http://i254.photobucket.com/albums/hh116/balthamossa2b/1290457745312.jpg
Can someone explain the flaw in this logic?
Can someone explain the flaw in this logic?
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 conventional length of the perimeter (or any differentiable curve), rather than the real length of the perimeter.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.
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.What do you mean by "real", Jarle? What makes the Euclidean norm more real than the taxicab norm?
What is the circumference of the curve x^{2}+y^{2}=r^{2} using the taxicab norm?
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.Another thing my friend pointed out is that, the perimeter he's calculating is always 4. Then he concludes that pi = 4. But pi is HALF the circumference. The perimeter of the circle would be 4, which means if 2*pi = perimeter then pi = (perimeter / 2) = (4 / 2) = 2. Silly of me not to notice that obvious problem.
But it sounds better to say pi = 4 for the purposes of confusing people, I suppose.
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.
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.
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.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.
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).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).
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).
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.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.
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.This 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 would appear that 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 same as 'a diagonal line'.
It does have a length of 2 -- using the taxicab, or L_{1}, norm, that is. What this troll physics shows is that the circumference of a circle using the taxicab norm is c=4d. That does not mean that pi is 4. It just means that different norms will yield different answers for length.This 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 would appear that the diagonal has a length of 2.
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.I don't really see any kind of paradox here. Just because you have a parametric curve [tex]f : [0,1] \rightarrow \Re^{2}[/tex] whose graph is a circle of radius r doesn't mean the arc length of f is [tex]2 \pi r[/tex]. For instance, f could 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.