I don't get why this troll physics is wrong.

[LogicX]

http://i254.photobucket.com/albums/hh116/balthamossa2b/1290457745312.jpg

Can someone explain the flaw in this logic?

 

[Integral]

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

 

[LogicX]

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

Makes sense.

 

[Hurkyl]

An integral is defined to be the limit of Riemann sums. Is perimeter defined to be the limit of approximating stairsteps?

 

[Grep]

haha That's a good one, I'll have to remember that next time I want to mess with someone.

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.

 

[FlexGunship]

No matter how many times you repeat the process, the circle is only touched tangentially four times (as defined at the start). To find the circumference you must touch it tangentially in all places. At no time will the function's rectilinear surface length resolve to a valid approximation of the circle for this reason (among others).

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.

 

[disregardthat]

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 conventional length of the perimeter (or any differentiable curve), rather than the real length of the perimeter.

 

[D H]

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 x²+y²=r² using the taxicab norm?

 

[disregardthat]

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 x²+y²=r² 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.

 

[Grep]

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.

 

[disregardthat]

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.

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.

 

[Grep]

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.

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.

 

[Grep]

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.

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).

 

[disregardthat]

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.

 

[disregardthat]

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).

 

[Grep]

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).

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... ).

 

[Dadface]

Consider say the left hand top quadrant of the circle.Now by drawing a series of alternate vertical and horizontal lines construct a series of steps which rises from the bottom of the left side of the quadrant and to the top.One can see that the total vertical distance covered by these steps=d/2 and the total horizontal distance=d/2.From this it is easy to see that the total distance covered by such steps surrounding the circle =4d as is given in the question.It is 4d regardless of how many steps are used and what their sizes are.

 

[yenchin]

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

 

[Curl]

how can anyone ever have a problem with this?

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.

 

[disregardthat]

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".

 

[lugita15]

I don't really see any kind of paradox here. Just because you have a parametric curve $$f : [0,1] \rightarrow \Re^{2}$$ whose graph is a circle of radius r doesn't mean the arc length of f is $$2 \pi r$$. 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.

 

[DaveC426913]

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'.

 

[Dadface]

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'.

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 traveled in one step(vertical plus horizontal distance)=2/x and total distance traveled 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.

 

[D H]

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.

It does have a length of 2 -- using the taxicab, or L₁, 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.

 

[disregardthat]

I don't really see any kind of paradox here. Just because you have a parametric curve $$f : [0,1] \rightarrow \Re^{2}$$ whose graph is a circle of radius r doesn't mean the arc length of f is $$2 \pi r$$. 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.

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.

 

[Filip Larsen]

Perhaps one may also regard area enclosed by the jagged line as having a fractal boundary? Although, I'm not sure if fractal is the right concept, since it doesn't seem self-similar. Its more like a partial fractal combined with zooming.

Regarding describing the area I got a bit stuck. Writing area as an infinite sum of contributions, you quickly end up with a series for $\pi^2$, but it was not one I could find in my normal handbook. Perhaps someone better at doing math with infinite series can confirm that you do end up with a "known" series for $\pi^2$?

 

[Dadface]

DH,I don't know what the taxicab norm is,but the method used by the "troll" shows that the perimeter of the jagged line surrounding the circle is 4,it does not show that the circumference of the circle is 4.
Jarle,the length of the jagged path does not converge,the length is equal to 4 and remains at 4 no matter how many zig zags are used.

 

[disregardthat]

Jarle,the length of the jagged path does not converge,the length is equal to 4 and remains at 4 no matter how many zig zags are used.

You have misread. I said the jagged paths converge to the circle, but their lengths does not.

 

[DaveC426913]

It does have a length of 2 -- using the taxicab, or L₁, 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.

Well, the "diagonal" of a square is a straight line between two opposing corners, so it will not have a length of 2.

But using this trickery, it looks to our macroscopic eye like we've drawn a true diagonal of length 2.

 

[disregardthat]

Well, the "diagonal" of a square is a straight line between two opposing corners, so it will not have a length of 2.

There are different definitions of length in the plane using different norms (and even different definitions of length using the same norm: the length of a curve can easily be defined as the limit of the lengths of jagged paths in the euclidean norm). The length of a line is even relative to this. Its length is 2 in the taxi-cab-norm, but not in the euclidean norm. pi is also defined in terms of the euclidean norm, different norms give different pi's, so to say, which the troll has shown.

Note that the limit of these jagged paths is the true diagonal; they converge uniformly to the line in the euclidean norm.

 

[DaveC426913]

Note that the limit of these jagged paths is the true diagonal; they converge uniformly to the line in the euclidean norm.

But they don't converge on the euclidean norm.

No matter how small you make the increment, the "taxi-cab diagonal" is always exactly 2.

 

[D H]

It does have a length of 2 -- using the taxicab, or L₁, 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.

Well, the "diagonal" of a square is a straight line between two opposing corners, so it will not have a length of 2.

That diagonal line does have a length of 2 using the L₁ norm.

The L_p norm of a vector x=[x₁,x₂,...,x_n] is defined as

$$||{\mathbf x}||_p \equiv \left(\sqrt{\sum_i |x_i|^p}\right)^{1/p}$$

For p≥1, this definition

• Is scalable, ||ax||_p = |a| ||x||_p,
• Is positive definite, ||x||_p>0 for all x but x=0, in which case ||x||_p=0, and
• Obeys the triangle inequality, ||x+y||_p≤||x||_p+||y||_p

In other words, it is a norm.

The L₁ norm of this vector is

$$||\mathbf x}||_1 = \sum_i |x_i|$$

This is also called the taxicab norm or Manhattan norm. Think of how a taxicab must drive in New York City. Taxicabs cannot follow the route taken by a crow flying from point A to point B.

 

[disregardthat]

But they don't converge on the euclidean norm.

No matter how small you make the increment, the "taxi-cab diagonal" is always exactly 2.

All in the euclidean norm (nevermind the taxi-cab-norm for a moment):
The lengths does not, but the curves do. The limit curve is the circle, or in your example, the diagonal. The limit path does not resemble the elements of the sequence. It is not "infinitely jagged" or anything of the sort. That is why things like these cause so much confusion. You have no guarantee for that the lengths will converge. For this the curves will need to satisfy stronger conditions, e.g. being differentiable and having uniformly convergent derivative-curves.

In the taxi-cab-norm:
The troll and your example illustrates a perfectly proper way of measuring the perimeter of the circle and the diagonal in this norm, and the limiting length of the curves does converge what you intend to measure. We can say that pi = 4 in the taxi-cab-norm (though pi conventionally refers to the limit in the euclidean norm under the conventional definition of the length of a curve), and the diagonal has length 4.

Bear in mind that in this context the jags only illustrate which norm are being used, and are not the approximating curves themselves.

 

[Dadface]

I just don't get this .Is it being claimed that by drawing a stepped line of length 4 around a circle of diameter 1,the said line touching the circle at different places,then the perimeter of the circle is equal to the perimeter of the line or that the perimeter of the line ,which is independant of the number of steps used,converges to the perimeter of the circle?Are there different ways of defining length or perimeter and that in a taxi cab norm definition pi=4?Where does that leave things such as Coulombs law?
As far as I understand it the stepped line has nothing to do with the dimensions of the circle other than it surrounds it and touches it at different points.I also think that a simple problem is being overcomplicated.I am probably misreading or misunderstanding so can someone please clarify.

 

[D H]

Pi is not 4. It is defined in terms of the Euclidean norm in Euclidean space. Use some other norm and you will get a different ratio of circumference to diameter. The ratio might not even be constant, as is the case in many non-Euclidean geometries. That doesn't mean pi has changed to some other constant or isn't even a constant. Pi is 3.141592653589793..., period. All that getting a different, and possibly non-constant, ratio of circumference to diameter means just means you are using a different norm than the Euclidean norm or a different geometry than Euclidean geometry. The resolution of this troll physics is that it isn't using the Euclidean norm.