Geometric convergence is tricky. Specifically, the issue here is that uniform convergence does not imply convergence in length or area or any other such measures. (see figure 6 here: http://www.sci.utah.edu/~etiene/publications/verifiable-vis.... . disclaimer - I'm a co-author)
Imagine a circle and its diameter. Now imagine two circles with half of the diameter, lined up so that the diameter lines align. Now split those two in four, etc. The circle becomes a snaking line whose total length doesn't change, and the snaking line converges uniformly to the line. Clearly, however, pi is not 1.
What you need is convergence in position _and_ angle. A curve that converges in position and angle _does_ converge in length: the reference I know which shows this is reference [9] on the above-mentioned paper. (edit: in case you don't want to download the gigantic file --- yay for publishing in graphicsy places --- the reference is: K. Hildebrandt, K. Polthier, and M. Wardetzky. On the convergence of metric and geometric properties of polyhedral surfaces. Geometriae Dediacata, (123):89–112, 2006.)
>>...uniform convergence does not imply convergence in length or area...
Under uniform convergence, the limit of the integrals is equal to the integral of the limit. So, this works fine for areas, or are you talking about something else?
I was trying to say that convergence in area for two-dimensional surfaces in R^3 requires convergence in normals in the same way that convergence in length for one-dimensional curves in R^2 requires convergence in normals.
For area in R^2, volume in R^3, and so on, you're definitely right.
In a similar way, you can "prove" that sqrt(2) = 2 by approximating the diagonal of a unit square (length = sqrt(2)) by iteratively dividing two adjacent edges of the square (length = 2).
One interesting and useless law that this reveals is that the perimiter of any convex blob of pixels is equal to the perimiter of its bounding rectangle.
In fact, just a few days ago, I discovered this law while working on my master's thesis. I'm modelling cells in a Monte Carlo model on a square grid, and I needed the contact area with other cells. The contact area seemed to be to large on diagonal interfaces. After thinking about it a bit, I found the problem, and I started using a different measure for contact area.
Just because the conclusion is wrong it does not mean that there is no point here. What’s described is exactly what one has to deal with in digital image analysis. Here is a write-up and a link to a paper: http://inperc.com/wiki/index.php?title=Lengths_of_curves. The paper proves essentially that there is no good way to approximate lengths of curves with any grid, even if the size of the mesh goes to 0.
Nice and fun, and brings into focus the definition of "distance" and how it interacts with the concept of a metric. Using the L<sub>1</sub> metric gives a different value of pi than using the L<sub>2</sub> metric. Obviously.
It's easy enough with arguments like this to show the "length" to be anything you choose. Such demonstrations are instructive.
EDIT: Changed lower-case ell to upper-case for clarity.
> “The use of Manhattan distance leads to a strange concept: when the resolution of the Taxicab geometry is made larger, approaching infinity (the size of division of the axis approaches 0), it seems intuitive that the Manhattan distance would approach the Euclidean metric [...] but it does not. This is essentially a consequence of being forced to adhere to single-axis movement: when following the Manhattan metric, one cannot move diagonally (in more than one axis simultaneously).”
Yes, you've provided the most succinct and accurate explanation here. The root observation of this thought experiment revolves around properties of the l1 norm, and is an important set of concepts to understand property in implementing distance metric-oriented ML algorithms like k-means clustering.
You can "disprove" the Pythagorean equation the same way, by taking the limit of the Manhattan distance between two points as the grid size approaches zero. It approaches d_x + d_y, not sqrt(d_x^2 + d_y^2).
Understanding you to be a distinguished algebraist (that is, distinguished from other algebraists by different face, different height, etc.), I beg to submit to you a difficulty which distresses me much.
If x and y are each equal to 1, it is plain that
2 * (x^2 - y^2) = 0, and also that 5 * (x - y) = 0.
Hence 2 * (x^2 - y^2) = 5 * (x - y).
Now divide each side of this equation by (x - y).
Then 2 * (x + y) = 5.
But (x + y) = (1 + 1), i.e. = 2. So that 2 * 2 = 5.
Ever since this painful fact has been forced upon me, I have not slept more than 8 hours a night, and have not been able to eat more than 3 meals a day.
I trust you will pity me and will kindly explain the difficulty to Your obliged,
that's not really the point because we are dealing with algebra and not the numberical values. The error is the assumption that (x-y)^2 equals (x^2-y^2), which is not the case.
Hmm, no. Look again. It's using the fact that (x^2 - y^2) = (x - y)(x + y), which is the case, and a common enough identity (“the difference of two squares“) that it's used without remark here.
The problem really is that you can't divide by zero, even in an algebraic expression.
A simpler example of this phenomenon (which blew my mind when I first encountered it) occurs with the equation x = x^2. If you divide by x, you get x = 1, which is a solution to the equation, but where did the other solution x = 0 go??
Whenever you divide an equation by an algebraic expression, you need to consider the possibility of that expression being zero and treat it as a special case. So in the case of x = x^2, you can reason as follows: maybe x = 0, in which case … what … ah yes, that's a solution! Or maybe x ≠ 0, in which case we can divide by it and get x = 1. That doesn't contradict the assumption x ≠ 0, so it's okay, and x = 1 is the other solution.
It is the case, since it was a premise that x and y are both 1. So the two things you list are both 0, so they are equal, given the premise.
So, the actual problem is dividing by zero. Your assumption that "we are dealing with algebra and not numerical values" is false because it completely ignores the "if x and y = 1" part.
Good one, took me a minute to realize that the mistake occurs in the step with the squares--you have to square the units as well. So what you really get is:
When I first saw a "proof" like this, the explanation of its incorrectness was something like "limits don't preserve curve length". I wasn't satisfied with that answer until I took a first course in real analysis, which explained (some of) the reasons behind calculus.
Real analysis was very satisfying, in somewhat the same way that building low-level software or libraries is satisfying -- I got to understand the guts. It was also fun to learn about erroneous historical assumptions made due to insufficient rigor. IIRC, until at least the 1870s, it was believed that any continuous function must be differentiable almost everywhere, and that in fact this should be obvious. It turns out that one can construct continuous functions that are nowhere differentiable!
> Real analysis was very satisfying, in somewhat the same way that building low-level software or libraries is satisfying -- I got to understand the guts.
That is exactly the reason why taking a course in real analysis made me switch majors. I was trying to get that feeling from physics, but only found it to my satisfaction in math.
What is even more tricky is that the set of functions differentiable in at least one point is first category, and thus nowhere differentiable continuous functions are dense in continuous functions space.
mblase's explanation was a very clear way of explaining the problem, with the two parallel lines never quite touching because you can't get a line to zero length by cutting it in half repeatedly, no matter how many times you do it. Thanks for sharing.
I actually found mblase's explanation to be just as inaccurate as the one in the original story. "Infinity is not a number and you can never get there" is never a good defence in well-formed arguments about limits. It just sidesteps the question and says that it is meaningless, incidentally doing the same to all of mathematical analysis.
"Just because a series of curves tends to a limit curve doesn't mean that the series of lengths of those curves tends to the length of the limit curve" is the whole of the explanation.
For any given n, all the points of the "right angled fractal beast" lie within some distance epsilon(n) of the circle. As n increases, epsilon(n) approaches zero, so in the limit, you have a circle, and at every step along the way, the length is 4. It's just that the length of the resulting shape isn't the limit of the lengths of the sequence of shapes.
If we write down area and perimeter error e_A and e_P as a function of n, in the limit of infinite n e_A is proportional to 1/n. However, e_P is independent of n, since the perimeter does not change at any step. Thus, if the perimeter was unequal to begin (clear by inspection), it does not get closer through this approximation.
The mistake is in accepting that anything that looks like a circle must be a circle. The fractal beast has area arbitrarily close to that of a circle, but it's clear once we look at e_A and e_P that it is not an approximation of its perimeter.
Pointwise, the limit is a circle. Every point on the enclosing shape gets mapped to a sequence of points. Each of these sequences has a limit, that limit is on the circle. The resulting implied mapping of the original square to the circle is a continuous bijection. By every sense that we usually talk of the convergence of lines, this enclosing shape does approach the circle in the limit.
The point is that the mapping involved is not a length-preserving mapping.
You make a good point intuitively, but I still don't quite buy your argument.
The circle is continuously differentiable along its circumference. The fractal is not---it's still got a large number of discontinuities (offhand, it looks like the number of discontinuities is equal to 2^(n-1) after n steps).
The area enclosed by this curve approaches that of a circle, but the curve is not a circle: it cannot be described by x^2+y^2=1 because its derivatives are not equal to those of same.
Every point on the limit of the fractal is within epsilon of some point on x^2 + y^2 = 1, for every epsilon > 0. So, if some point on the limit of the fractal were not on the circle, it would be some finite distance d from the circle. By choosing epsilon = d/2 you have a contradiction.
The circumference of the limit doesn't equal the limit of the circumferences.
That doesn't follow from what I wrote. Every shape in the sequence of shapes has a circumference/length of 4, and is not a circle. The limiting shape is a circle and does not have a circumference/length of 4, but of pi.
We can have a bijection from the original square to the circle. Why is this a surprise? It's not length preserving. We can have a bijection from the interval [0,1] to the interval [0,2]. That's not length preserving either.
If I've misunderstood you then perhaps you could explain your thinking in more detail. I don't understand where you think there are two versions of a circle.
Only countably many points "arrive" at their destination in a finite number of steps. There are uncountably many points that never "arrive," but whose limit is on the circle.
The usual picture given to non-mathematicians about sequences and limits can end up being strongly misleading in cases like this. Just because a sequence never "gets there," the limit is still the limit. This is the same kind of murky area that talks about 0.999... recurring never "getting to" 1. It doesn't have to "get to" 1 because it's never travelling.
It's also the kind of problems that arise when talking about proof by induction. Talking about dominoes falling down is, in the longer term, very misleading. We prove P(1), and we prove that P(n) => P(n+1), then they are all true. They don't become true one by one, they are simply all true - it is what it is.
I hope that helps.
I'm thinking of starting a blog to talk about things like this - it falls between the levels of the non-mathematician and true researcher.
I think I see what you're getting at, but I guess I still don't fully understand. Any thoughts on proving that the perimeter is pi without using the knowledge that the limiting shape is a circle? Does that question make sense? To me it seems strange that for any integer number N (number of times this chunking operation is applied), the perimeter is 4, but somehow in the limit the perimeter is not 4.
Edit: Nevermind, reading cousin_it's posting I think I've got a handle on it. My confusion was exactly the difference between a sequence of approximations, and the limit itself.
Here is another way to see what is happening, that may be more clear from an intuitive sense:
From the puzzle, name the square object S and the circle inside it C_1.
Imagine another circle C_2 that circumscribes the square S from the puzzle. i.e. the corners of the square lie on the circle C_2. Then for each step, when we invert the outer-most corners of the square, we constrict C_2 such that the circle lies on the new outer-most points of S.
What happens is that as you repeat this process more and more the outer circle C_2 gets smaller and smaller, approaching the size of the original circle inside the square, C_1.
Also you can infer that the area of S is equal to the area of C_1 and C_2 since (area C_2) -> (area C_1) and (area C_1) <= (area S) <= (area C_2). Which makes sense intuitively, too, since they all enclose the same space.
This tells you nothing about the relationships between the circumference of the objects, though.
What... ouch. This is not math, this is... I don't even.
Read the post closely. The author introduces a set S of squarey curves approximating a circle. This set has an obvious correspondence with the natural numbers: there is curve #1, curve #2, etc. Then the author defines a function f: S->S that takes curve #n to curve #n+1. Wait, that doesn't sound right! The function f has no interesting structure whatsoever, it's exactly equivalent to defining f(n)=n+1 on the naturals. Of course, taking the "limit" of a function f: S->S makes no sense at all.
What would make sense is taking the limit of a certain function N->C, where N is the naturals and C is the set of all curves on the plane. That is, the limit of a sequence of curves (not of a function from curves to curves as the OP tried to say). To talk about such limits, you need to define what it means for a sequence of curves to converge - a "topology" on C. There are many ways to do that, some more outlandish than others. One way is pointwise convergence: assume a parameterization t->C_i(t) on each curves in the sequence, and require that C_i(t) converges co C_lim(t) for each t separately.
Now, pretty much any reasonable notion of "convergence" on the space of all curves has to imply pointwise convergence. That is, pointwise convergence is a very "weak" notion of convergence: if we have a sequence of curves that has a limit in some reasonable sense, then it had better coincide with the pointwise limit, dontcha think?
And here we come to the second facepalm moment in the post. Under pointwise convergence, the sequence of squarey curves under discussion does not converge to some "right angled fractal beast". It converges to the circle. As n grows, every point on S_n comes closer and closer to some point on the circle. Ain't nothing more to it.
Now the correct explanation for the original puzzle. Pointwise convergence of curves doesn't imply that their lengths converge to the limit's length. Hell, we don't even need 2-dimensional space to show that! A simple straight line will do. Imagine a human traveling a straight road of 1km length in this fashion: he takes two steps forward, then one step back, then repeats. In the end he will have traveled about 3km instead of 1km. As we make the human and his steps tinier and tinier, his movement looks smoother and smoother to an external observer, but he still travels 3km in total instead of 1. Or maybe (going back to the 2D space) the human could take a step left, then forward, then right; this would make his path look like a fine comb that approximates the straight line more and more closely, but it's always 3x longer. Something like this is happening in the original puzzle.
Finishing touch: there are notions of convergence where it's true that the length of the curves in the sequence always converges to the length of the limit. One such notion says that the direction of travel (velocity vector) must also pointwise converge to the velocity vector of the limit curve. Under this definition of convergence, the original sequence of curves does not converge, because it makes too many sharp turns.
On a less trollish note, I had always been puzzled by the related question you discuss: why Manhattan Distance does not converge to the Euclidean distance.
I'm pretty uneducated about math, so please let me know what trap I'm falling into.
Your answer isn't yet helping me because you just note that Manhattan Distance remains the same no matter how many twists and turns there are. Yes, this is the premise of the question!
But turn up the twistiness to infinity. Now the Manhattan Distance line is identical to the hypotenuse. There is no point on the Manhattan distance line isn't also on the hypotenuse and vice versa. They both have an infinite number of points, of the same aleph-number, I think.
You mention a hypothetical path that doubled back on itself -- it is easy to see why that would result in a different answer. But the Manhattan Distance line is not doubling back on itself. Every point is a step towards the goal, and it doesn't cross itself. All of its deviations from the simple hypotenuse are infinitely small.
There's no such thing as "infinitely small" deviation. The Manhattan Distance limit line is the hypotenuse, and its length is the length of the hypotenuse. The question is not about the limit line, which does not exhibit any weird behavior. The question is about the sequence of approximations to that line: why the length of the approximations doesn't converge to the length of the limit. And the answer is that it doesn't have to, because even though the approximations are very similar to the limit line in one respect (geometric closeness), they are all very different from it in another respect (directions and angles of travel). If we had a sequence of approximations whose direction of travel converged correctly, the length would converge correctly too.
Playing with infinities is dangerous business, and it's very easy to yield paradoxes like the OP. But note that before the 20th century, mathematicians like Gauss would avoid playing with such ill-defined objects, for precisely this reason[1]. Their rule was that you should first derive an expression in the finite case, then take the limit to infinity at the end.
In this case, it's mathematically simple that a Manhattan line should have the same length, no matter what N (the number of steps) is.
I think nowadays most mathematicians are "comfortable" with such aberrant results, but that was not always the case. And there's still a chance that future generations of mathematicians will dismiss this work.
Perhaps I could more easily answer your question if you tried to explain why you think the length of the zig-zag line should approach the length of the straight line. It's zig-zagging - why should it end up the same length? You say that in the Manhattan Distance every point is a step towards the goal, but it's not directly towards the goal, therefore you always get a bit more than you want.
The number of points isn't relevant - the Cantor Set (also known as Cantor Dust) is an uncountably large subset of the real line, and yet its length is zero.
I guess the only insight I can offer at this point is that length has nothing to do with points.
I think he is not asking what he wants to know. He wants to know how we know why Archimedes' solution does converge to the right length. Couldn't there be a problem with his solution that we're just not seeing yet? i.e. what makes Archimedes right and the troll wrong?
Here's a thought: Archimedes's approximation approaches the target. That is, the perimeter of each subsequent approximation is closer to that of the circle than all that came before.
The side of a regular polygon is √(sin²(τ/n)+(1−cos(τ/n))²)=√(2(1−cos(τ/n))), while the corresponding arc is τ/n.
d/dn τ/n−√(2(1−cos(τ/n))) = −(τ/n²+sin(τ/n)/√(2(1−cos(τ/n)))) < 0, so the difference is strictly decreasing (note that sin(τ/n)>0 when n>2).
So there's your lower bound for the difference. You'd still have to prove the limit is at 0, though.
That doesn't explain why. Why is it that if tangents approach the circle (whatever that means precisely) that the length of the approximation approaches the length of the circle?
Loosely speaking, the total path length is defined as the sum of a bunch of infintesimal segments that have both position and direction. For the sum (integral) of those infintesimals to approach the sum of the similar infintisemals around the actual circle, those of the approximation path have to approach those of the circle in the limit, and of course this means in both position and direction.
The reason it won't converge is that a very large number of the points will ALWAYS lie off of the line. While you might be able to make something that looks like a line on a computer screen when you make it smaller and smaller segments of manhattan distance. it will in fact only ever be able to intersect the real line (they are infinitely thin remember) at N+1 points at most (where N is the number of turns you make when following it along the grid in manhattan distance). because of this you will always have a very large portion of your approximation lying off of the line.
What you (simcop2387) say doesn't help, as I can produce curves that do as you describe and yet they do converge. It's possible to have convergence even when all the points lie off the line (in the only sense I can understand you to mean) so your comment isn't really an explanation.
> The reason it won't converge is that a very large number of the points will ALWAYS lie off of the line.
No they won't. It moves zero distance horizontally and zero distance vertically. All the points of the infinitely zigzagging line are on the hypotenuse.
Of course, now we're back to Zeno; if it's not moving off the line, and it only makes progress when it's off the line, how can it get anywhere?
For every fractally-type enclosing shape, yes they will. And when you talk about the limit, then you have a circle.
You can't talk about moving "zero distance horizontally and zero distance vertically" becuase then, as you rightly say, we're back to Zeno. You don't have an infinitely zigzagging line. That way of trying to think about things is a dead end, and unhelpful.
Better answers are given elsewhere in this thread.
Put another way, the right angles approximation as N->infinity puts a countable infinity of points exactly on the line.
The true line has an uncountable infinity of points on the line, and the difference between an uncountable infinity and a countable one is an uncountable infinity, so we're still way the hell off.
+ There are families of curves where at no stage do any of the points lie on the circle, and yet the limit is the circle, and the limit of the length is the length of the circle.
In short, your comment isn't really explaining anything.
ADDED IN EDIT: Whoa - cool - a downvote! Please, let me know where I'm wrong. I thought I'd explained clearly why the comment was wrong, and I have explained elsewhere more about what's really going on. No doubt you, the downvoter, have moved on now, but whoever you are, I'd love more information as to what you think I've got wrong.
You can also demonstrate this is incorrect by calculating the area of the parts of the square cut off, which is the sum of an infinite geometric series. You will find it doesn't equal the area of the unit square minus the area of the inscribed circle.
Here, let's try: the square is 1x1. Consider one quarter of the circle: radius of the circle is 1/2, half a diagonal of the square is sqrt(2) / 2, diagonal of the removed square is (rt(2) - 1) / 2. Area of the square removed is ((rt(2) - 1) / 2) ^ 2. (This is trivial via the pythagorean theorem, saving some math.)
Alright, that's the first square we accumulate. Now the magic happens: every step, we cut the square's side in half, but make two of them. Agree with me so far? Good. If we cut the side of a square in half, we cut its area to a quarter, but since we have two squares now the total area is 1/2 of the last square. Agree with me so far? Good. We can trivially sum infinite geometric series: t1 / (1 - r), where t1 is the first term and r is the fraction each term gets multiplied by. In this case, it turns out that in any one quadrant the sum of the series of squares removed is 2 * ((rt(2) - 1) / 2) ^ 2, or just (rt(2) - 1)^2 / 2.
Multiply by 4 to get the picture over all four quadrants, and we get 2 * (rt(2) - 1) ^ 2. A little simplification and we get 2 * (2 - 2 rt(2) + 1) = 6 - 4 rt(2)
So, we've got a unit square, so the area of the square is 1. If we subtract the area of the infinite series of squares, we get 4 rt(2) - 5 =~ .657. We expect the area of the circle to be pi / 4 =~ 0.7853975. Thus, the square minus and infinite series of squares doesn't approximate the known area of a circle at all.
This isn't quite right; when you divide up your problem, your "quarter circle" doesn't look the same after each iteration, and so the sizes of the squares cut off don't follow a geometric sequence. In fact, I'm sure that the area of the figure does approximate the area of a circle (because no point inside the circle is ever removed, yet every point outside of it will be removed eventually).
The problem lies not with the area, but the perimeter; the figure's area turns out to be unrelated for this problem.
Hmh. Its 1 AM here. Give me some time to sleep and think over that. You might be right: I can't remember why I thought the squares would go geometric. I thought I had a better reason than "It looks that way on the picture, kinda...", but it eludes me.
If A_n is a sequence of sets bounded by these square like curves, then m(A_n) -> pi, where m is a Lebesgue measure. Each set is clearly measurable, since it is closed in Euclidean topology, and thus Borel. You can prove this sequence converges by seeing that each A_n is enclosed in some circle of diameter 1+e for some e, and, as n -> \infty, e -> 0.
Right. This way, you aren’t approximating the circle. And if you were, the area would converge to the right number. Areas are good this way, unlike lengths which is what the post is about.
The paradox lies in "infinity" and "never". Achilles will overtake the tortoise when he's one atom away from the tortoise, and similarly at one point your corner removal will reach the atom level, where you can no longer reduce it and maintain a square shape.
At atom level, when your squares consist of three atoms in a L pattern, you can't reduce it further without distorting the squares
Assuming, of course, that atoms are the smallest particles.
I think the analogy is flawed. Using limits we can resolve Zeno's arrow paradox. For the situation described here limits cannot solve the problem because it turns out that it is not an accurate model of a circle.
Because in Infinity you can't predict things with sight. Actually, it seems like the rectangles have become a lined curve, but in reality they aren't. They are just too small to be noticeable.
The easiest way to informally prove that the demonstration is false is to imagine starting with a circle in a triangle instead of a circle in a square. Or with any other weird shape around the circle and follow the same "cutting" algorithm to infinity. This way you can prove that pi is equal to anything greater than pi.
try to find the perimeter of australia..
depending on the zoom level you find a different perimeter.
:)
if the 'troll' from the article adds smoothing to his 'ad infinitum' squarish circle then he finds pi.
Imagine a circle and its diameter. Now imagine two circles with half of the diameter, lined up so that the diameter lines align. Now split those two in four, etc. The circle becomes a snaking line whose total length doesn't change, and the snaking line converges uniformly to the line. Clearly, however, pi is not 1.
What you need is convergence in position _and_ angle. A curve that converges in position and angle _does_ converge in length: the reference I know which shows this is reference [9] on the above-mentioned paper. (edit: in case you don't want to download the gigantic file --- yay for publishing in graphicsy places --- the reference is: K. Hildebrandt, K. Polthier, and M. Wardetzky. On the convergence of metric and geometric properties of polyhedral surfaces. Geometriae Dediacata, (123):89–112, 2006.)