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

[disregardthat]

Are there different ways of defining length or perimeter and that in a taxi cab norm definition pi=4?

Yes, the length of a curve can be defined relative to different norms, but normally the euclidean norm is used. If we say that pi is the ratio between the perimeter and diameter of a circle relative to the taxi-cab norm then we will get that pi = 4. Pi is however defined relative to the euclidean norm, not the taxi-cab norm, so the conclusion is wrong.

 

[G037H3]

Because Manhattan/taxicab geometry doesn't apply to Euclidean geometry.

 

[D H]

Exactly. While both Euclidean geometry and taxicab geometry "fit" on R², taxicab geometry violates some key characteristics of Euclidean geometry, one of them being the Pythagorean theorem. An AMS article on taxicab geometry: http://www.ams.org/samplings/feature-column/fcarc-taxi

 

[Dadface]

Hello DH and Jarle,thanks for clarifying.Of course I know that pi is 3.142 etc but I was mainly wondering why a definition in terms of taxicab geometry is made.Perhaps the definition can have useful applications.There is more than one way to answer many question and I think that in this thread the question has been answered with and without reference to taxicab geometry.
DH,I really like the look of the content in the link you posted above and I have put it on my list of "things to read".

 

[D H]

I was mainly wondering why a definition in terms of taxicab geometry is made.Perhaps the definition can have useful applications.

In mathematics there are many definitions of 'distance' other than Euclidean distance. Some have names because they are rather useful in some circumstances. A couple of examples in addition to the taxicab norm discussed in this thread:

• Chebyshev distance, or L_∞ norm, $$||{\boldsymbol x}||_{\infty} = \max_i |x_i|$$.
  This is at the opposite end of the L_p famility of norms from the taxicab norm. Suppose you and a coworker are each asked to come up with a simple approximation of some complicated function. Your coworker uses a root mean square approach to minimize the error while you seek to minimize the worst-case error. In many cases, the function that minimizes the absolute error is deemed to be a better fit than a least squares "best fit".
• Mahalanobis distance, $$||{\boldsymbol x}||_S=\sqrt{{\boldsymbol x}^T S^{-1} {\boldsymbol x}}$$.
  Suppose you are told that a good-sized meteorite is bearing down on your city. Odds of living if you are within 100 meters of the impact are rather low. Outside a kilometer away, not much of an impact at all. The impact is estimated to occur within an ellipse 50 km long by 1 km wide. You live 10 km from the center of the ellipse. Should you get out of Dodge? The answer depends on direction as well as Euclidean distance. If that 10 km is along the long axis of the ellipse, going somewhere else is a rather good idea. If that 10 km is along the short axis of the ellipse you might as well stay home. The odds of getting in an accident due to all the idiots on the road trying to escape the meteorite are a lot higher than you getting hit by the meteorite in this case. The Mahalanobis distance is a much more meaningful metric in this case than is Euclidean distance.

 

[Edgardo]

An article on the diagonal paradox which has been mentioned before can be found on http://mathworld.wolfram.com/DiagonalParadox.html" .
Another interesting paradox involving sine functions converging to zero is described on http://www128.pair.com/r3d4k7/Mathematicae4.html" .

 

[OmCheeto]

An article on the diagonal paradox which has been mentioned before can be found on http://mathworld.wolfram.com/DiagonalParadox.html" .
Another interesting paradox involving sine functions converging to zero is described on http://www128.pair.com/r3d4k7/Mathematicae4.html" .

The diagonal paradox also references the http://mathworld.wolfram.com/CoastlineParadox.html" , which, from the link listed earlier:

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.

Makes me wonder if it was correct to correct the author:

The limit of f as n approaches infinity is not a circle, it’s some right angled fractal beast [edit: I've been corrected, the limit is a circle, but the length of the curve is defined in terms of the derivatives, which means that it is not defined on the corners], π does not equal 4, proceed to infinity with caution.

It strikes me as a fractal beast. At least as far as I understand fractals. You can hypothetically take the limit to infinity, but the circle would then be infinitely jagged. Which in my mind, must violate some rule of integration.

Help! It's been 30 years since I've studied Calculus!

 

[disregardthat]

It strikes me as a fractal beast. At least as far as I understand fractals. You can hypothetically take the limit to infinity, but the circle would then be infinitely jagged. Which in my mind, must violate some rule of integration.

Help! It's been 30 years since I've studied Calculus!

The limit path is the circle. The curves converge uniformly to the circle, and not some "fractal beast". When you "zoom in" on the limit path you will never see jags, it has a constant curvature. It is not "infinitely jagged", whatever it means. But in order to be certain of that length is preserved under the limit you will need the curves to be differentiable, and the differentiated curves will have to be continuous and converge uniformly as well. This is why the length of the sine curves (being differentiable with continuous derivatives) in the other link does not converge to the length of the unit interval [0,1]; the derivatives does not have a limit. But the curves do converge to the unit interval.

What rules of integration do you think it violates?

 

[OmCheeto]

The limit path is the circle. The curves converge uniformly to the circle, and not some "fractal beast". When you "zoom in" on the limit path you will never see jags, it has a constant curvature. So it is not infinitely jagged. But in order to be certain of that length is preserved under the limit you will need them to be differentiable, and the differentiated curves will have to converge uniformly as well. This is why the length of the sine curves (being differentiable) in the other link does not converge to the length of the unit interval [0,1]; the derivatives does not have a limit. But the curves do converge to the unit interval.

I disagree. You can go all the way to infinity, and it will still be jagged. Otherwise, pi would equal 4.

And can someone point me to the definition of "limit path". I'm not familiar with the term.

 

[disregardthat]

I disagree. You can go all the way to infinity, and it will still be jagged. Otherwise, pi would equal 4.

And can someone point me to the definition of "limit path". I'm not familiar with the term.

This is wrong, the limit of the curves is the circle, but the lengths does not converge to the length of the limit curve. It's that simple.

A sequence of functions $$(f_n)$$ converge uniformly to the function f if $$\sup||f_n(x)-f(x)|| \to 0$$ as $$n \to \infty$$, where the supremum is taken for x over the domain. In this case the relevant functions would be parametrizations of the curves approximating the circle.

Explain what you mean by infinitely jagged.

 

[OmCheeto]

This is wrong, the limit of the curves is the circle, but the lengths does not converge to the length of the limit curve. It's that simple.

A sequence of functions $$(f_n)$$ converge uniformly to the function f if $$\sup||f_n(x)-f(x)|| \to 0$$ as $$n \to \infty$$, where the supremum is taken for x over the domain. In this case the relevant functions would be parametrizations of the curves approximating the circle.

Explain what you mean by infinitely jagged.

I think you are wrong. I see no curves. The function represented in reducing the square to a circle consists of an infinite number of non-continuous functions. Isn't there some rule about the function needing to be continuous?

But perhaps I don't speak maths well enough to explain myself.

ps. I went to Wolframs and could find no definition of "limit path". It also shows up only 6 times when googling: calculus "limit path" wolfram.
(One of which being your usage.)

Are you sure you used the correct term? Or is it an ellipsis?

 

[disregardthat]

I think you are wrong. I see no curves. The function represented in reducing the square to a circle consists of an infinite number of non-continuous functions. Isn't there some rule about the function needing to be continuous?

But perhaps I don't speak maths well enough to explain myself.

ps. I went to Wolframs and could find no definition of "limit path". It also shows up only 6 times when googling: calculus "limit path" wolfram.
(One of which being your usage.)

Are you sure you used the correct term? Or it an ellipsis?

It's a limiting function under the uniform norm which happens to be a path/curve, therefore I call it a limit path/curve, or limiting path/curve if you will. I have explained what I meant, it should provide no further confusion.

See http://en.wikipedia.org/wiki/Uniform_convergence

Of course there are curves. The jagged curves around the circle are approximating the circle. There are no non-continuous functions here.

You may believe I am wrong, but I can't see any evidence from your side demonstrating it.

 

[OmCheeto]

It's a limiting function which happens to be a path, therefore I call it a limit path, or limiting path if you will. I have explained what I meant, it should provide no further confusion.

See http://en.wikipedia.org/wiki/Uniform_convergence

Of course there are curves. The jagged curves around the circle are approximating the circle. There are no non-continuous functions here.

You may believe I am wrong, but I can't see any evidence from your side demonstrating it.

The function of the jaggedness of the functions defining the broken square approaching the circle is: f(x)=4*2^x-4

As x approaches infinity, the jaggedness approaches infinity even faster.

If that even makes sense.

 

[disregardthat]

The function of the jaggedness of the functions defining the broken square approaching the circle is: f(x)=4*2^x-4

As x approaches infinity, the jaggedness approaches infinity even faster.

If that even makes sense.

That does not mean the limiting function has an infinite number of jags. That is faulty logic. It just means that the number of jags are increasing without bound for the functions in the sequence. The limiting function need not resemble these.

Consider a similar variant along these lines: You iteratively pick some arbitrary number from the stack of rational numbers. Each pick leaves a new disjoint interval in the rational number line. As you continue, the number of disjoint intervals increase. Still, the "limit" of this process will exhaust the rational numbers since they are countable (given that you actually pick from a list of the rational numbers), leaving no such intervals.

 

[D H]

This is wrong, the limit of the curves is the circle, but the lengths does not converge to the length of the limit curve. It's that simple.

No, its not that simple. The limit of the steps is a curve that converges uniformly to the circle but is nowhere differentiable. In short, it is not the circle.

 

[OmCheeto]

That does not mean the limiting function has an infinite number of jags. That is faulty logic. It just means that the number of jags are increasing without bound for the functions in the sequence. The limiting function need not resemble these.

Consider a similar variant along these lines: You iteratively pick some arbitrary number from the stack of rational numbers. Each pick leaves a new disjoint interval in the rational number line. As you continue, the number of disjoint intervals increase. Still, the "limit" of this process will exhaust the rational numbers since they are countable, leaving no such intervals.

Exhaust?

Please.

I may be daft, but I'm not stupid.

Where is the wizard?

 

[disregardthat]

Exhaust?

Please.

I may be daft, but I'm not stupid.

Where is the wizard?

Don't you know the word?
Definitions of exhaust on the Web:
consume: use up (resources or materials); "this car consumes a lot of gas"; "We exhausted our savings"; "They run through 20 bottles of wine a week"
run down: deplete; "exhaust one's savings"; "We quickly played out our strength"
use up the whole supply of; "We have exhausted the food supplies".

Please explain what genuine questions you have if any.

 

[disregardthat]

No, its not that simple. The limit of the steps is a curve that converges uniformly to the circle but is nowhere differentiable. In short, it is not the circle.

If you are right then that is if you insist on the induced parametrization of the circle. I may have been sloppy and written path when I meant curve and vice versa. Sometimes it matter. Only the range (curve) of this limit function is relevant though, and that is the circle. Why would you say "it is not the circle" when you say it converges uniformly towards the circle? The circle does not depend on any parametrization and does not have a preferred (differentiable) one.

As far as I know the length of a curve is defined independently of any particular parametrization, and is only applicable if a differentiable parametrization exists. Hence the length of the limit curve is not necessarily calculated using the induced parametrization. If I have understood you right.

 

[OmCheeto]

Don't you know the word?
Definitions of exhaust on the Web:
consume: use up (resources or materials); "this car consumes a lot of gas"; "We exhausted our savings"; "They run through 20 bottles of wine a week"
run down: deplete; "exhaust one's savings"; "We quickly played out our strength"
use up the whole supply of; "We have exhausted the food supplies".

Please explain what genuine questions you have if any.

Since there are an infinite number of rational numbers, how long do we have to wait before they are exhausted? Or is that the next question? "Prove that there are an infinite number of rational numbers."

I must say, is General Math always this lively?

 

[disregardthat]

Since there are an infinite number of rational numbers, how long do we have to wait before they are exhausted? Or is that the next question? "Prove that there are an infinite number of rational numbers."

I must say, is General Math always this lively?

Note that is said the limit of this process. A limit is never "reached". I don't see what you are getting at.

 

[alphachapmtl]

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

Can someone explain the flaw in this logic?

In the same way, a square diagonal will lead to 1+1=sqrt(2).

 

[qwerty4056]

The correct way to approximate pi would be to take the diagonals of the "bumps", which are illustrated in the attached image (View attachment Pi aproximation.bmp). Now you add up all the diagonals' lengths and then you get some number under pi.

 

[1MileCrash]

Maybe I'm not seeing it, but I picture repeating this process "infinitely" just yields a diamond or sideways square, never a circle.

 

[lurflurf]

The main point (what makes circumference difficult at times) is that circumference is a local property. That is a small (in some metric) deviation can cause a large difference in circumference. Our jagged circle thing is almost a circle in some sense, but it has a very different circumference.

 

[coolul007]

An interesting book containing "proofs" that are wrong is:

The Mathematical Recreations of Lewis Carroll: Pillow Problems and a Tangled Tale

He is a master at them.

 

[Tac-Tics]

Limits don't always commute with other operations.

In other words, if s_i is a sequence, it is it not necessarily true that f(lim s_i) = lim f(s_i). In this case in particular, the limit of the perimeters isn't necessarily the perimeter of the limit.

Why is this? It just is.

Let's put it this way. You expect there should be a law that says the perimeter of the limit is the limit of the perimeters. But you can't just assume it's true. You have to PROVE it's true.

The problem is that you can't prove it to be true, because it's false. Why is it false? Because we can find a counter example. What counter example is that? The circle-square limits.

Another trivial example is to take the sign function: sgn(x) = 1 if x is positive, -1 if x is negative, or 0 if x is 0.

Now, take the sequence s_i = (-1)^i (1/i). So the sequence starts off -1, 1/2, -1/3, 1/4.

The limit of s_i is 0, because the numbers get closer and closer to zero, as close as we want if we go far enough. Thus, sgn(lim s_i) = 0.

BUT, the sign of each element in the sequence alternates. sgn(s_1) = -1, sgn(s_2) = 1, sgn(s_3) = -1. In fact, the sequence sgn(s_i) doesn't even converge! It has no limit. Thus, sgn(lim s_i) cannot equal lim sgn(s_i).

 

[Unrest]

Maybe I'm not seeing it, but I picture repeating this process "infinitely" just yields a diamond or sideways square, never a circle.

It does if you cut out squares, then you can get another trick about the lengths of sides of a square or right-tiangle, like alphachapmtl said.

But here the diagram shows you have to use rectangles so the new internal corners touch the circle.

 

[guss]

EDIT: THIS IS WRONG. Working on the correct one now.

There is always going to be area between where the "ridges" are and where the outside of the circle lies. That area is given by:

$$A = 1 - \pi\ - \frac{4n+4}{n(1 - 2\sqrt{2})}$$

Where n is like this (and A is the blue area):

The limit of the above function as n -> infinity is .0461.

P.S. My constant of $$1 - \pi\$$ may be off. If someone wants to check/correct me on that equation that would be great, I'm a little tired at the moment. If not, I can go over this tomorrow.

 

[Deveno]

it is not true that lim(lengths) = length(limits).

the (euclidean) length of a curve is not well-defined on the set of all possible curves. the notion of length, can be counter-intuitive, it depends on 2 things: what you are measuring, and how you measure it. the various fractal curves give examples of how length can be "worse than it looks" (the koch snowflake, for example. its length doesn't appear to be infinite).

if you were to produce a specific parametrization (piece-wise) for the "boxed" approximation, you would find that in the limit, it is not differentiable. the differentiability of the parametrizations are a key hypothesis in proving this independence.

 

[disregardthat]

if you were to produce a specific parametrization (piece-wise) for the "boxed" approximation, you would find that in the limit, it is not differentiable. the differentiability of the parametrizations are a key hypothesis in proving this independence.

Not only that, the derivatives of the parametrizations must themselves converge uniformly to a continuous function. At least this is a sufficient condition, and counter-examples to "limit(length) = length(limit)" where this is not satisfied (while your condition is) exists. E.g: One could easily make the jagged lines around the circle smooth by substituting the tip with a quarter of a circle or something similar, but the lengths would still not converge to the correct value.

And it is not true, as suggested by someone eariler, that the jagged curves around the circle does not converge to the circle. The curves does converge uniformly to the circle, the problem is just that the lengths does not.

 

[guss]

Ugh, finally got it. Maybe. Could someone try to do this out themselves?

n is the number of points in each quadrant where there is a point touching the circle. A is the area of the blue. See below image for sample. Basically, there is always going to be area surrounding the circle that is blue.

Equation:
[PLAIN]http://www4b.wolframalpha.com/Calculate/MSP/MSP67019f5i3id7f23f46800006a6bfeede1b8g0ih?MSPStoreType=image/gif&s=30&w=215&h=61

lim x->infinity = 1-(pi/4) = original area of blue when n=0. But I don't understand how that could be the case. I think the above equation must be wrong. I'll have another look tomorrow. I suppose it could make sense when there are thousands of points touching the circle, but I don't think so.

Graph:
[PLAIN]http://www4b.wolframalpha.com/Calculate/MSP/MSP96719f5hfca6ccfc80700004fie2ha6eb8fca2b?MSPStoreType=image/gif&s=10&w=300&h=166&cdf=Coordinates&cdf=Tooltips
http://www.wolframalpha.com/input/?...sqrt(x^2+1)-x)^2)/(2+(x^2+1))+from+x=0+to+x=5

 

[Yuqing]

I'm quite interested in seeing a derivation of that formula if it's correct. From my knowledge this is quite a difficult problem. In each removal, the areas of the squares removed are non-uniform. For example, for the third "cycle" (n=7 according to your notation), the area of the squares removed at the edges of each corner are larger than the squares removed at the interiors of the corner and this effect builds up as n increases. Did you manage to account for that?

edit: Upon reading your post again, it may seem like I've misunderstood. That formula seems to suggest that the area of the blue surrounding does not decrease which is certainly not the case. I believe that the area of blue surrounding does converge towards the circle and its just the perimeter that does not.

 

[guss]

Alright, I'll post the derivation I did. I'm almost 100% sure there's a mistake in here, so please try to help.

Let's say that n is the number of times a white "corner" touches the black circle in one of it's quadrants. See previous image above for a guide. For simplicity's sake, let's only focus on one quadrant for now. We can multiply it by 4 to get the total area lost later.

Now, to get the area lost (in each quadrant) by having n number of touches (in each quadrant).

I THINK we can break the whiteness up into even squares. The amount of squares along the top row is simply n, and the amount of squares along the outside column is also n. This means (I THINK) we can get the total number of squares for touches n by using the triangular number formula (but I could be wrong because the circle being a curved surface might change this).

$$\frac{n^2+n}{2}$$

So that is the total number of squares lost in each quadrant. Now, let's get the area of each of those squares. Let's focus now on the corner-to-circle touch on the edge of the quadrant. (in quadrant 1, it would be the farthest right touch).

First, we should simply get the angle on the circle where this touch is. cos(theta) gives us the distance from that touch to where the top edge of the original square was (if n=0 and there were no touches). sin(theta) gives us the distance from that touch to the right edge of the original square was (in quadrant 1, but that's all we have to worry about for now). That means, to get theta, all we have to do is:

$$n(sin(\theta)) = cos(\theta)$$

Now, let's get the distance from where theta lies on the outside of the circle to the edge of where the original square would be. This will give us the length of each side of each square. This equation is:

$$L = \frac{1}{2} - \frac{cos(\theta)}{2}$$

After some simplifying and combining the previous two equations, we get

$$L = \frac{1}{2} - \frac{n}{2\sqrt{n^2-1}}$$

The original area with just the original square and circle is just 1- pi/4. Let's square the L in the equation above to get the area of each square, then multiply it by the triangular number formula (giving us the total area subtracted from each quadrant), then multiply that by 4 to get the resulting area subtracted from the entire figure. After all this, we get this:
WolframAlpha equation

I'm almost certain my math is correct, it's just the concept. I'm not sure if we can count every square as having the same area, and I'm not sure if we can simply use the triangular number equation due to the nature of the circle being curved.

If someone could take a look at this that would be great, because I'm stuck, and not to mention kind of sick of working on this problem myself.

Thanks.

 

[Yuqing]

As I've mentioned, for each iteration of removing squares (which I will call cycles), the area will differ. (i.e. the second and third squares removed will be smaller than the first square removed). In fact it gets more complicated than that. For each removal cycle you will remove 2 squares near the edge of the quadrant and you will remove squares interior to the quadrant. These squares will also have different areas in fact. This effect will build up and for later cycles there may be several different sizes in a single cycle (the two most interior squares share an area, then pairwise as you progress outwards of the quadrant). You cannot simply take every square to be the same size.

 

[guss]

As I've mentioned, for each iteration of removing squares (which I will call cycles), the area will differ. (i.e. the second and third squares removed will be smaller than the first square removed). In fact it gets more complicated than that. For each removal cycle you will remove 2 squares near the edge of the quadrant and you will remove squares interior to the quadrant. These squares will also have different areas in fact. This effect will build up and for later cycles there may be several different sizes in a single cycle (the two most interior squares share an area, then pairwise as you progress outwards of the quadrant). You cannot simply take every square to be the same size.

My equation takes all of that into account, except for the fact that the squares change size. If someone could derive the right equation I'd love to see it.