Traveling across a square
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Traveling across a square
This might be an easy question for all of you, but I can't figure out what I'm missing.
Picture a square with side length one. For our purposes, let's call a side length one mile because someone will be walking across it. Now, if a person walks the perimeter from one corner to the opposite corner, they've traveled 2 miles. If they walk across the diagonal, they've walked the square root of 2 miles. Simple enough.
Now, let's say the inside of the square is marked out like a chessboard. If a person walks from one corner to the other, but instead of going straight across the diagonal, they trace around the squares that the diagonal runs through (either all black squares or all white squares). So they keep turning at right angles as they work their way across the square, being very close to the diagonal, but never actually walking on it. Now, I know that as long as they turn at right angles, they never cut out any distance, and the distance traveled will still be two miles. (That seems to be a common fallacy, that cutting across a shape making rightangle turns somehow saves distance.)
Where I get stuck is by imagining more turns being made at smaller intervals. Say the square had 256 squares instead of 64 and the individual again walks around the squares cutting through the diagonal. The line is still a bit jagged, but is approximating the diagonal a bit more. Divide the square up into a million squares and now the path traced across the square (though all right angle turns) looks almost exactly like the diagonal.
My question is: If you divide the square up into smaller and smaller squares (approaching an infinite number of squares), it seems like the path traveled will begin to map onto the diagonal and make the distance traveled root 2 miles. But if all the turns made are right angles, no matter how many are made( even infinite?), then the distance traveled will always be 2 miles. These two ways of looking at it don't reconcile, so I'm obviously missing something. What is it?
By the way, I'm leaving on a trip for a few days, so I may not be able to respond for a little while. I hope my description makes sense.
Picture a square with side length one. For our purposes, let's call a side length one mile because someone will be walking across it. Now, if a person walks the perimeter from one corner to the opposite corner, they've traveled 2 miles. If they walk across the diagonal, they've walked the square root of 2 miles. Simple enough.
Now, let's say the inside of the square is marked out like a chessboard. If a person walks from one corner to the other, but instead of going straight across the diagonal, they trace around the squares that the diagonal runs through (either all black squares or all white squares). So they keep turning at right angles as they work their way across the square, being very close to the diagonal, but never actually walking on it. Now, I know that as long as they turn at right angles, they never cut out any distance, and the distance traveled will still be two miles. (That seems to be a common fallacy, that cutting across a shape making rightangle turns somehow saves distance.)
Where I get stuck is by imagining more turns being made at smaller intervals. Say the square had 256 squares instead of 64 and the individual again walks around the squares cutting through the diagonal. The line is still a bit jagged, but is approximating the diagonal a bit more. Divide the square up into a million squares and now the path traced across the square (though all right angle turns) looks almost exactly like the diagonal.
My question is: If you divide the square up into smaller and smaller squares (approaching an infinite number of squares), it seems like the path traveled will begin to map onto the diagonal and make the distance traveled root 2 miles. But if all the turns made are right angles, no matter how many are made( even infinite?), then the distance traveled will always be 2 miles. These two ways of looking at it don't reconcile, so I'm obviously missing something. What is it?
By the way, I'm leaving on a trip for a few days, so I may not be able to respond for a little while. I hope my description makes sense.
Re: Traveling across a square
This is covered pretty well here.
Re: Traveling across a square
Being a short distance from the diagonal is a different condition than approximating its length. In fact, to be assured that you approximate the length of a curve the normal vectors of your line segments must approach the normal vectors of the curve. There is a standard limit/geometry problem which demonstrates that the same thing happens to 2d surfaces in 3d. It also demonstrates a problem with taking a limit of two variables.
Take a right circular cylinder. You can tessellate the curved portion with [imath]n[/imath] rows of [imath]m[/imath] triangles. However you take both [imath]n[/imath] and [imath]m[/imath] to infinity, the tessellation gets closer to the surface in terms of distance. However, depending on how you take [imath]m[/imath] and [imath]n[/imath] to infinity, the limit of the area can be anything between the actual area of the curved portion and [imath]+\infty[/imath] (inclusive).
Take a right circular cylinder. You can tessellate the curved portion with [imath]n[/imath] rows of [imath]m[/imath] triangles. However you take both [imath]n[/imath] and [imath]m[/imath] to infinity, the tessellation gets closer to the surface in terms of distance. However, depending on how you take [imath]m[/imath] and [imath]n[/imath] to infinity, the limit of the area can be anything between the actual area of the curved portion and [imath]+\infty[/imath] (inclusive).
Re: Traveling across a square
The issue depends on how you define the "limit" of a sequence of curves as opposed to a sequence of numbers. In an appropriate sense, the limit is a fractal, and then the reason its length doesn't behave the way you would expect it to is that it is not, in a precise sense, either 1dimensional or 2dimensional.
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Re: Traveling across a square
Torajirou, how are you defining limit for the curve to converge to something other than the diagonal of the square?
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Re: Traveling across a square
let's think in algebra
if there are n amount of squares then the side of the squares is [imath]1 / \sqrt{n}[/imath]
then thanks to pythagoras we know that the diagonal of the squares is [imath]\sqrt{(1 / \sqrt{n})^2+(1 / \sqrt{n})^2}[/imath], or [imath]\sqrt{2/n}[/imath]
now the amount of squares on the diagonal will be equal to [imath]\sqrt{n}[/imath]
then times it by the diagonal length of the squares [imath]\sqrt{2/n} * \sqrt{n} = \sqrt{2n} / \sqrt{n}[/imath], or [imath]\sqrt{2}[/imath]
anyway if you go using right angles then you would be going [imath]\sqrt{n} * 1 / \sqrt{n} = 1[/imath] in both x and y directions
if there are n amount of squares then the side of the squares is [imath]1 / \sqrt{n}[/imath]
then thanks to pythagoras we know that the diagonal of the squares is [imath]\sqrt{(1 / \sqrt{n})^2+(1 / \sqrt{n})^2}[/imath], or [imath]\sqrt{2/n}[/imath]
now the amount of squares on the diagonal will be equal to [imath]\sqrt{n}[/imath]
then times it by the diagonal length of the squares [imath]\sqrt{2/n} * \sqrt{n} = \sqrt{2n} / \sqrt{n}[/imath], or [imath]\sqrt{2}[/imath]
anyway if you go using right angles then you would be going [imath]\sqrt{n} * 1 / \sqrt{n} = 1[/imath] in both x and y directions
Last edited by keeperofdakeys on Mon Feb 23, 2009 6:02 am UTC, edited 1 time in total.
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Re: Traveling across a square
I look at the problem this way:
If you are never technically moving in a strictly diagonal direction, then your x distance is still in intervals of dx and the y distance in intervals of dy. The sum of all the dx intervals is still 1 mile, and the same for the dy intervals.
It's kind of like a fractal then?
If you are never technically moving in a strictly diagonal direction, then your x distance is still in intervals of dx and the y distance in intervals of dy. The sum of all the dx intervals is still 1 mile, and the same for the dy intervals.
It's kind of like a fractal then?
Re: Traveling across a square
It's simple. "Length" is not a continuous function on curves. You might as well be surprised that the floor of 1e is 0 for small e>0, but the floor of 1 is 1.
Not all useful functions are continuous. This is especially true when you start taking limits of "big" things like curves.
Not all useful functions are continuous. This is especially true when you start taking limits of "big" things like curves.
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Re: Traveling across a square
antonfire wrote:It's simple. "Length" is not a continuous function on curves. You might as well be surprised that the floor of 1e is 0 for small e>0, but the floor of 1 is 1.
Not all useful functions are continuous. This is especially true when you start taking limits of "big" things like curves.
Unl[imath]\varepsilon[/imath]ss you'r[imath]\varepsilon[/imath] using a nonstandard d[imath]\varepsilon[/imath]finition of 1, e, "', or floor, [imath]\lfloor 1e \rfloor = 2 \neq 0[/imath]....
***edited all things that should be 'e' into [imath]\varepsilon[/imath], and left all things that should be [imath]\varepsilon[/imath] as 'e', just because I can.
Last edited by Yesila on Mon Feb 23, 2009 3:58 pm UTC, edited 1 time in total.
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Re: Traveling across a square
I think by 'e', he means [imath]\epsilon[/imath].
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Re: Traveling across a square
More important is to actually consider the topology of the space of paths. It seems obvious that lim (sawtooth curves) = diagonal, but what do you actually mean by limit? Most people intuitively use the C^{0} topology (that is, with the sup norm). But length isn't even defined on C^{0}, as many curves have infinite length. Furthermore, even where it is defined, length is not continuous.
But you're using the wrong topology. If you use the C^{1} topology, you get all the right answers. Of course, since length is usually defined to be [imath]\int_a^b f'(x) dx[/imath], it's natural to consider length as a function on C^{1} (or piecewise C^{1}), and also to use the C^{1} topology. Here, length is a continuous function.
However, lim (finer sawtooth functions) doesn't exist here. You might think it's limit is the diagonal, but since the derivative of the diagonal is constantly in the diagonal direction, and the derivative of the sawtooth is constantly either vertical or horizontal, the diagonal is not the C^{1} limit.
The whole question, "What is the right topology to put on my infinite dimensional space of functions?" is functional analysis. Limits of functions have nontrivial technicalities, depending on what structure you're trying to preserve.
But you're using the wrong topology. If you use the C^{1} topology, you get all the right answers. Of course, since length is usually defined to be [imath]\int_a^b f'(x) dx[/imath], it's natural to consider length as a function on C^{1} (or piecewise C^{1}), and also to use the C^{1} topology. Here, length is a continuous function.
However, lim (finer sawtooth functions) doesn't exist here. You might think it's limit is the diagonal, but since the derivative of the diagonal is constantly in the diagonal direction, and the derivative of the sawtooth is constantly either vertical or horizontal, the diagonal is not the C^{1} limit.
The whole question, "What is the right topology to put on my infinite dimensional space of functions?" is functional analysis. Limits of functions have nontrivial technicalities, depending on what structure you're trying to preserve.
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Re: Traveling across a square
I must admit ignorance when it comes to topology... so Cycle's post went a fair bit over my head. When I think of the limit of the staircases, I think:
Let [imath]f_n : [0,1] \to [0,1]^2[/imath] (for natural n > 0) be the staircase path from (0,0) to (1,1) with n "steps". For example:[math]f_1(x) = \begin{cases} (0,2x) & x < 0.5 \\ (2x  1,1) & x \ge 0.5 \end{cases}[/math][math]f_2(x) = \begin{cases} (0,2x) & x < 0.25 \\ (2x  0.5,0.5) & 0.25 \le x < 0.5 \\ (0.5,2x  0.5) & 0.5 \le x < 0.75 \\ (2x  1,1) & x \ge 0.75 \end{cases}[/math]And so on.
Then we can call the "limit" of these curves [imath]f_\infty(x) = \lim_{n \to \infty} f_n(x)[/imath]... for instance, [imath]f_\infty(1/3)[/imath] would be the limit of f_{1}(1/3), f_{2}(1/3), etc, etc... Now, with some pokery, I'm pretty sure it'd be possible to prove that [imath]f_\infty(x) = (x,x)[/imath]... (possible proof outline: I think that each coordinate of f_{n}(x) will be within 1/n of x... since it'll be in the step of the staircase that steps over (x,x), and the size of that step is 1/n... I could make this more rigorous if I wasn't lazy, but it's kinda a side point).
So by this, [imath]f_\infty(x)[/imath] is the basic diagonal... just what intuition asks for.
Then we have:[math]arclen(\lim_{n \to \infty} f_n(x)) = arclen(f_\infty(x)) = \sqrt 2 \text{ (since it's a simple diagonal)}[/math][math]\lim_{n \to \infty} arclen(f_n(x)) = 2 \text{ (since it's 2 for any finite n)}[/math]So arclen simply isn't continuous.
Cycle: is this definition of the limit of a sequence of functions equivalent to one of the definitions you mention? Or is it something completely different and wrong?
Let [imath]f_n : [0,1] \to [0,1]^2[/imath] (for natural n > 0) be the staircase path from (0,0) to (1,1) with n "steps". For example:[math]f_1(x) = \begin{cases} (0,2x) & x < 0.5 \\ (2x  1,1) & x \ge 0.5 \end{cases}[/math][math]f_2(x) = \begin{cases} (0,2x) & x < 0.25 \\ (2x  0.5,0.5) & 0.25 \le x < 0.5 \\ (0.5,2x  0.5) & 0.5 \le x < 0.75 \\ (2x  1,1) & x \ge 0.75 \end{cases}[/math]And so on.
Then we can call the "limit" of these curves [imath]f_\infty(x) = \lim_{n \to \infty} f_n(x)[/imath]... for instance, [imath]f_\infty(1/3)[/imath] would be the limit of f_{1}(1/3), f_{2}(1/3), etc, etc... Now, with some pokery, I'm pretty sure it'd be possible to prove that [imath]f_\infty(x) = (x,x)[/imath]... (possible proof outline: I think that each coordinate of f_{n}(x) will be within 1/n of x... since it'll be in the step of the staircase that steps over (x,x), and the size of that step is 1/n... I could make this more rigorous if I wasn't lazy, but it's kinda a side point).
So by this, [imath]f_\infty(x)[/imath] is the basic diagonal... just what intuition asks for.
Then we have:[math]arclen(\lim_{n \to \infty} f_n(x)) = arclen(f_\infty(x)) = \sqrt 2 \text{ (since it's a simple diagonal)}[/math][math]\lim_{n \to \infty} arclen(f_n(x)) = 2 \text{ (since it's 2 for any finite n)}[/math]So arclen simply isn't continuous.
Cycle: is this definition of the limit of a sequence of functions equivalent to one of the definitions you mention? Or is it something completely different and wrong?
Last edited by phlip on Mon Feb 23, 2009 9:07 pm UTC, edited 1 time in total.
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Re: Traveling across a square
phlip wrote:is this definition of the limit of a sequence of functions equivalent to one of the definitions you mention? Or is it something completely different and wrong?
It's different, but not necessarily "wrong." There are many ways of defining the limit of a sequence of functions. Your definition is known as pointwise convergence: [imath]f_n \to f[/imath] pointwise if and only if [imath]f_n(x) \to f(x)[/imath] for all [imath]x[/imath].
But pointwise convergence is not always good enough. For example, the pointwise limit of continuous functions may not be continuous. A stronger notion of the limit for functions is uniform convergence: [imath]f_n \to f[/imath] uniformly if and only if [imath]\sup_xf_n(x)  f(x) \to 0[/imath] which means that the functions must eventually be close to the limit function everywhere in the domain (at the same time). Uniform convergence preserves continuity.
But it does not preserve arc length, as we've seen. For this, we'll need an even stronger notion of limit, so we'll restrict ourselves to differentiable curves (or piecewise differentiable curves) and require that not only the functions converge uniformly, but that their derivatives do as well. This is C^{1} convergence.
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1+1=3 for large values of 1.
Re: Traveling across a square
t0rajir0u wrote:The issue depends on how you define the "limit" of a sequence of curves as opposed to a sequence of numbers. In an appropriate sense, the limit is a fractal, and then the reason its length doesn't behave the way you would expect it to is that it is not, in a precise sense, either 1dimensional or 2dimensional.
It isn't really fractal because there isn't really self similarity under expansion, it is more that, simply put, lines that wiggle are longer than lines which are straight, and smaller tighter faster wiggles can expand the length by just as much as a slow wiggles. Also note that the Hausdorff dimension of everything in sight here is 1. Er, well, everything in my cylinder example has a dimension of 2.
In order to discuss the limit of a sequence of sets it is easiest to have some sort of distance measurement in mind. (There are other ways to define limits, but...) The Hausdorff distance is perhaps the most obvious. In [imath]\mathbb{R}^2[/imath] area is continuous with respect to this distance. In [imath]\mathbb{R}^3[/imath] volume is. Therefore things like the disk and washer pictures in Calc II are in some sense justified. Also the boxes in the Riemann integral. However, area is not a continuous function of the Hausdorff distance in [imath]\mathbb{R}^3[/imath], nor is length in [imath]\mathbb{R}^2[/imath], and therefore you have to be more careful when measuring these lower dimensional structures.
Edited to add:
Bah, now I think that my statement that area is continuous under the Hausdorff distance needs some qualification. Consider say the distance between a space filling curve in the unit square and the unit square. I think it just need to be restricted to closed sets, but it is late and perhaps I need to reread some geometric measure theory tommarrow.
Re: Traveling across a square
Mathmagic wrote:I think by 'e', he means [imath]\epsilon[/imath].
I know, but I like to point out other peoples "errors" by being a jerk. Maybe it would have been more appropriate to indicate that there is a way to typeset the [imath]\varepsilon[/imath], but wheres the fun in that?
**edit; I found the fun in that... well I found fun in doing it that way AND the way I did before. Previous post edited to reflect said fun.
Re: Traveling across a square
Well, if the meanings of symbols are contextindependent, the permittivity of what material are you talking about, Yesila?
TeX takes longer to type and makes the page slow to load (and jerky while it's loading), so I try to avoid it when it's not necessary. The use of epsilon in that context is conventional, but using another variable doesn't change the meaning, as I hope everyone reading this thread is aware.
TeX takes longer to type and makes the page slow to load (and jerky while it's loading), so I try to avoid it when it's not necessary. The use of epsilon in that context is conventional, but using another variable doesn't change the meaning, as I hope everyone reading this thread is aware.
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Re: Traveling across a square
I remember seeing this problem somewhere as a teenager. Possibly in Asimov on Numbers?
Re: Traveling across a square
This is pretty funny because I came up with this walking home from school one day and I've been puzzled about it ever since! On the other hand, I don't get much of the math in this thread :p
Re: Traveling across a square
You mathematicians always overthink everything. If you're walking strictly horizontally part of the time, and strictly vertically part of the time, sum all the displacement vectors with the same direction, and you get d_{x}=1 and d_{y}=1. 1+1=2.
Re: Traveling across a square
"We mathematicians" also think about convergence properly. You're assuming that total displacement is continuous, and there've been a lot of very intelligent comments in this thread about what kind of lengths are and aren't continuous in various topologies we could place on the space of paths.
Re: Traveling across a square
But it does not preserve arc length, as we've seen. For this, we'll need an even stronger notion of limit, so we'll restrict ourselves to differentiable curves (or piecewise differentiable curves) and require that not only the functions converge uniformly, but that their derivatives do as well. This is C1 convergence.
So if C1 functions f_{n} converge uniformly to f, then the arclength of f is the limit of the arclengths of f_{n}? If so, are those the strictest conditions?
EDIT: Completely misunderstood the above quote when I read it initially. _
Last edited by z4lis on Tue Feb 24, 2009 5:55 am UTC, edited 1 time in total.
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Re: Traveling across a square
z4lis wrote:But it does not preserve arc length, as we've seen. For this, we'll need an even stronger notion of limit, so we'll restrict ourselves to differentiable curves (or piecewise differentiable curves) and require that not only the functions converge uniformly, but that their derivatives do as well. This is C1 convergence.
So if C1 functions f_{n} converge uniformly to f, then the arclength of f is the limit of the arclengths of f_{n}? If so, are those the strictest conditions?
What do you mean by strictest? When [imath]f[/imath] is differentiable, we can find arclength with the integral [imath]\int \sqrt{1+f'(x)} dx[/imath]. Now, [imath]C^1[/imath] convergence ensures convergence of this functional, almost by definition. So the condition is sufficient. I do not believe that it is necessary. Both in the sense that one could also have some sequence which just happen to be convergent, and in the sense that there are various nontrivial relaxations of this condition studied in geometric measure theory: fractional powered derivatives and such.
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Re: Traveling across a square
z4lis wrote:So if C1 functions f_{n} converge uniformly to f, then the arclength of f is the limit of the arclengths of f_{n}? If so, are those the strictest conditions?
EDIT: Completely misunderstood the above quote when I read it initially. _
No. It's not enough for C^{1} functions to converge uniformly to f, they must converge in the C^{1} metric d(f,g)=max(sup(fg),sup(f'g')). Those surely aren't the most general conditions under which the statement is true, however, since you can produce examples where f_{n} > f pointwise, and the lengths converge as well, without f_{n} > f in the C^{1} metric.
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Re: Traveling across a square
Wow, I think about this one all the time. xkcd related places are really creepy deja vu for me.
Basically, I think of it in terms of a velocity.
For my example, 1m by 1m square, you move at 1m/s
If you use "taxicab" movement, it takes you 2 seconds to do, correct?
Even if you keep making the movements smaller, it still takes you 2 seconds to do, because you keep crossing the diagonal, back and forth, smaller and smaller, until it seems you are going completely diagonal.
However, if you go diagonally, it takes rt(2) seconds to travel, right?
This makes much more sense in my head, I'm sorry. x_x
Basically, I think of it in terms of a velocity.
For my example, 1m by 1m square, you move at 1m/s
If you use "taxicab" movement, it takes you 2 seconds to do, correct?
Even if you keep making the movements smaller, it still takes you 2 seconds to do, because you keep crossing the diagonal, back and forth, smaller and smaller, until it seems you are going completely diagonal.
However, if you go diagonally, it takes rt(2) seconds to travel, right?
This makes much more sense in my head, I'm sorry. x_x
Re: Traveling across a square
A much better generality I think is something like Lipschitz functions, or absolutely continuous functions. Probably with a metric like [math]\sup_{x \in [0,1]} f(x)g(x) + \limsup_{\epsilon \to 0} \sup_{xy< \epsilon} \frac{f(x)  g(x)}{xy}.[/math] I say something like that metric, because I'm not sure that's natural, or even well defined. You can fix it, the standard ways to make a norm bounded. Then you can define length as [math]L(f) = \sup_{0=x_0<x_1<...<x_n=1}\sum_{i=1}^n f(x_i)f(x_{i1}).[/math] (Or you can take the easy way out and just take 1 dimensional Hausdorff measure.) It's a much larger class of curves, with a normed linear space topology, so that length is continuous function to R. I don't know if that's the best you can get though.
Hausdorff measure, in any dimension, is not continuous on the Hausdorff (unfortunate naming scheme here) topology on the set of compact sets. (Did you read Edgar too?) Consider, for example, the sequence A_{n} = {q \in [0,1] ; q is rational and the denominator of q is less than n} converges to the set [0,1].
Hausdorff measure, in any dimension, is not continuous on the Hausdorff (unfortunate naming scheme here) topology on the set of compact sets. (Did you read Edgar too?) Consider, for example, the sequence A_{n} = {q \in [0,1] ; q is rational and the denominator of q is less than n} converges to the set [0,1].
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