do something with fractal geometry, I believe you can make a object that looks like a line segment (infinitely thin), yet it can have any arbitrary length, even infinite.

Quote: Original post by szecs
Having fun?

Yessir.

Quote: the construction in the OP still seems to fulfill the usual definition of a circle: all the points in a plane that are a given distance away from a given point

Maybe that's not the case. Again, if you just zoom in, it's stair-steps, it's not a circle. The shape on your screen isn't really a circle either, given that it's also painted onto such a grid.

The regular polygon isn't a circle either, and I think we can all agree on that.

Thing is, the regular polygon is designed to reduce its perimeter toward that of a circle while this, isn't.

The whole point of reducing is to approximate so that the figure's properties approach a circle, and this one fails because it doesn't attempt to do this in whole. It attempts to reduce one property, its area, while inflating the other, as a joke. If you look at the other property, you do get Pi, which is consistent with its circleish appearance.

[Edited by - JoeCooper on December 2, 2010 3:29:47 AM]