Pi=4

Originally posted by Soothfast
What I was trying to convey is that there will be points on the circle that are never incorporated in the range of any function f_n in the sequence.

But they are at the limit, which is what we're talking about. That you can't reach it in a finite number of steps is of no consequence.

Originally posted by Palynka
You seem to imply that the "standard way" is the only way (or else why would it be relevant?)" which is obviously wrong. I also am baffled how you believe the Hausdorff metric is irrelevant when talking about distances between metric spaces. Also, the Dirichlet theorem is not irrelevant, because if you put the origin on the lower left corner of the square, t cle. Thank you for now agreeing with me, although I don't see why you take it so personal.

I think I've stressed many times that the "standard way" is not the only way, but I guess I should add that it, or some trivial variant of it, is indeed the only way if we want to keep pi the way it is and not arrive at pi=4.

As for the Hausdorff metric, it simply is not needed in this analysis. The sequence {f_n} converges to a circle. The arclength of each f_n is 4. The arclength of the circle, on the other hand, is 3.14159265... And so what do we say to that? Only that we do not usually define the arclength of a circle by the means illustrated in the OP, but rather by other means that keep pi equal to 3.14... Nothing else is relevant, because at this juncture everything seems pretty clear. The Hausdorff metric as a device for defining distances between sets in a metric space just isn't something we need to take out of the tool box. You keep bringing it up, but I'm not interested in it.

As for the Dirichlet theorem, that's something that I guess you brought up in conversation with someone else around here, but I'm not involved in that conversation so I'm not going to address it. It's interesting, of course, but not needed for the basic analysis of what appears to be a paradox presented in the OP.

Now, once again, I admit it: I didn't think the sequence converged to a circle. I was in an airport and I wasn't thinking it through. Clearly though, it does converge to a circle. Never actually attains a circle, of course, but that doesn't matter in a limit process.

I take it personal because I saw how personal you took your fight with bbarr a couple weeks ago. In that instance you seemed to be telling a professional philosopher he didn't know what he was talking about on a philosophical topic. You have a history.

Originally posted by Palynka
But they are at the limit, which is what we're talking about. That you can't reach it in a finite number of steps is of no consequence.

Thank you, but I actually understand that funnily enough. But I'm not creating my posts exclusively for you to read, so I'm trying to bring these matters out in the open because it might help others think about the situation.

Originally posted by Palynka
If anyone is using a non-standard definitions, that is you...

So this quote by you is what really got my goat. I mean, among mathematicians, to accuse someone of using "non-standard definitions" is about as close to fightin' words as you can get. 😞

Whatever helps you cope, buddy. Thanks for agreeing with me, all while making a gigantic effort to appear you're not.

Is there a way I can derive pi from the sequence of polygons, without knowing that the limit is in fact a circle? ie can we derive the limit and its perimeter directly from the sequence?

I guess the sequence contains the circle as part of its definition.

Originally posted by twhitehead
Is there a way I can derive pi from the sequence of polygons, without knowing that the limit is in fact a circle? ie can we derive the limit and its perimeter directly from the sequence?

I guess the sequence contains the circle as part of its definition.

From this sequence, you can't. Basically, since the sequences preserve arc length, in this case the arc length of the limit is not the limit of the arc lengths.

This is why it is important for Archimedes' method of approximating Pi using regular polygons, that you also show that the inscribed polygon also converges to Pi. In fact, the method of Archimedes works because you get an upper bound and a lower bound and can show they converge to the same number: pi. If you only used the outer polygon, you would only get an upper bound. In our case, we only show that 4 is an upper bound.

Originally posted by Palynka
But that is also true for the sequence: 0.9, 0.99, 0.999,... which is smaller than 1 at any finite step. That does not mean that this is true at the limit.

For any epsilon, I can find an Nth iteration such that the distance between any point in the circle and the figure is smaller than epsilon. Do you disagree?

But the re-iteration of that sequence, it seems to me, suppose you had a way of graphing in real time each iteration, like looking at a fractal, more magnification, the triangles get smaller, the points get closer together, you up the magnification one step at a time and no matter how small the triangles get, they never converge on the point to point connection of the theoretical circle line. It is always a triangle. I don't see how just taking that to infinitely smaller size would change that.

Originally posted by sonhouse
But the re-iteration of that sequence, it seems to me, suppose you had a way of graphing in real time each iteration, like looking at a fractal, more magnification, the triangles get smaller, the points get closer together, you up the magnification one step at a time and no matter how small the triangles get, they never converge on the point to point connec ...[text shortened]... ways a triangle. I don't see how just taking that to infinitely smaller size would change that.

I agree that it is always a set of triangles at every step, just that the set of triangles gets arbitrarily close to the line. How are you defining the limit?

Originally posted by Palynka
From this sequence, you can't. Basically, since the sequences preserve arc length, in this case the arc length of the limit is not the limit of the arc lengths.

This is why it is important for Archimedes' method of approximating Pi using regular polygons, that you also show that the inscribed polygon also converges to Pi. In fact, the method of Archimede ...[text shortened]... olygon, you would only get an upper bound. In our case, we only show that 4 is an upper bound.

It is trivial to create in inscribed sequence of polygons that also converges to the circle, but also has a perimeter of 4. (essentially we just draw the triangles on the other side of the arc.)
What am I missing?

Edit: I am guessing that Archimedes method must rely on the fact that the shortest distance between two points is a straight line, so straight lines between points on the circle provide a lower bound.

Originally posted by twhitehead
It is trivial to create in inscribed sequence of polygons that also converges to the circle, but also has a perimeter of 4. (essentially we just draw the triangles on the other side of the arc.)
What am I missing?

Edit: I am guessing that Archimedes method must rely on the fact that the shortest distance between two points is a straight line, so straight lines between points on the circle provide a lower bound.

Nothing, that would also be an upper bound of the perimeter.

Originally posted by Palynka
Whatever helps you cope, buddy. Thanks for agreeing with me, all while making a gigantic effort to appear you're not.

I agree with the mathematics. The mathematics is all that counts, and there cannot be any escape from its logical conclusions. That you wish to make it all about agreeing with YOU, personally, speaks volumes about you. As I've said, you have a history. A history of needing, psychologically, to have everyone agree with YOU. Well that is fine, and we all have such a need at some level I'm sure, but you elevate the enterprise to a high art. It doesn't make you bad, but it makes you, well, unpleasant to converse with because of your constant browbeating and introduction of extraneous side issues.

Originally posted by Soothfast
I agree with the mathematics. The mathematics is all that counts, and there cannot be any escape from its logical conclusions. That you wish to make it all about agreeing with YOU, personally, speaks volumes about you. As I've said, you have a history. A history of needing, psychologically, to have everyone agree with YOU. Well that is fine, and we al ...[text shortened]... onverse with because of your constant browbeating and introduction of extraneous side issues.

Yawn. I was not the one having problems with disagreement.

Originally posted by Soothfast
I think I've stressed many times that the "standard way" is not the only way, but I guess I should add that it, or some trivial variant of it, is indeed the only way if we want to keep pi the way it is and not arrive at pi=4.

As for the Hausdorff metric, it simply is not needed in this analysis. The sequence {f_n} converges to a circle. The arc ...[text shortened]... idn't know what he was talking about on a philosophical topic. You have a history.

I was thinking there might be another way to express this whole thing. Why not reduce this to a discussion of the two quantitative measures involved, perimeter and diameter?

The expression "converges to a circle" is imprecise because there is more than one parameter involved. One parameter of the figure converges to a parameter of the inscribed circle, and the other does not.

The first parameter is the maximum distance d' which I define as twice the distance from the center of the inscribed circle to the most distant point on the perimeter of the figure. As a measure, d' converges to (as a once-accredited HS math teacher I would have said 'converges toward'😉 d, the diameter of the circle.

The second one is the perimeter of the figure undergoing change. Initially, the perimeter is the sum of the lengths of the 4 line segments making up the figure; it is equal to 4d where d is the diameter of the inscribed circle. We could go blah blah from here about the specific recursive operation, apply it once, and figure out what the sum of the 12 line segments is after the initial iteration, and so on, but we know they will add up to 4d. So the simple question is, as the iterations of the operation proceed, does p converge to a different value than it is initially? No. It is 4d. The perimeter, as a measure, does not "converge to (that of) a circle."

The behavior of p/d' might be of interest to some bright students.

Originally posted by JS357
I was thinking there might be another way to express this whole thing. Why not reduce this to a discussion of the two quantitative measures involved, perimeter and diameter?

The expression "converges to a circle" is imprecise because there is more than one parameter involved. One parameter of the figure converges to a parameter of the inscribed circle, and ...[text shortened]... at of) a circle."

The behavior of p/d' might be of interest to some bright students.

You bring up some intriguing ideas.

But first, I feel compelled to stress that there is a very precise meaning to the idea of "converging to a circle" here.

In general, if we have a sequence of functions {f_n} -- that is, f_1, f_2, f_3, ... -- that map from a set S to the set of real numbers, then to say that the sequence converges uniformly to a function g (which also maps S to real numbers) means this:

For every e>0 there exists some positive integer N such that, for all x in S and n>N, |f_n(x)-g(x)|<e.

The idea is that e (usually represented by Greek letter epsilon) can be made arbitrarily small, but no matter how small it is, we can find a sufficiently large integer N for which all functions f_n with n>N will have a graph that fits within a "band" of width 2e containing the graph of g.

A picture would help here, because the idea is not easy unless you're used to it, but it is absolutely precise.