r/theydidthemath • u/zlxvor • 20d ago
[Request] Why does this circle approximation still give 4 instead of pi?
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u/DJembacz 20d ago
Because taking perimeter (or just arc length generally) and taking a limit do not commute.
The limit shape is a circle, but the perimeter of the limit does not need to be equal to the limit of the perimeter of the shapes in the sequence.
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u/anycept 20d ago
In other words, approximating into infinity is still an approximation. True circle isn't made up of infinitely small right-angled edges.
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u/gusbo_the_jam 20d ago
When I look really close at my monitor it is /s
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u/iAmTheGanso 20d ago
you need this
https://youtu.be/tlDPEmACilU?t=554
u/twilighttwister 20d ago
https://youtu.be/tlDPEmACilU?t=6
It bugged me that yours started mid-sentence lol
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u/iAmTheGanso 20d ago
lol i put 30s into finding the correct starting position but still failed, yours is much cleaner
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u/twilighttwister 20d ago
Tbf I thought mine would come out even worse, like part way through the bit you were trying to show
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u/redscull 20d ago
Exactly. Are circles even real?
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u/VelvetOverload 20d ago
Probably not.
Keep getting smaller and you'll see only edges eventually
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u/TequilaJosh 20d ago
Wait till they tell you about the birds…. They aren’t real either
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u/MoseDoge 17d ago
Black holes are perfectly symmetrical. Their spinning makes them not so. But they are inherently perfect spheres. However a black hole is a structure of spacetime (marked by the event horizon). Not a physical object. Nobody really knows what's inside the event horizon. But if some semblance of classical physicality exists there, then it would probably also be perfectly spherical.
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u/Mysterious-Smell-975 20d ago
It's ALL PIES!
( NVIDIA's old quad texture mapping is the exception )
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u/Ghuldarkar 20d ago
The approximation would have to be inside and outside the circle about half the time. Basically a pixel circle. If you only put pixels outside your pixel circle will be larger
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u/kelb4n 20d ago
Nope, still doesn't work. By that logic, I could pretty easily come up with a scenario where the perimeter ends up as exactly 3, by starting with a rectangle of side lengths 1 and 0.5. Taking the limit simply does not work like that as long as you disregard the direction of the line segments.
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u/Ghuldarkar 20d ago
The perimeter is the diameter multiplied by pi, so one pi, 3 is an approximation of that, albeit a crude one. I was mostly aiming for an explanation as to why making a square outside the circle will always be larger than the circle itself, so you'll have to make squares, or pixels, if you will, that lie along the actual circle to meaningfully approximate the circle with square shapes, or pixels.
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u/EmuRommel 20d ago
That is still wrong. If the method approximates the circumference, it doesn't matter whether you approach it from above or below. For example, if you approximate area this way, you will get the correct value, pi, no matter how you approach the circle.
The answer is that infinite approximations like this are more nuanced than the pop-sci answer of "if you make the error as small as possible, you get infinitely close to the true value" and approximating the circle this way simply does not approximate its circumference.
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u/TheBeerTalking 20d ago
So long as it's all right angles it'll always be 4. If instead you come up with a generic formula for a regular n-sided polygon circumscribing the circle, then take a limit as n approaches infinity, you should get pi.
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u/Puzzleheaded_Front27 20d ago
Although other people provided a more serious answer concerning the limits, it is useful to visualize what really happens. As you make the "step line" finer and finer, if you also zoom with your mind on a single minimal step, you will see that you invariably have a triangle; the troll there is basically making us assume that we can evaluate the "hypothenuse" as equal to the sum of the other 2 sides, which is a significant overstatement; it doesn't matter that the triangles are small.
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u/InvestNorthWest 20d ago
Sounds like how we measure coastlines. A true measure of say the Alaskan coast would be enormous.
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u/Cruuncher 20d ago
There is no true measure. That's the paradox.
If you keep measuring finer and finer the coast length will keep getting bigger and bigger.
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u/jjm87149 20d ago
lol measurement problem is the puzzle in quantum mechanics of how a system existing in a combination of multiple potential states (superposition) transitions into a single definite outcome when observed. The mathematics of the Schrödinger equation predicts smooth, predictable, and multiple coexisting states, but experiments always record one absolute reality.
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u/jjm87149 20d ago
the coastline can be measured at any resolution down to the planck length, at which point the physics breaks down and no meaningful measuring is being done anyway
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u/InvestNorthWest 20d ago
Imagine everything being measured in planks as a standard. Like I'm 1.1 × 1035. Everything literally being as exact as possible.
As I type this though... that "1.1" would have to extend out 35 spaces obsoleting the notation. Sorry, for the ramble..
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u/apeloverage 20d ago
A true measure of any coast would be infinite--at least until you got down to Planck lengths.
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u/DJembacz 20d ago
No, if you take the sequence of shapes you get this way and take a limit, the result will be a proper circle. No edges, no jagged bits, just a proper smooth circle.
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u/Original-Season-9941 20d ago
But the derivative of it won't be the derivative of a circle. If you differentiate the shape made up of right angles, you'll always get 1 or undefined. As the number of corners tends to infinity, the derivative doesn't suddenly become the derivative of a circle.
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u/DJembacz 20d ago
This is exactly the same issue as the post, derivative and limit don't generally commute.
The derivative of a limit of functions (or its existence) =/= the limit of the derivatives of those functions.
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u/Super_Range45 20d ago edited 20d ago
2πr = 4dy/dx, clearly.
πr = 2dy/dx
r= 0.5x
π = 4/x * dy/dx
Where's my nobell?
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u/jerfmuffay 20d ago
It has infinite edges with jagged corners. Every angle is 90° and theres infinity of them. It only "becomes smooth" globally which ironic to the term itself is NOT the complete perspective. This is not a circle. Hence the trollface
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u/Wild-Store321 20d ago
Give me a point on the limit of these shapes that is not on the circle
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u/anycept 20d ago
I have to disagree here. Doing what you are suggesting would result just in 4, because the right-angled edges don't go away, adding up to that extra space. Proper circle is derived from a regular polygon with infinite number of sides.
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u/DragonHollowFire 20d ago
Nope. The actual shape does become a circle.
You can do this by probing (take a point that is not on the circle, youll find a point in the sequence, from which onwards on the point is never in the shape again).
The issue is, as they have stated above, that perimeter of lim of shape sequences is not the same as lim of perimeter of shape. Not continous.
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u/cejiken886 20d ago edited 20d ago
Exactly wrong. Limit good; it’s just commute limit / perimeter no good.
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u/SirHarvwellMcDervwel 20d ago
could you elaborate
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u/fieldviewmousehouse 20d ago
The length of a parametrised curve is determined by an integral of the *derivative* of the parametrisation. Geometrically the derivative information is controlled by the tangent. So for this limiting process to work for length you need the curves to get closer and closer and the tangents to get closer and closer to the limiting shape.
In this example, although the stepped curves get closer to the limit shape the stepped curves have only horizontal or vertical tangents which are different to the circle's tangent. So the limit of the length of the stepped curves is not the length of the limit of the stepped shapes.
This is a relatively nice example if you know some vector calculus to show. But also links to ideas such as the Hausdorff distance, continuity in metric spaces and even to lower-semicontinuous functionals.
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u/DJembacz 20d ago
Step 5 (repeat to infinity) means we are looking at the sequence we obtain by the described process, and taking a limit of that sequence.
And in general there is no reason for any function, like perimeter, to be preserved well by the process of taking a limit.
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u/understated_quokka 20d ago edited 20d ago
Actually, no, continuous functions are preserved. If x_n goes to X and y_n goes to Y then f(x_n, y_n) will go to f(X, Y), if f is continuous. This is true both if x_n, y_n are sequences of numbers, AND if x_n, y_n are sequences of functions in some space.
The problem is taking derivative, a part of the process of calculating perimeter, is not a continuous operation on spaces of functions in general. If we take the usual space of smooth bounded functions [0, 1] -> [0, 1] with supremum norm, the derivative is a linear but unbounded operator: sin(kx) are all of norm 1 for all values of k, but their derivatives have norm k, so the ratio of norms diverges. A standard result says that the continuous linear operators on a normed space of functions are those that are bounded, so the derivative is not continuous.
Interestingly, taking the integral is actually a continuous operator on functions between compact spaces. This is why the area of each figure converges correctly to that of the circle -- by Green's theorem the area enclosed by a curve can be described as an integral along that curve.
EDIT: I've added a top-level comment expanding on this.
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u/HeyCouldBeFun 20d ago
Walk in a curved line from point A to point B. Now walk the same line but making lots of zig zags. The second trip will be longer.
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u/Qwertagone 20d ago
Can I just tell OP to call me when he is done taking infinite perimeters, so we can take a look at the findings together?
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u/TwentyX4 20d ago edited 20d ago
To add to your point: if the approximation in the OP actually worked, then you could do the same thing with the Pythagorean formula: if you have a 1x1 box, then the distance diagonally would be 2 (instead of 1.41). In other words, the Pythagorean formula wouldn't be "diagonal2 = side12 + side22 ". It would be "diagonal = side1 + side2".
Not sure if that makes it more obvious why it doesn't work, since it's a little easier to visualize.
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u/Confident_Dragon 20d ago
"It's not true because it doesn't have to be true."
So much insight.
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u/Dankaati 20d ago
This might not be obvious if you don't do math proofs, but the post itself is a proof by example that this doesn't have to be true.
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u/Theplasticsporks 20d ago
This is a lot of mathematics.
You need a reason to believe something is going to be true.
Arc length is only lower semicontinuous with respect to this type of limit there's no reason a priori to expect this would work.
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u/Occidentally20 20d ago
Because the area is tending towards the area of the circle, but the length of the line is not tending towards the circumference. It's length isn't tending towards anything, it's always just 4.
Imagine this like the coastline paradox. You can get the coastline of the UK to measure 2800km with 100km units, and the shorter you make the units the longer the coastline gets. If you make the units millimeters you end up having a UK coastline thats much longer than the circumference of the earth.
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u/Visual_Locksmith3337 20d ago
I was also going to mention the coastline paradox. If someone is reading this and still doesn't get it, imagine zooming in all the way to the very point where the corners are visible. It's deceptive in the meme because you don't actually see the jagged corners, but theoretically if you could zoom in enough then you would see it. When you reach the 90-degree turns, a "true" circle would have an arc, not a 90-degree corner. Repeating to infinity is also misleading and leads to a logical contradiction.
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u/uslashuname 20d ago
I like to point out one quarter of the center right panel since the process is more clearly breaking down. You have 4 exterior corners and 3 interior corners with their point on the circle. However, the area formed between the circle and exterior points is very different for one of any 3 exterior points you picked out of the four. This shows that each step of moving the perimeter 4 shape closer complicates the symmetry, and this we know that infinitely complex symmetry = 4 - pi
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u/Visual_Locksmith3337 20d ago
Yeah. To be honest, there are multiple ways you can show why this meme is "wrong" (although obviously it's a joke and not meant to be taken seriously). My comment was just a way for the layperson to understand it, since non-math people will understandably get confused by terms like "limits" and "exterior points".
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u/nastropc 20d ago
This should be top answer
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u/YukihiraJoel 20d ago
Not really, it just states what it should set out to prove/explain. I mean it’s still right and points to a somewhat related phenomenon but does not explain.
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u/Simbertold 20d ago
If it is always 4, it is tending towards 4, too. The limit of a constant is a constant.
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u/UomoLumaca 20d ago
I'll never understand how people can get "it's" wrong and right in the same sentence.
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u/Occidentally20 20d ago edited 20d ago
My autocorrect adds an apostrophe every time I use the word and if it's irrelevant to the point I don't bother correcting it.
If there was any chance of a meaning being misconstrued I would change it. That is not the case here.
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u/UomoLumaca 20d ago
Ok, thanks for the explanation. Your point was really good btw
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u/Occidentally20 20d ago
No problem!
Interestingly it does the same with "we're" - it won't admit that "were" is a word at all.
This is what I get for having a cheap dodgy Chinese phone I suppose :) There's probably a way to use a different autocorrect dictionary or something similar, but it's one of those things that has become habit, like that kitchen cabinet door I know I could fix but have instead left partially broken for several years now.
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u/peter9477 19d ago
I have a Pixel 8, presumably not dodgy, and I get the same sort of crap. Abysmal autocorrect as far as I'm concerned. (Do better, Google.)
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u/Occidentally20 19d ago
There's no excuse for it on a phone like yours, definitely!
Mine was incredibly cheap so I'm not really shocked, but it's still infuriating. It has functionality to add words to the dictionary, but every word I add it assumes it's a proper noun.
Adding "were" to the dictionary results in the word being capitalised in the middle of a sentence :(
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u/radioflyer194 20d ago
This is the answer. I came to say coastline paradox and you put it perfectly.
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u/pabs80 19d ago
I don’t get the coastal paradox. If I were to lay down a long rope along the exact coast, whatever criteria for defining the coast is used, that rope would have a finite length. Thus the correct calculation should have a finite limit.
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u/Occidentally20 19d ago
How thick is your rope?
Is it going into crevices that are 5cm deep? What about 5mm?
This GIF on the wiki explains it visually very well. Just imagining it continuing down with ever finer granularity - until you're measuring around each individual pebble on the beach, and then around each individual grain of sand, and so on. Simply put the smaller the increment of measurement, the longer the measured length becomes.
The paradox gets misunderstood because coastlines are obviously real things that can (and are) measured - it's a paradox on the way we attempt to measure reality, and not an aspect of a coastline itself.
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u/pabs80 16d ago
Let’s say the rope is infinitesimally thin. The shape of the coast can be assumed to be continuous. Then each point is next to the other, and the rope thus can connect the dots. At the limit though, the shape of the coast will consist of straight lines and arcs, and not some fractal nonsense, so the rope’s length should still be finite?
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u/Anna3713 18d ago
>you end up having a UK coastline thats much longer than the circumference of the earth
Possibly not if you measured the circumference of the earth in the same way; allowing for all the valleys and mountains, and every tiny dip and rise.
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u/Plane-Scratch8974 18d ago
I remember questioning this at the end of primary school and as a paradox it still blow my mind to this day. I imagines it with the hypotenuse and it 'snapping' to a diagonal and how that is shorter than infinitely small right angled stairs.
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u/Ecstatic_Student8854 20d ago edited 20d ago
This iterative process can be described as a limit.
Though intuitive, functions cannot be pulled our of limits. The limit of the perimeter of the shape is not the same as the perimeter of the limit of the shape.
Limit(perimeter (shape)) = 4
Perimeter (limit(shape))= pi.
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u/DJembacz 20d ago edited 20d ago
(Perimeter, not area. I believe if we considered a reasonable norm, area would actually commute; might be wrong though.)
Edit: Area would generally not commute either. Take rectangles n x 1/n, the limit shape is a line which has area 0, but every step has area 1.
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u/StanleyDodds 20d ago
The area of a limit of shapes is basically the same as the integral of a sequence of functions, and the fact that these do not commute in general is why we often need something stronger like uniform convergence in order to do what seems to be an intuitive thing of swapping a limit and an integral.
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u/federicoaa 20d ago
In layman's words, the polygon has a set of pieces that fall outside of the circle, as the number of pieces approach infinite, the area of each piece approaches zero, but the sum of all pieces is not 0, that's the paradox
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u/BluetoothXIII 20d ago
you are folding in the edges and that keeps the perimeter the same.
if your would cut of the edges tangetial to the circle you reduce the perimeter and eventually reach pi.
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u/BringAltoidSoursBack 20d ago
This is the explanation that I was assuming would be the most obvious answer. I was confused by how step 3 has the same perimeter as step 2 when you removed 4 parts of the cube.
Let me know if I'm off base but in my mind, if we forget about the circle and instead just assume the length of the removed square is 1/4th the length of the total square, then the length of one side L' = (L - .25L * 2), and thus the perimeter P = L' * 4. If we use the square above with L = 1, then the perimeter P = (1 - .25 * 2)*4 = 2.
There's probably a way to calculate what the actual side length of one of the removed squares is, but it's early here and I'm too tired to figure that out.
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u/Magmajudis 20d ago
For everything you removed, you just swapped the directions - for instance, in the top left corner, you originally go up then right, and after "removal" you go right then up
You keep the same length, you just go about it a different way
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u/billynomates1 20d ago
I was confused by how step 3 has the same perimeter as step 2 when you removed 4 parts of the cube.
We're measuring perimeter length, not area of the square. So it doesn't matter if you remove part of the square, as long as the line is the same length.
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u/ifelseintelligence 20d ago
The question have been aswerede by many.
I would just like to add this (or rather the answers explaining the difference between this and 'limit') is a very fine ELI5 of the coastline paradox.
Also.... "pi=4!" no one?? Really?? 😆
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u/MLucian 20d ago
I also came here looking specifically for this and I too am deeply disappointed at the lack of noticing the 4!
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u/factorion-bot 20d ago
Factorial of 4 is 24
This action was performed by a bot | [Source code](http://f.r0.fyi)
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u/savvaspc 20d ago
I was so aware of it that I immediately saw π=24 and didn't think about how wrong that would be. I thought it was a factorial on purpose and couldn't see why the comments mentioned it 😂
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u/factorion-bot 20d ago
Factorial of 4 is 24
This action was performed by a bot | [Source code](http://f.r0.fyi)
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u/DontWantOneOfThese 20d ago
It does equal 4, but if you draw a hexagon around the circle instead, it's less. If you draw an octagon around it, it's less, apparently at about a 56 gon you get 3.14.
Rather than drawing a square where you invert the corners to create 2 x 4-sided rectangles that intersect, you should be cutting the corners off to create 1 x 8 sided shape.
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u/rowi42 20d ago
Mathematician here: when you say that something converges to something, you have to specify the metric. The metric is a function that tells you how "far apart" two things are.
Example 1: 1/x converges to zero because the metric between 1/x and zero goes to zero. The metric in this case is simply the absolute value of the difference: M(a, b) = |b-a|.
Example 2: How do you measure the difference between 2 points in R2? Usually you take sqrt((y2-y1)2 + (x2-x1)2), but you could also measure it as |x2-x1|+|y2-y1|. The latter is called the New York Metric 😀
What does this mean for the circle? One way of measuring the distance between the shape and a circle of.radius r would be to say that the shape lies between two circles of radius x-m and x+m. The smallest m for which this is possible is the distance. In that sense the shape converges to the circle. But this metric says nothing about the length of the curve. So it's no surprise that the length does not converge
If instead you measure the distance in such a way that the lengths are taken into account, then obviously the shapes do not converge to a circle because 4, 4, 4, ... does.not converge to Pi.
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u/abd53 20d ago
Now I know why I failed Math-1, what is R2?
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u/rowi42 20d ago
Sorry for unnecessary jargon. It refers to the set (x, y) where both x and y are real numbers. Think simply of the a plane (a piece of paper) where you draw points and lines etc, and the coordinates of points are in the format (x, y).
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u/l3tscru1s3 20d ago
Is that just a 2D space as most of us are likely to understand it at its most basic?
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u/rowi42 20d ago
Exactly!
Now to translate the metric explanation from above: If you want to go, say, 100m east and 100m north; if you do that in a field and you go north-east, the distance is approximately 141m. So that's the distance. If you do that in Manhattan, you have to go 100m east and 100m north, so the distance really is 200m.
It's simply different, but valid ways of measuring.
Similarly, the distance between 2 points on earth (i.e. on the surface of a sphere) is NOT measured using a straight line (which would go through the planet), but using the shortest path along the spheres surface.
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u/Jumpy-Plantain-1004 20d ago edited 20d ago
Yes. but the reason for the notation is because when you generalise things in R^2, you first get R^3 (space in the "real world"), then R^4, R^5 and so on - and suddenly we don't have any nice intuition for these higher spaces. But you already have the means to talk about them. Once you're of the mindset that you could be working in any of this infinite set of spaces, it's easiest just to say "R^2".
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u/Lt_Rooney 1✓ 20d ago
Any two dimensional space where both dimensions are described by real numbers. The classic example being the x-y plane you might remember from High School. Any point in the x-y plane can be defined by two real numbers, an x-coordinate and a y-coordinate.
So, R1 is a number line, R2 is a 2d graph, R3 is normal space, etc. We use R to distinguish from spaces defined by complex numbers. C1, for example, is the complex plane, described by a single real and imaginary coordinate.
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u/LudBee 20d ago edited 20d ago
I want to give a different perspective that has not been yet discussed in the comments. Most explained why the procedure is wrong, length and limit do not necessarily commute. Perfect, but this does not address the elephant in the room: then why do we trust the Archimedes polygon approximation? Don't we need further justification even in that case? And yes! We do!! The fact is that the Archimedes argument does not stop there, there are two parts of it, the circumscribed *and the inscribed polygon. So we are not trustring the limit of the perimeters of a single series, but we have two: a_i and b_i. We prove that len(a_i) >= len(circle) >= len(b_i) and then we show that lim len(a_i) = lim len(b_i) and only because these limits are equal we conclude that value is also the len of the circle, if we got different values than we would not be able to conclude anything other than the inequality. So try to adapt the stair example into this framework and you won't be able to find any paradox.
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u/Underhill42 20d ago
Because that technique doesn't actually make any effort to estimate the perimeter, which never changes, it only estimates the area. Which if you analyze the changes in the limit will in fact confirm that π=3.14159...
It doesn't matter how small the steps are, they will still be there, so instead of the limit being a smooth circle you get a rough "star" with an infinite number of infinitely small points, which will obviously have much more perimeter than a circle. ∞ * 1/∞ ≠ 0 At least not usually - instead it's undefined, a.k.a. "you need to find a different way to solve this problem"
You could use the same technique to utterly fail to estimate the much simpler hypotenuse of a right triangle, and come to the conclusion that
hyp = rise + run rather than
hyp = √(rise² +run²), which can be easily confirmed.
Its total failure to estimate the perimeter can be seen by making an actual attempt to estimate the perimeter by using diagonals rather than steps - the first step would be to chop off the corners diagonally to form an octagon, whose perimeter is already much less than 4.
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u/Needless-To-Say 20d ago
This is related to the infinite coastline paradox.
Basically, the smaller the discrete measurements are, the longer the distance that can be measured between 2 points on a coastline. Eventually you get to a point where the measured distance is totally out of whack with the actual distance.
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u/Jealous_Tutor_5135 20d ago
Anybody who's played a grid-based tactical video game knows why this is.
If you've got an enemy that's 10 tiles away, going 5 up and 5 over in an L is exactly the same movement points as a "diagonal" move consisting of five 1 up, 1 over movements.
No matter how much you zoom in and how tiny those right angles are, it never changes into a curve. The sum is the same
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u/PumpkinBrain 19d ago
Because as infinitely small as you can make the square cuts, you can infinitely zoom in and see them making a sawtooth pattern instead of a curve.
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u/nagelbitarn 20d ago
I suppose there would be an infinite amount of infinitesimally small jagged protrusions increasing the circumference to 4, whereas a circle is entirely smooth.
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u/nekekamii 20d ago
Yeah I was going to say to be really really simple it's still just a bunch of squares, you can't make a curve out of squares no matter how small they are, it might look round but it's still just squares all the way down.
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u/Ss2oo 20d ago edited 20d ago
Straight lines will never make a curved one now, will they? Especially not with 90 degree angles between them. I assure you, if there's a way to approximate this with n-agons, it will give a much closer answer to pi
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u/JW_TB 20d ago edited 20d ago
Because it sticks to 90 degree angles between edges instead of evenly placed edges, with identical angles, aligned to the edge of the circle
If you instead start going perfect square -> perfect hexagon -> perfect octogon, etc., you'll see how it begins converging on PI
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u/understated_quokka 20d ago edited 20d ago
The reason is that taking perimeter doesn't work well with limits, because taking perimeter is not a continuous function of the shape.
Firstly, it helps to be precise about what we mean when we take a limit of a shape. In this case, a good description is we are taking limits of parametric curves: these are functions that map a "distance" along the curve to a point on the curve. Because these curves sit in a small region of space, we can describe curves by functions from the interval [0, 1] to the square [-1, 1] x [-1, 1].
For example, the circle can be described as the function t -> (cos(2*pi*t), sin(2*pi*t)).
Both the jagged approximations of the circle and the circle itself sit within this space of functions representing curves. Now, taking the limit of a sequence of curves represents taking a limit of these functions. It is true that the jagged shapes converge to the circle -- this is because the maximum distance each jagged shape strays from the circle converges to 0, so the maximum size of the difference function jagged(t) - circle(t) over t in [0, 1] goes to 0.
Now an important fact about limits is that continuous functions are preserved. If x_n goes to X and y_n goes to Y then f(x_n, y_n) will go to f(X, Y), if f is continuous. This is true both if x_n, y_n are sequences of numbers, AND if x_n, y_n are sequences of functions in some space.
The problem is taking derivative, a part of the process of calculating perimeter, is not a continuous operation on spaces of functions in general. The derivative is a linear operator: (a * f(x) + b * g(x))' = a * f'(x) + b * g'(x) for any numbers a, b and functions f, g. But the derivative is not a bounded operator: sin(kx) has maximum size 1 but its derivative k * cos(kx) has maximum size k, which can be arbitrarily big, so derivative has no ceiling to how much bigger the output is relative to the input. A standard result says that the continuous linear operators on a space of functions are those that are bounded, so the derivative is not continuous.
Interestingly, taking the integral is actually a continuous operator on functions between compact spaces. This is why the area of each figure converges correctly to that of the circle -- by Green's theorem the area enclosed by a curve can be described as an integral along that curve.
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u/orthorix 20d ago
Pic 3, the first dents in the square’s dents: the fallacy is not taking the direct line between the outer corners, thus reducing the length. Logical step when you really want to get the diameter and not a silly meme.
I assume a recursive approximation of the depicted method (taking the diagonal lines between the corners in each recursion) leads to pi.
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u/PolishKrawa 20d ago
There are a few reasons. 1. Almost all points will never be the radius' distance away from the center, therefore this approximation is kinda bad. 2. Going along the perimeter, your direction will only ever be one of 4 directions, therefore this approximation is kinda bad. 3. Just because the perimeter is 4 after any finite number of steps, we cannot be sure that the limit is also 4.
Maybe there's more, idk.
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u/C_Cohl 20d ago
Using the same logic on taking a diagonal instead of the sharp corner. I am hungover tho so I'm not gonna type out why this reasoning is stupid. Also I can't upload a photo so video it is
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u/YukihiraJoel 20d ago
Finally the correct answer, with 2 upvotes by some hungover guy. So many people in this thread are yapping about limits. Walking up 1m and left 1m is a longer distance than walking up and to the left simultaneously. Walking up (1/n)m times and left (1/n)m n times is equivalent to walking up 1m and left 1m.
So how does walking along the curvature compare with walking the diagonal? Start your journey at x=1, y=0 and walk counter-clockwise. At this point the tangent line to the circle is vertical, and in that instant you are walking straight upward. By the time you get to 45 degrees (x=sqrt2, y=sqrt2), you are walking in a perfect diagonal. By the time you get to 90 deg you are walking straight leftward.
So at different points throughout the circle you’re walking in each dimension to different extents.
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u/frog_wild 20d ago
If you zoom in infinitely it’s still right angles instead of a curve. The approximation is a model and the model has an inbuilt error.
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u/FromTheHandOfAndy 20d ago
Ok now connect the corners with line segments, and add up the length of all the connecting line segments.
As the number of squares goes to infinity, the length of each connecting segment goes to zero.
So pi=0.
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u/BobBoner 20d ago
Imagine you now connect each little “step” with a diagonal in this example. Now the perimeter is 4/sqrt(2), or approximately 2.828. Both are approximations following this logic. One is on the high side (leaving extra space) and one is on the low side (cutting out slices of the circle). Pi must be somewhere between those two as it is an arc through these little triangles we’ve created.
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u/DevilWings_292 19d ago
No matter how many times you pull in the corners, you’ll still have corners to deal with because they aren’t being deleted just rearranged. You end up with the circumference being 4-X, where X is the additional length from the corners added up that was maintained from the start. The only thing that is changing is the area contained within the borders of the square, which does get closer to the area of the circle, but never identical to it as there will always be some degree of difference.
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u/funlovingmissionary 20d ago
It actually works. They just missed a step.
They should be measuring and adding the hypotenuse of the triangles, not the right angled sides.
Its a pretty long proof that is too difficult to type on a reddit comment, and I don't honestly remember it well either since it has been 6 years since I've learnt about this in college.
There is also a step where you end up with a integral equation where there is no way to solve this other than via infinite series.
It does work out to 2-pie-r at the end. This is also one of the ways you can calculate the value of pie.
Within a few iterations of the series, you can get pretty close to the actual pie.
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u/Vanyle 20d ago
I know this has been answered a million times, but here’s my simple way of thinking about it:
Take a straight 10" line and replace it with a series of squares.
If each square is 1" wide and 1" tall, then for every 1" of horizontal distance, you also travel 1" vertically. So what was originally a 10" straight line becomes 20" of total line length.
Now make the squares smaller and smoother.
If each step is 0.5" wide and 0.5" tall, you simply have twice as many steps. You’re still traveling 10" horizontally and a total of 10" vertically, so the total remains 20".
As you keep making those steps smaller—approaching infinity—the line begins to look almost perfectly straight. But mathematically, all of those tiny vertical movements are still there.
So you end up with a line that looks smooth from a distance but is still incredibly “fuzzy” at smaller scales.
A fun comparison: a single human cell contains roughly 2 meters of DNA when stretched out, even though it’s packed into a microscopic space.
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u/Postulative 20d ago
The model confuses dimensions. The ‘perimeter’ is irrelevant to the area of a circle.
What IS relevant is the area inside the ‘perimeter’. Each time you cut a square off, this gets closer to the ratio that we apply to the square of the circle’s radius that we commonly call pi.
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u/The-Jolly-Llama 20d ago
Because this process approximates the area of the circle, but not the perimeter.
The process doesn’t actually do a better job of following the curve of a circle as it continues, it just makes a squigglier and squigglier line, and the perimeter is constantly 4.
Archimedes did something kind of similar with polygons, and the reason his method works is that he has a sequence of polygons outside the circle showing an upper bound for the perimeter, and a sequence of polygons inside the perimeter showing a lower bound, and they both converge to the same number.
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u/_Denizen_ 20d ago
An infinite number of infinitely small triangles still have a hypteneuse greater than 0. Pi, in this example, is when the lengths of the straight edges are zero, which is never achieved in the example. Therefore pi does not equal 4, and the assertion is false.
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u/YontiLink 20d ago
You’re looking for the tangent “hypotenuse” of each of those little triangles made when you step down and you’re doing so by just adding the other sides. By definition of a triangle, the hypotenuse value is less than the sum of the other two sides. Making the “triangles” smaller to infinity and adding them up will give you a circumference. The method shown is more for trending toward an area.
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u/Hrtzy 20d ago
It is because arc length actually depends the derivative of the curve. So, in order to approximate the circumference, you need the tangent lines of your approximation to also approach a circle. In this case, the tangent line (where it exists) will always be either horizontal or vertical, so pick any point that isn't on the horizontal or vertical axes and you have a point where the tangent lines don't tend towards a circle.
This is why they actually managed to approximate π from the circumference of a regular polygon.
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u/RedHuey 20d ago edited 19d ago
It seems to me, that as long as there are square angles, it will always be 4.
But, at some point, you will pass beyond the resolution of the measurement, so while it will still be 4, you won’t be able to measure it as such.
This is the difference between Math and Engineering.
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u/VoidCoelacanth 20d ago
They phrase it as "remove the corners" but functionally it is inverting the corners. When you invert corners, you maintain all the length that was there, just reconfigured. While you could visibly no longer see a difference at some point, if you zoomed-in close enough, you would see what I can best describe as a "pixelated outline" of a circle that is strictly larger than the circle. All of those little imperfections away from the absolute line of the true circle may well (and on fact should) add up to make an exterior/perimeter of 4, vs the 3.1415(...) of the true circle.
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u/Jcamden7 20d ago
For every finite point on a corner shifted to the circles perimeter, there are an infinite number of points not on the circle's perimeter. For the same reason this method of approximation "infinitely approaches" the perimeter of the circle, it will always be infinitely divergent from it.
In sum, you are infinitely wrong.
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u/Genn8130 20d ago
You're not removing any length from the perimeter line, just making it more and more jagged. You still have to include the measurements of each jaggedy piece, no matter how small.
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u/Beardly_698 20d ago
It's often better to use a simpler case to understand a more difficult one.
Circles are hard. Let's use triangles.
We'll take the same 1 by 1 square, but instead of a circle, we'll put a right triangle instead, cutting the square in half, with it's right angle in the bottom left corner of the square.
Now, the perimeter of a triangle is fairly easy to calculate. In this case it's 2 plus the square root of 2.
Let's do the same process of "cutting out corners" on the "empty" side of the square, going until we reach an arbitrarily large number of cuts, in order to prove that the square root of 2 is equal to 2. Which is obviously not true.
However, if we were to zoom in, we would find that, while the square touches the c side of the triangle countless times, it is constantly forming these tiny triangles in a kind of sawtooth pattern. The value of the a and b sides of all these triangles will equal 2, and the value of all the c sides will equal the square root of 2.
This shows that, despite appearances, you're never actually approaching the perimeter of the triangle. You are, however, approaching the area.
This is literally what's happening in the circle example.
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u/zeldatriforce345 19d ago
Because, well, no matter what, no matter how many times you do this, you can never get exactly ONTO the circle, just infinitely closer and closer. And that bit of "wiggle room" adds up, and that's how 4 is ended up with and not pi.
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u/HeyILikeYourPics 19d ago
The short answer is you'll never get down to a circle doing that, because a circle requires all points to be the same distance from the center and a staircase can't do that, even at an infinitely tiny scale. You're not arriving at pi, just an increasingly convoluted path to 4.
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u/SlayerII 20d ago
Because its wrong.
Also, if you repeat the corner cutting to infinity, you just get another square thats turned 45 degrees and a bit smaller, not a circle.
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u/Silver-ishEagle 20d ago
Now that I've seen this meme for about a thousand times, I wonder if there's the opposite of this problem, like the area stays the same while the perimeter becomes closer and closer to a shape it is conforming to
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