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Can someone explain the flaw in this logic?

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- Thread starter LogicX
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- #1

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Can someone explain the flaw in this logic?

- #2

Integral

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I see no proof that the construction will ever exactly equal the circle.

- #3

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Could you be more specific? Just like an integral is the riemann sum of n number of rectangles as n goes to infinity. I assume that is the same reasoning being used here.

EDIT: http://www.axiomaticdoubt.com/?p=504 [Broken]

Makes sense.

EDIT: http://www.axiomaticdoubt.com/?p=504 [Broken]

Makes sense.

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- #4

Hurkyl

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- #5

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I think of it this way. Since "removing" the corners like that doesn't change the perimeter at all, it will fail to converge on the perimeter of a circle. So it's rather unlike, say, increasing the number of sides of a polygon inside the circle. That one converges on the real perimeter. His example does not.

The fact that the perimeter never changes as he removes the corners is pretty much proof that the technique will never work. It needs to converge to a smooth curve (to it's limit) that equals the perimeter of the circle, which this will never do (being jagged).

Not sure if that's 100% clear, someone else can probably put it better.

- #6

FlexGunship

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Furthermore, we know the process never enters the perimeter of the circle, so we can conclude that the final perimeter will have to be larger than the actual circumference. I would guess you'd get an answer ~4ish.

- #7

disregardthat

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I think of it this way. Since "removing" the corners like that doesn't change the perimeter at all, it will fail to converge on the perimeter of a circle. So it's rather unlike, say, increasing the number of sides of a polygon inside the circle. That one converges on the real perimeter. His example does not.

There is no such thing as the "real" perimeter of the circle independent from the (arbitrary or not) definition of it as the limiting sum of the uniformly decreasing length chords. The limit in the example is not the same as the

- #8

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What is the circumference of the curve

- #9

disregardthat

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What is the circumference of the curvex^{2}+y^{2}=r^{2}using the taxicab norm?

I mean what he calls "real" is more correctly put as "conventional", since there is no pre-existing platonic length of the circle independent of our ways of "finding" it. The choice of the euclidean norm is conventional as a measure of length (which is commonly understood by length unless otherwise is stated), it is not measuring real lengths as opposed to other norms. Similarly, approximating with regular polygons doesn't measure real length as opposed to other limiting sums. They measure different things, but none of them are more real than the other.

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But it sounds better to say pi = 4 for the purposes of confusing people, I suppose.

- #11

disregardthat

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But it sounds better to say pi = 4 for the purposes of confusing people, I suppose.

The diameter in the drawing is set as 1. Besides, the resulting ratio must be larger than pi since the measured length is constantly larger.

- #12

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I kind of see your point. But it's, IMO, nitpicking of the highest order. Let's assume I have a perfect unit circle drawn out (or to well within the tolerances with which I will measure). I can also measure the circumference to a certain precision. Which means I can empirically determine PI to whatever the limits of my measurement and my ability to get a perfect unit circle. Won't get me there exactly, but neither will any other method since it's an irrational number. All I can do is compute it to a certain number of decimal points.I mean what he calls "real" is more correctly put as "conventional", since there is no pre-existing platonic length of the circle independent of our ways of "finding" it. The choice of the euclidean norm is conventional as a measure of length (which is commonly understood by length unless otherwise is stated), it is not measuring real lengths as opposed to other norms. Similarly, approximating with regular polygons doesn't measure real length as opposed to other limiting sums. They measure different things, but none of them are more real than the other.

It's real in the sense that if I compute it and it disagrees with my empirical measurement within my level of accuracy, I would have to conclude that my computation is wrong. Either that, or define a circle as something which has no relation to the real world, which wouldn't be very useful. And I mean "real" in that sense, and that sense only.

- #13

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And? That means the sides of the initial square are of length 1. Which means it's perimeter is 4 throughout. And since the perimeter will always be 4, we would have to derive a pi which is half that, following his logic (which is obviously totally wrong).The diameter in the drawing is set as 1. Besides, the resulting ratio must be larger than pi since the measured length is constantly larger.

- #14

disregardthat

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I kind of see your point. But it's, IMO, nitpicking of the highest order. Let's assume I have a perfect unit circle drawn out (or to well within the tolerances with which I will measure). I can also measure the circumference to a certain precision. Which means I can empirically determine PI to whatever the limits of my measurement and my ability to get a perfect unit circle. Won't get me there exactly, but neither will any other method since it's an irrational number. All I can do is compute it to a certain number of decimal points.

It's real in the sense that if I compute it and it disagrees with my empirical measurement within my level of accuracy, I would have to conclude that my computation is wrong. Either that, or define a circle as something which has no relation to the real world, which wouldn't be very useful. And I mean "real" in that sense, and that sense only.

It might well be nitpicking, but then again it might not since I believe there is a presupposition of something platonically real about some preferred definitions of distance (and area for that matter). I am not commenting on imprecise mathematical formulations here.

What you physically measure is also based upon your choice of measure. In any case I am referring to the mathematics, not physics.

When one somewhat arbitrarily (though physically applicable) generalize distance to differentiable curves in some particular way one cannot expect (and one can never do so) that this is the "real" way to do it as opposed to alternative ways. They are only different. As is alternative norms.

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- #15

disregardthat

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And? That means the sides of the initial square are of length 1. Which means it's perimeter is 4 throughout. And since the perimeter will always be 4, we would have to derive a pi which is half that, following his logic (which is obviously totally wrong).

No, since pi is defined as the perimeter divided by the diameter. In this case the perimeter equals pi, since the diameter is 1. pi is only only half the perimeter if the diameter is 2 (2*r*pi =d*pi = 1*pi = pi).

- #16

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Apologies, on that one I have to agree with you 100%. I clearly mixed up r and d. My fault for assuming my friend got it right and not thinking it through for myself (not a regular habit I assure you... ).No, since pi is defined as the perimeter divided by the diameter. In this case the perimeter equals pi, since the diameter is 1. pi is only only half the perimeter if the diameter is 2 (2*r*pi =d*pi = 1*pi = pi).

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- #18

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Somewhat related: http://www.mathpages.com/home/kmath063.htm

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It's an infinitely zigzag path, of course the length is longer. You can make a zigzag path of infinite length but still be finite in "size" if you will.

- #20

disregardthat

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It's an infinitely zigzag path, of course the length is longer. You can make a zigzag path of infinite length but still be finite in "size" if you will.

It's not, whatever it might mean, it's the limit of paths with a increasing but finite number of zig-zags, and this limit path is the circle. The confusion arise in that even though the paths uniformly converges to the circle path, their lengths does not.

Following your logic the circle is an infinitely zigzag path and hence "of course" has a longer length than the circle...

It might well be so that the example of the zig-zag path of infinite length but finite size you have in mind suffers from the same confusion between the form of the elements in the sequence of paths and the path they converge to.

If I am not mistaken it is sufficient that the paths in the sequence are differentiable and the derivatives converges uniformly. In this case this condition is not satisfied; they are not differentiable in the "corners".

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- #22

DaveC426913

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Start with a square 1 unit on a side.

You can perform the same staircasing, and it would

The key is that 'a zig zag line with an arbitrarily large number of vanishingly small zigs and zags'

- #23

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Start with a square 1 unit on a side.

You can perform the same staircasing, and it wouldappearthat the diagonal has a length of 2.

The key is that 'a zig zag line with an arbitrarily large number of vanishingly small zigs and zags'is not the sameas 'a diagonal line'.

This is similar to what I described above but presented more clearly.Following Daves example if you take x equal sized right angled steps to the top,the distance travelled in one step(vertical plus horizontal distance)=2/x and total distance travelled for all steps=2/X*x=2.It comes out to the same value whatever the value of X even when X tends to infinity.In the case of the circle the stepped(zig zag) line surrounding the circle does not have the same length as the circumference of the circle.The length of the stepped line is equal to the sum of the horizontal lengths of all the steps plus the sum of the vertical lengths of all the steps and it comes out to the same value no matter how many steps there are.

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It does have a length of 2 -- using the taxicab, or LThis is a simpler problem wrapped in a complex cloak.

Start with a square 1 unit on a side.

You can perform the same staircasing, and it wouldappearthat the diagonal has a length of 2.

- #25

disregardthat

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rdoesn't mean the arc length offis [tex]2 \pi r[/tex]. For instance,fcould go around the unit circle multiple times. So it may be true that the sequence of jagged figures "converges" in some sense to a parametrization of a circle, but it can have a different arc length.

In this case the parametrization goes around the circle only once. The length of the curve the jagged paths are converging to is the same as the perimeter of the circle. The problem here is not that we get a different parametrization of the circle, the problem is that the jagged paths does not have derivatives which converge, and hence we cannot be sure that the length of the jagged paths converge to the length of the circle.

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