So far, I havnt really seen any 'explanations' that go beyond 'but the polygon method confirms to our expectations but this one does not!'

How are they different? Both reduce the distance between the curve and the circle arbitrarily. Both are equally piecewise linear.

I think the fundamental difference is that the one arbitrarily closely approximates the tangent of the circle while the other does not. A sufficiently fractal curve could have any length while satisfying the above properties, while one with constraints on its tangent direction could not.

Still not a proof, but I think its a correct intuition.

Quote: Original post by Eelco
How are they different? Both reduce the distance between the curve and the circle arbitrarily. Both are equally piecewise linear.

They're different because it doesn't reduce the perimeter, and the perimeter is what we're measuring. Therefore nothing measured is being reduced. Nothing at all. We might as well not do any reduction, just leave it as a square, and refute any complaints with "I don't see any 'explainations' other than a square isn't a circle."

It's the area that's reduced, so you're only using the reduction method if you measure that.

Quote: Original post by JoeCooper

Quote: Original post by Eelco
How are they different? Both reduce the distance between the curve and the circle arbitrarily. Both are equally piecewise linear.

They're different because it doesn't reduce the perimeter, and the perimeter is what we're measuring. No reducing is done. The figure's perimeter isn't changed at all. You might as well not change the figure.

The area is, and if you measure that, you'll find it most certainly does give us Pi.

Is that a valid distinction though? We can conceive of perimeters that do change, yet do not converge to the correct value; say, a grid outline on one side and a polyhedron on the other, or a more subtle instance of such a curve. Just because it changes, doesnt make it right, nor does the fact that it doesnt change tell you anything, without a preconceived notion of the quantity you are trying to figure out.

We do have preconceived notions, assumptions and specifications, which is perfectly OK and in fact mandatory. If we can't agree on what a circle is and what Pi is supposed to do, than we might as well skip the diagrams, make up numbers and call it a day.

This approach is only being questioned because it looks like the reduction approach. Since the perimeter is not reduced, it is not the reduction approach, and is therefore one of an infinite number of totally random activities that also don't do anything relevant or useful, like watching Stargate. Your arguments are equally valid in support of me watching Stargate, counting the minutes, adding a break to refill my coffee, dividing it by 9 and calling it Pi; I can't tell that this is the wrong approach unless I have some idea of what I'm looking for.

We know the circle has Pi*diameter circumference and Pi*(0.5*diameter)^2 area.

The reduced square will always have an area greater than the circle's, plus it will always have a circumference of 4*diameter (also greater than the circle's).

If we take the points of the reduce square which are touching the circle's edge, and join those points into a polygon, then that polygon will always have an area and circumference less than the circle's.

One gives us maximum values, the other minimum values. Pi is somewhere between the two.
[edit]Another way to realise that this reduce square thingy is obviously going to have a large circumference is to take the idea to the extreme... Imagine a circle, then pick a point on it and travel around clockwise. You travel along in infinitely small steps, at each step, you move the edge inwards and outwards, creating a bump of an infinitely small distance. We just added an infinite number of bumps to the edge, which have a greater than zero length... What's the new circumference? Is 1/infinity * infinity equal to 1? If so, the new circumference is... ~Pi*Diameter+1??

[Edited by - Hodgman on December 2, 2010 8:37:03 AM]

Quote: Original post by JoeCooper
We do have preconceived notions, assumptions and specifications, which is perfectly OK and in fact mandatory. If we can't agree on what a circle is and what Pi is supposed to do, than we might as well skip the diagrams, make up numbers and call it a day.

This approach is only being questioned because it looks like the reduction approach. Since the perimeter is not reduced, it is not the reduction approach, and is therefore one of an infinite number of totally random activities that also don't do anything relevant or useful, like watching Stargate. Your arguments are equally valid in support of me watching Stargate, counting the minutes, adding a break to refill my coffee, dividing it by 9 and calling it Pi; I can't tell that this is the wrong approach unless I have some idea of what I'm looking for.

Right. Pi is indeed a well defined concept, and we can determine that some procedures should converge to it, and some should not. Its just that you have so far failed to enumerate what the characteristics of such a procedure should be, and the characteristic 'its not pi for any of the iterations I have inspected' is not very helpful in general. It is infact sufficient an observation for fixed iterations like this, but is rather useless for all other conceivable algorithms, correct or incorrect, none of which produce pi for any n.

In other words; 'its not pi' is not much of an answer to the question 'why isnt this pi?'

Can you answer why my Stargate method for producing Pi doesn't yield Pi without falling back on the fact that it's not designed to find Pi?

Can you at least show why the perimeter-of-a-square method should work?

Quote: In other words; 'its not pi' is not much of an answer to the question 'why isnt this pi?'

That's not my answer. My answer is that I can't use it for what Pi is used for. I can't take Pi=4 and get the object's area without, at some point, winding up with the 3.14etc. figure somewhere else in the equation, at which point we're only renaming things.

Quote: failed to enumerate what the characteristics of such a procedure should be

It's a useless activity. You earlier enumerated the characteristics and only succeeded in suggesting that an intentionally broken procedure should work. The whole joke is designed to play on the idea that you'll look at it in those terms, in order to mock you for doing so. It's recognizable as looking like a correct procedure in the same sense that the lunar maria are recognizable as a face.

[Edited by - JoeCooper on December 2, 2010 11:04:57 AM]

When you switch from a finite number of iterations to an infinite number of iterations, the square with cut corners becomes a perfect circle and the perimeter changes from 4 to pi without warning.

From http://news.ycombinator.net/item?id=1927096:
"The question is about the sequence of approximations to that line: why the length of the approximations doesn't converge to the length of the limit. And the answer is that it doesn't have to, because even though the approximations are very similar to the limit line in one respect (geometric closeness), they are all very different from it in another respect (directions and angles of travel). If we had a sequence of approximations whose direction of travel converged correctly, the length would converge correctly too."

Quote: Original post by JoeCooper
Can you answer why my Stargate method for producing Pi doesn't yield Pi without falling back on the fact that it's not designed to find Pi?

I gave a stab at a general rule, by saying it had to do with the lack of convergence of its tangent direction. Fractals can have any length.

Quote: Can you at least show why the perimeter-of-a-square method should work?

Why would I?

Quote:

Quote: In other words; 'its not pi' is not much of an answer to the question 'why isnt this pi?'

That's not my answer. My answer is that I can't use it for what Pi is used for. I can't take Pi=4 and get the object's area without, at some point, winding up with the 3.14etc. figure somewhere else in the equation, at which point we're only renaming things.

Indeed you are renaming things, because again you give that exact same answer: 'its not pi'.

Quote: It's a useless activity. You earlier enumerated the characteristics and only succeeded in suggesting that an intentionally broken procedure should work.

This starts to smell like trolling. I never said it should work, I said you failed to distinguish what sets it apart from working methods. (that is, aside from the trivial answer; it not working)

Quote: Original post by JoeCooper

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.

Sorry, I wasn't precise enough in what I was saying. It is a circle if we interpret "repeat to infinity" as "take the limit as this sequence is iterated to infinity" or, if we number the steps and consider the nth step, "take the limit as n approaches infinity".

Quote:
We do have preconceived notions, assumptions and specifications, which is perfectly OK and in fact mandatory. If we can't agree on what a circle is and what Pi is supposed to do, than we might as well skip the diagrams, make up numbers and call it a day.

This isn't a very good way to proceed. You still haven't explained why the one is correct and the other isn't. That is, why the procedure that yields 3.141... is correct and the one that yields 4 isn't. The only explanation you have is that you knew beforehand that 3.141... is the correct answer, but by what procedure did you come up with that number? How did you decide that that procedure yielded the correct answer?

This is important because in math and science you can often get two different values for an unknown quantity and you need to determine which procedure is correct.

As for why the original construction "should" produce pi, intuitively, if curve A converges to curve B, then the length of curve A should converge to the length of curve B. So, if we know the what length curve A converges to and curve B is a circle of known diameter, then we should be able to calculate pi from its primary definition: the ratio of a circle's perimeter to its diameter. In the original construction, B is a circle with a diameter of 1 and A is a curve that converges to a circle and whose length converges to 4.

The problem is that the "intuitive" part there is wrong.

Quote: Original post by BlueSalamander
"If we had a sequence of approximations whose direction of travel converged correctly, the length would converge correctly too."

So, an obvious necessary condition is that the curve itself converges, but that's apparently not sufficient. Given that the curve converges, is a converging gradient sufficient?