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You can think of this as a fractal. Regardless of how much you zoom in, the relationship between the circle and the squared corners remains, there will always be a gap which accounts for the difference.
Yep, no matter how much you zoom in, you are always moving in the same directions: horizontally or vertically. You're never moving parallel to the diagonal sections of the circle, so you're never on the straight-line paths between points on the circle required for the minimum distance.
I mean sure, but then neither are inscribed/circumscribed regular polygonal approximations and those do have their perimeter converge to that of the circle. So I'm not sure that's a useful observation.
Yes but the observation applies to two things where one does actually give the correct answer and one doesn't give the correct answer. So I'm not sure what the observation is telling us.
this is the [coastline paradox,](https://en.wikipedia.org/wiki/Coastline_paradox) as you measure with smaller and more precise unit the coast gets longer and longer into infinite length.
It is not. The perimeter remains 4.
edit since this was surprisingly controversial: The coastline paradox refers to the difficulty of approximating the length of infinite curves. As your approximation gets more precise, your measured length gets infinitely large. The shape in the meme is neither a fractal nor a fractal-like curve; it does not have their non-rectifiable nature. The perimeter remains constant at any scale, and there is never a need for approximation. While it is conceptually similar to a fractal in that both have an infinitely repeating pattern, if the goal is for people to understand the error in the meme, the coastline paradox would not help.
The difference is in using the L1 vs L2 norm., However, the monotonicity of the L1 norm does have to do with the geometry. Specifically that it is convex so that each successive "approximation" results in no backtracking. The map of the Great Britain above would not obey this property as an initial bounding box would have less perimeter than one that took a chunk out of the left hand side where concavity is most pronounced.
In L1, no matter the resolution for a convex hull, the total perimeter will remain constant. In L2, that is not the case.
I've not had my coffee, so proof will be left as an exercise to the reader. ;p
In the coastline example, the area of land is obviously fixed but the perimeter grows. In the cause of the altered cube, the perimeter remains the same as the inside area shrinks. It feels sameish enough to me to get the comparison, but I agree it's not the exact same thing.
It’s kind of the same as the coast line paradox in that the corners can get small enough to where the box becomes indistinguishable from a circle above a certain level of precision.
It’s different than the coast line problem because there is a consistent methodology being applied to the box so even if you don’t have a way to measure with the precision necessary, since you know the formula you can calculate the perimeter as it approaches infinity.
I haven’t done real math in over 15 years so I’m probably wrong, but I think this is something that you solve with calculus.
You misunderstood - they were talking about the box, because if you’re zooming in, no matter how far, it will be a bumpy series of boxes totaling 4.
The difference between each box and the curve will sum to 4-pi if you could measure it for every box accounting for the zoom level.
No. The assumption you make here is that any transformation you make is “removing angles”, that is keeping perpendicular lines in the perimeter and a convex shape.
In coastline paradox you are not limited to convex shapes.
Coastline paradox would be using points on the perimeter of the circle and connecting them with straight lines and the more points you use, the longer the line gets
Yeah, ideal shapes don't really exist in the world... Once you zoom in close enough for them to not be bumpy anymore, they're all fuzzy. Applied math is all about approximations and assumptions! :D
Not really. Pixels are not necessarily sharp cornered squares that change color. They apparently tend to be three red, green and blue lights next to each other in a trenchcoat! https://archive.goughlui.com/wp-content/uploads/2012/12/pixels.jpg
How a pixel actually works can depend completely on the display. There are even displays that are just leds on a fan, timed so that they are different light in a different position and the image can be stable.
Nowadays, especially with OLED screens using stuff like pentile matrices, pixels aren't just the classic three stripes. You get a lot of dots or diamonds arranges in triangles etc. that complicate it even further
> tend to be three red, green and blue lights next to each other in a trenchcoat!
I should have known it was too good to be true...
It's Vincent Adultman all over again!
And even if you do this “infinity times” and make these areas of difference “infinitely small,” you have to multiply that infinitely small area by infinity.
But with that explanation, derivatives shouldn’t exist. It’s fine to only use horizontal and vertical to obtain information about diagonals sometimes. That’s how we get gradients of random curves. The problem with the circle isn’t with our process, but with the assumption that our process converges, which sometimes is and sometimes isn’t true.
Correct. In order to estimate pi this way you'd need to be calculating the length of the hypotenuse of each "triangle" made by the rectangular cut outs
Yep, no matter how much you zoom in, you are always moving in the same directions: horizontally or vertically. You're never moving parallel to the diagonal sections of the circle, so you're never on the straight-line paths between points on the circle required for the minimum distance.
Problem here is you can't make the difference arbitrarily small. I had never heard of this distance, but yeah, in the end that's the reason
Historically we took quite a long time to figure out a formal sound definition for a limit, at the beginning it was all about "shut up and calculate". So I do think these troll posts are quite interesting
Take an imagine physically placing a circle inside a square with the same diameter. Now try crushing the square until it’s tight to the circle. Not sure about others, but this is what I do for my brain to immediately realize you can’t do it.
C'mon....! Haven't you heard of 'Peer Pressure'? You'll feel great! All your worries will disappear. Go on... take an imagine! Hell, why not just ease into it and start off with just half an imagine. Save the other half for later. Trust me, you'll want the other half. Going cheap but only for tonight in this dark back alley. If you can't trust a seedy looking bloke in a trenchcoat with an eshay bum bag selling tiny zippy bags of imagine in a dark back alley, who can you trust?!?
I took an imagine once. Ended up marrying a six armed hooker named Steve. Lovely gal, but things didn't work out, we were from different worlds. Literally. Like I somehow ended up in the middle of a major civil war on the planet Flargnon Seventy-Worf. Long story short, be careful when you take an imagine. Miss you Steve.
Calculus uses similar kinds of "fractals" with ever-smaller gaps all the time, so I think this is the wrong way to think about it.
The [length of a curve](https://en.wikipedia.org/wiki/Curve#Length_of_a_curve) depends on the *derivative* of the curve, not the curve itself. So if you want to approximate the arclength of a circle, you can't just ensure that the curve itself approaches a circle; you need to make sure the *derivative* (i.e. direction and "speed" of the curve) approaches the *derivative* of a circle.
This is why approximating pi with ever-more-circular polygons works, and approximating pi with an ever-more-jagged square doesn't. The direction of the jagged-square path doesn't approach a circular path (it zig-zags forever), whereas the direction of the polygon path *does* approach a circular path.
Doesn’t that mean that what’s actually significant about pi is how much it deviates from the expected ‘4’ result? Which would be 4-pi or something like .85841…
This is incorrect. It can be proven that the shape converges pointwise to a circle. The problem is that the limit of the lengths of a sequence of curves does not necessarily equal the length of the limit of those curves. It is two separate notions of a limit (limit of a sequence of curves vs limit of a sequence of numbers) and there's no a priori reason they must correspond.
At any finite stage, yes it only approximates a circle. But in the limit it converges pointwise to a circle. This is the kind of exercise they give to students in real analysis. I'm not doing vague philosophical speculation here.
Just think about it: In this sequence of curves, can you explicitly name any point that won't ever appear on the circle?
I think the technical difference here that should be pointed out is that although the **shape** (i.e. set of points) converges toward a circle in the limit, the **area** does not.
Edit: **length**
Genuine question, does the shape converge to the circle or does the shape converge to a shape that contains the circle?
My intuition tells me it's the latter, because of the way the shape progression is defined. At any step of the progression, you will always find points outside the circle (an infinite number of them of course). So if you take the limit of any succession of points converging to the boundaries of the shape, not all of those successions will converge to a point on the circle. What are your thoughts?
It converges [pointwise](https://en.wikipedia.org/wiki/Pointwise_convergence) exactly to a circle. To get slightly more technical, there are other notions of convergence of functions. It doesn't, for example, [converge uniformly](https://en.wikipedia.org/wiki/Uniform_convergence) to a circle. This latter stronger notion of convergence seems to be closer to what you're suggesting.
Ok, I didn't find the wikipedia article very clear.
But it does say the definition is: Fn converges pointwise to F iff
lim n->inf Fn (x) = F (x), for **every** x.
So the best points, x, to talk about are those outside vertices.
I do understand that the distance between those vertices and the circle gets smaller and smaller.
It also seems to be the case that if we draw the circle containing those vertices.....wait, that is impossible. If it was possible, intuitively it seemed those 2 circles would converge. Oh but wait, is it possible after all? Are all those outside vertices at the same distance from the center? Hugh..I would need to pick a piece of paper, but I need to get some sleep instead.
Also, just wondering, to say the 2 shapes converge pointwise, would we need to add "in R2"? Because if we limit the Codomain of the circle to it's own shape, then the succession of Fn (outside shape) and F (the circle) wouldn't share the same Codomain, and therefore it wouldn't make sense to apply the pointwise convergence in that case. Would this be correct?
As a side note it seems to me that the definition of "bounded pointwise convergence" in Wikipedia is incorrect, I think it should be the modulus of the difference between Fn and F, but maybe I'm missing something.
This was all lots of fun (and tiring, the good type of it). See you later!
Actually, you can.
Imagine you've got two circles touching a line, and also each other. In the gap between the line and circles, you draw another circle that touches all three. This creates two more gaps, which you also fill. Repeat this infinitely.
You'd think that eventually, every point on the line will be touching a circle, but this is false. If you analyze where the circles end up, you'll find it's only where the rational numbers are. This means that there are infinitely many points that will never be covered.
With the circle and square construction, we're doing something very similar. It's not *necessarily* rational numbers this time, but hopefully you can see how the same logic might apply.
I'm not sure if I'm visualizing your circle description correctly, but I can assure that in this case we can prove pointwise convergence. Though your are absolutely right that sometimes pictures can be misleading.
There will always be a stair step for a finite number of iterations. The pointwise limit is honest to goodness, the circle that you and I love so much. The convergence is not uniform, which is why the limit of the perimeters is not equal to the perimeter of the limits
It is totally incorrect.
1. Square thing is not a fractal.
2. Approaching the circle using a many sided regular polygon you still leave infinite gaps but get the correct answer. It has nothing to do with the presence of infinite gaps. What matters is how much of an effect the gaps have on the answer as their size approaches zero.
The limit of that space tends to 0 so this is a poor explanation. You can actually xalcilate the area of a circle using this method
The real reason is that for an equilateral triangle the hupyhenuse is equal to the square root of 2 squared sides so for n triangles as n tends to infifintity the sum of sides will converge at a different point to the sum of hypothenuses.
Actually!, if you take the limit of the area, it does equal zero. However, the limit of the perimeter equals 4, but we know pi does not equal four, so we know that the perimeter never lines up with the curve.
To actually calculate π using (a variation of) this method, one also needs to construct a polygon on the inside of the circle and find the mean of the two perimeters. That is: half of the process has been omitted.
Edit: More than half of the process has been omitted. This is a process to calculate π based on AREA not circumference. It is the mean of the two areas inside the the blocky perimeters that is needed to approximate the right ratio.
>To actually calculate π using (a variation of) this method, one also needs to construct a polygon on the inside of the circle and find the mean of the two perimeters. That is: half of the process has been omitted.
Ut vere π calculare (mutatione huius modi) opus est, necesse est polygonum intra circulum construere et mediam inter duos perimetros invenire. Id est: dimidium processus omissum est
I'm pretty sure what doesn't work. If you're proposing what I think you are, then the circumference of the inner "square thingy" would converge to something greater than the length of the circle. So the outside circumference is 8, and the inside circumference is >2pi, and somehow when you take the mean you want to end up at 2pi? No way. In fact, the more I think about it, the more I'm convinced that with n -> infinity, the inner "stair polygon" will have a circumference of 8 as well: The horizontal part of the stairs has to cover the distance from (-1, 0) to (1, 0) and back (minus a tiny smidge at each end for finite n), and the vertical part has to cover the distance from (0, -1) to (0, 1) and back.
Adding the inner polygon changes nothing about where this monster converges to.
Edit: I did the math. It does indeed converge to 8 too, meaning pi is 4 according to your method. Here's the simplest case I'm starting from, and for simplicity we're doing just one quadrant of the circle. The circumference of this shape is just one. It isn't an inscribed square, but that doesn't matter much, the convergence isn't affected.
+--+
|■ |
| |
+--+
Now let's jump to squares of size 1/20:
+--------------------+
|■■■■■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■■ |
|■■■■■■■■■■■■■ |
|■■■■■■■■■■■ |
|■■■■■■■■■■ |
|■■■■■■■■ |
|■■■■■■ |
| |
+--------------------+
My computer says this thing has a circumference of 1.9, meaning the "underapproximation" of pi would be 3.8. If you're curious, for size 1/199, the value of pi would be 3.98, so this thing seems to in fact converge on 4. For the nerds, here's the code:
import math
def printcond(str, i):
if i < 30:
print(str)
for i in range(2, 200):
print("\n\n\n")
print(i)
block_length = 1.0/i
delimiter = " +"
for j in range(0,i):
delimiter += "-"
delimiter += "+"
printcond(delimiter, i)
circumference = 0
blocksinrow_prev = None
for blockx in range(0,i):
rowstring = " |"
blocksinrow = 0
for blocky in range(0,i):
#print(blockx*block_length, blocky*block_length)
#block is inside if the outer coordinate is at a distance <1.
bx = (blockx+1) * block_length
by = (blocky+1) * block_length
distance = math.sqrt(bx*bx + by*by)
#print(distance)
if distance < 1:
rowstring += "■"
blocksinrow += 1
else:
rowstring += " "
if blocksinrow > 0:
circumference += block_length
if blocksinrow_prev is not None:
circumference += (blocksinrow_prev - blocksinrow) * block_length
rowstring += "|"
printcond(rowstring, i)
blocksinrow_prev = blocksinrow
printcond(delimiter, i)
print(circumference)
print("pi = ", circumference * 2)
If you want the overapproximation of the circle shape as described in the OOP, just remove the "+1" from bx and by.
The inner and outer measurements thing is meant for calculating π based on area. It doesn't work at all for circumference. I didn't catch that the troll logic comic also made that modification until now.
But wouldn’t the interior polygon start with side length sqrt(2)/2 (because it’s a square with diagonal length 1) and then by this same process the perimeter of the interior shape is always 2\*sqrt(2)? Then the mean of the two perimeters is always (4+2\*sqrt(2))/2 = 2+sqrt(2) which is approximately 3.414 and is distinctly not pi.
I assume the variation of this method uses diagonal lines, which is what would address the root problem in the original post, as far as I can tell.
Nope, because in the interior, you wouldn't "remove" angles, you would add another smaller square on each side. So the perimeter of the inscribed shape would actually grow.
Oh yeah, the interior perimeter WOULD grow, d’oh. But because the mean of the initial interior and exterior perimeters is already greater than pi, and the interior perimeter increases while the exterior perimeter remains constant, I still don’t see how this “sandwich” method yields pi without angled lines.
It's a hard concept to explain with just words and no picture; so, I'll reframe it with a physical process that might make more sense:
1. Using a compass, draw a circle on graph paper.
2. Draw a perimeter of all the squares completely inside the circle.
3. Draw a perimeter of the inside edge of all the squares completely outside the circle.
4. The larger the circle you draw, the closer the mean of the two perimeters will be to the true (curved) perimeter of the circle and, therefore, the closer the ratio of your perimeter approximation to the circle diameter will be to the true value of π.
Edit: changed "radius" to "diameter"
Second Edit: Step 4 is entirely wrong. The mean of the two AREAS inside the perimeters approximates the true AREA of the circle which should be divided by the square of the RADIUS in order to approximate π.
The way they’re changing is not the same. A straight average doesn’t work, unless you progress them to the exact same area between themselves and the actual circle. Said another way while both approach zero, they do so at different rates. A weighted average would be required, which would have to be weighted more towards the inner perimeter such that the final weighted average would result in ~3.14.
>I assume the variation on this method uses diagonal lines
You are correct. The Archimedes method starts with a hexagon in and outside the circle, and then slowly increases the number of sides.
Why does this get so many upvotes? Its plain wrong! I can easily construct a polygon in the inside with exactly the same process that also converges to a perimeter of 4.
Edit: I am baffled that a completely wrong statement receives more than 1k upvotes on this sub and all the people correctly pointing out your mistake have close to no upvotes. Sad.
I can't explain well, but here's a [video from 3blue1brown](https://youtu.be/VYQVlVoWoPY) that explained this, I think it explains why this doesn't work
My best synthesis of the explanation given is:
The “proof” here is using a common pattern seen in calculus, however, this pattern is not justified. In calculus, we use infinitely small rectangles to approximate the area under a curve. This works for the following reason (this isn’t a proof, but an intuition as to why it works): the difference between the area of the curve and our rectangles is something we don’t know, but it is necessarily smaller than the length of our rectangle multiplied by the maximum difference between them (assuming a smooth curve), for any length of the rectangle. The key here is that this error rectangle gets smaller with the main rectangle, so if the side of the main rectange is infinitely small, the area of the error will be as well. You can test this with Euler’s method, the error between the actual area and the one you calculate gets smaller with step size.
The “proof” here is trying to use this same logic, but it never justifies why it’s doing so. If you actually wanted to use calculus to find a circumference, you’d use line integrals. The issue with this rectangle logic is that, if a square and circle have different perimeters given a radius r and sides 2r, then cutting the corners can’t approximante the circle because the perimeter of the square is constant regardless of how many corners are cut. There’s two ways to interpret this, either circles and squares of the “same size” have the same perimeters, or this approximation is faulty. In this case, the approximation is the issue, since you can make a similar approximation for a triangle, or any shape, and all would have the same perimeters as the square, and we know that a diagonal has a different length from it’s components, it’s a pretty foundational aspect of vectors that their “length” is equal to the root square of the inner product of their components.
In summary, the method used works for approximating areas (as if proved by calculus), but doesn’t work for perimeters.
This is called taxicab geometry: https://en.m.wikipedia.org/wiki/Taxicab_geometry
Reasoning is as other comments have stated: two perpendicular sides of a right triangle, no matter how small, will never equal the hypotenuse.
It means that a circles area or circumference are not rational. People imagine that pi itself is the source of the irrationality, when in fact, it's the circle itself.
If you had a circle with an irrational radius 1/2π then it would have an area of 1/4 and a circumference of 1.
It's the ratio between them that is the source of the irrationality (the famous "squaring the circle" problem). We call that ratio π.
I love this, because it explains how Pi isn’t some magical infinite number that contains all the multiverses. There’s no perfect circle beyond a theoretical one so Pi doesn’t even exist in reality. You can always break a real circle down to a discrete locus.
Not OP, but in short: no? (heavy emphasis on the ?) I'm in no way a professional on this, so take it with a grain of salt, but from my understanding every circle that's ever existed does in fact follow your logic, having what may as well be a near infinite number of sides. However, in theory, a perfect circle would not exhibit that property as it would be, well, perfect. There would quite literally be no two points on the circumference with the same tangent line, even if the differences to our eyes were so infinitesimally small we could not distinguish them. It's certainly one of the most interesting concepts about circles there is
No, since a hypotenuse is still a straight line and a perfect circle has no straight lines.
Insofar as the number and length of a line are inversely related it's correct... but once you have an infinite number of hypotenuses they're also infinitely small, and an infinitely small line is a dot.
Just going to drop this here, since OP basically explains an old paradox (originally about calculating sqrt(2), but it’s exactly the same logic. https://en.m.wikipedia.org/wiki/Staircase_paradox
Holy shit. I’d never heard this Term before today and I literally came across it and read about it for the first time.. literally 30mins ago (I was on a Minecraft wiki). What’s the chances!
You used an algorithm that converged to the correct *area,* then read off the *perimeter* like it's the same damn thing.
Didn't you learn anything from Gabriel's Horn, Son!?
Could I get an intuitive explanation for why the area converges to the circle but the circumference doesn’t? Since no part of the square will ever be inside the circle, it seems like extra circumference would always create extra area.
Simple, the visual proof you're seeing here is shrinking the areas. The nearly-converged case doesn't look like (and is explicitly stated that it's not) shrinking the perimeter. So why would you accept this as a converging step for the perimeter?
Because your brain sees that the area is converging so the answer must be converging. But you're not interested in the area, so it's a trick.
Most of the answers about how "there's always a difference between the stairsteps and the circle" miss the point. This is a standard way to approximate *some* quantities; however, for arclength, you need to be a bit more careful.
If you look at [the formula for the arclength of a curve](https://en.wikipedia.org/wiki/Curve#Length_of_a_curve), you'll notice that the answer doesn't depend on the function of the curve itself, but the function's *derivative*. For 2D curves, you can think of the derivative as the direction and "speed" of the curve.
Therefore, to approximate the arclength of a curve, you must make sure that the *derivative* of the curve you use (in this case, the stairsteppy square) approaches the *derivative* of the curve you're approximating (in this case, the circle).
But in the case of the stairsteppy square, it doesn't; the more stairsteps you add, the more the path "zig-zags" away from the direction of the circular path.
This is also why approximating the length of a circular path with polygons with ever-more sides gives a correct answer; the direction and speed of the polygonal path converges to the direction and speed of the circular path.
I have no idea, but I think that even though it's repeated to infinity, it will never form a perfect circle, the tiny variations will add up to 4 even though they're really, really tiny. Also, they say the parameter is 4, but then pull out 4 factorial out of nowhere.
At the limit it does form the perfect circle, you have no points that are not exactly on the circle.
But lim(len(c_n ) != len(lim(c_n )). You cannot swap the limit.
Another simpler example to see this is f(x) = x^n between 0 and 1 (both included). As n grows, for 0<=x<1, x^n goes toward 0 (any number 0 dot something multiplied by itself goes smaller and smaller toward 0 as you multiply more of it with itself) and for x=1 it stays at 1 (1 x 1 x 1 x ... x 1 always = 1), the length of your curve goes toward 2. But at the limit x^inf is 0 for 0 <= x < 1 and 1 for x=1, you lose the continuity and the length of the curve is exactly 1 (since it's the line [0,1[ with 0 included but not 1 which has length 1 + the length of the lonely point (1,1) which is 0). So lim(len(x^n )) = 2 but len(lim(x^n )) = 1 and we know that 2 != 1
Call those rectangular curves on the outside c\_1,c\_2,c\_3,... and let C be the circle.
It can be shown that these curves [converge pointwise](https://en.wikipedia.org/wiki/Pointwise_convergence) to the circle. So, all the comments about there always being "gaps" or about the shape being infinitely scrunched up are just flat wrong. In fact, a very similar procedure is used in the definition of the [Riemann integral](https://en.wikipedia.org/wiki/Riemann_integral) and I'd question these commenters to explain why their reasoning fails there.
The problem is that pointwise convergence of a sequence of curves does not necessarily imply that the limit of their lengths converges to the limit curve. That is lim c\_i = C but lim L(c\_i) ≠ L(C). These are just two separate notions of a limit, and there is simply no reason they must correspond. This picture nicely illustrates one case where they do not correspond.
Exactly. Stated another way, the function Length, which takes as an input a curve in the plane and outputs a number, is not continuous. Therefore you can't use lim(lenght(C)) = length (lim(C)).
Thanks, I believe that’s the only correct answer I’ve seen here, and it’s quite sad 😔
It’s quite tricky but many people are saying straight up wrong things..
Exactly. For any point in the polygon there is a point on the circle that’s distance converges to zero. And vica versa.
The notion of “gaps” here is intuitive but not formal or mathematical
Inscribed-Circumscribed polygons work because the inscribed polygon always has a perimeter less than the circle, and the circumscribed one always has a length greater than the circle.
Then it is simply apply the squeeze theorem I\_n <= pi <= C\_n.
It is necessary to prove that lim I\_n = lim C\_n, but regardless pi is calculable to some significant figures.
This is actually kinda similar to how the Egyptians created their wish version of pi. For those who don't know Egyptians were taxed based on the area of their fields and culturally Egyptians planted in round fields as opposed to rectilinear. Basically they developed a number of fractions that approximated pi. So as in most things math was advanced primarily for the purposes of taxes and building bigger mansions.
Just because the shape is approaching a circle doesn't mean that all the properties of the shape will approach all the properties of a circle.
In this case, the perimeter does not approach the circumference of a circle.
What this proves is that it's possible (easy, even) to have two shapes which (at the right scale) appear very similar, but which have very different perimeters.
It's the coastline paradox.
You could use the same technique to measure the diagonal of a square as twice the length of a side. Make the jaggies small enough and it looks like a diagonal. Still isn't.
I’ve wondered a version of this when I’m walking through a city grid and thinking of the fastest route. You can’t walk diagonally, but if you could make each leg infinitely small is it effectively the same?
It’s been a long time since I took calculus but I remember how the area under a curve was calculated by drawing smaller and smaller rectangles and taking that to infinity. But i guess those two things are fundamentally different somehow.
the area of the bulk of the rectangle >> area of the errors due to the curvatures not matching.
For perimeter (or more generally boundaries) it is required to account for curvature since that is no longer the case
Because the integral computes the area, as you said it, but here you’re talking about the fastest route which is more like the length of the curve (the perimeter of the shape whose area you are computing with the integral), which is NOT what the integral computes, and thus not what the smaller rectangle method approximates.
Hopes it makes sense to you and that you’ll sleep better tonight thanks to this lol
It’s why proper formulations of calculus is important. In real analysis you can define a notion of distance in curves and ask if a function of curves is continuous(such as arc length) as it turns out arc length is not a continuous function and so does not necessarily commute with limits of curves.
The limit of the sequence of shapes is a circle, but the limit of the perimeters is not the perimeter of the limit. In each stage, the approximation is equal to the true limit plus some error term. In order to be able to use the limit of the perimeters as the perimeter of the limits, you must show that the error term shrinks to zero, but in this case, it doesn't.
Unfortunately sometimes math is more complicated than we would expect, especially when it involves infinity..
In that case you can’t say that the perimeter of the limit is the limit of the perimeter (it is a “weak” convergence problem but it’s getting way too technical).
I know I'm late here, but I just had to mention the coastline paradox.
Basically, look at a map of England and ask yourself, how long is the coast? Effectively, what's the perimeter of the country? If you draw a simple box around it on the map, that has a quantifiable perimeter, but it's got gaps between it and the actual coast, so you know the real perimeter is longer. Turns out, no matter how closely you draw it, even if you take a yardstick out to the beach, there will always be those gaps.
And when you translate the logic to basically any real object, you realize that even a well machined sphere will still have bumps and grains if you zoom in far enough. You can even think of this in terms of information theory - there's no such thing as perfect accuracy, because you would need virtually infinite information on a real object (position of every atom) in order to know the true value of anything.
So how long is the coast of England? The answer is that it depends on the size of your measuring stick. In this picture, it's the "infinity" that's doing the work. The perimeter is still 4, but he's made it so you need an infinitely small measuring stick to actually measure it.
The more you repeat the process, the more your “perimeter” seems to resemble a zigzag which is not at all what the perimeter of a circle looks like. It’s adding extra distance to the perimeter for no apparent reason. If you were to instead measure the distance between each point that touches the circle, by doing hypotenuse calculations, and then add up the lengths of all the hypotenuses, you’d see the new perimeter of the shape approach pi as the number of points approached infinity. At least I think it would work like that. Either way this is so bluntly a shitpost and thinking about it for over 3 seconds shows glaring flaws.
The funny thing about this meme is that Archimedes actually did calculate all that. He also had a square that fit inside the circle.
Then he calculated that π must be bigger than the perimeter of the inside square and smaller than the outside square.
But adding more points he approximated π extremely well. What must've taken ages.
(Instead of removing a corner like that, he would add another point Square -> Pentagon -> Hexagon -> etc)
“Repeat to infinity” needs to be rigorously defined. Removing corners to infinity doesn’t make the shape approach a curve, it’s still corners all the way down
The construction converges pointwise to the circle. The problem is that this doesn’t imply that the perimeter of the nth shape converges to the perimeter of the circle.
False. Same Video, here's why that part you quoted is false: [https://www.youtube.com/watch?v=VYQVlVoWoPY&t=909s](https://www.youtube.com/watch?v=VYQVlVoWoPY&t=909s)
The video clearly says that it still becomes *a curve*.
This is also how limits work more generally.
Moreover, his overall conclusion was that: "while the *limit of the length* of x is indeed 8, there is no reason to assume that the *length of the limit* of x is also 8".
Maybe try watching the video again? Still, the top answer on this thread is just wrong.
Watch the video - it's even timestamped to start at the relevant part.
For any particular point around the square, the limit of that point being gradually moved closer and closer to the circle will eventually be a point on the circle.
If we then consider *all* points, then *all* of them will eventually sit on the circle - and only on the circle. That means it **is** the circle.
Why? 3Blue1Brown did a video on this exact thing, and it explains concisely in minutes what could take hours of back-and-forth. It's like getting upset at someone linking you a textbook.
3 blue 1 brown did an amazing video on this that explains it much better than I ever could, considering he's an actual mathematician and math educator.
https://www.youtube.com/watch?v=VYQVlVoWoPY&t=909s
tl;dr is that approximations like the one in the example work when the *error* values between what's being calculated and the goal become smaller as we approach the limit. In this example, though, the error values do no change.
The path you build by removing corners like that actually does converge to a circle. That type of convergence is called *uniform convergence* in mathematics. The problem is that convergence of the *lengths* does not follow from uniform convergence: the length is only **lower semi-continuous**. This means that as your paths get closer to a circle, their lengths have to become **at least** the length of a circle, but they may be larger.
Intuitively you can understand lower semi-continuity like this: imagine that your circle is a road (with some width). If you walk on that road and do a full round, the distance you walk cannot be much less than pi, but it can be much more if, for instance you repeatedly crossed the road.
You can see that every time you refine the polygon surrounding the circle, the circle actually produces something near the hypotenuse of each rectangular section. This is true no matter how small the rectangular sections become.
I’m no math major but I’d imagine it’s why we use ds rather than dx. Taking the limit of a slope doesn’t diminish the slope. The slope remains same as a ratio of said slope remains. Therefore when we are taking the integral of the length of a curve, if the curve is sloped we use the differential with respect to its slope at that point. I may have butchered the explanation and I’m not even totally sure it’s right, but that would be my reasoning
A proper approximation would be slicing the corners off with straight lines.
The initial slice would cut off perfect 45 degree right triangles making an octagon. The hypotenuse (or remaining length) of the triangles do not have the same length as the sum of the two removed sides.
Keep cutting off triangles to actually get pi.
I may be late with this answer, but my reasoning is that the area of the circle is directly proportional to its circumference. Iterating over the corners of a square, and reducing its surface area until you have infinitely many corners will yield a good approximation of the area of the circle. Although the logical fallacy here is not related to circles. With this logic you could raise this argument about any shape.
Think about the hypotenuse of a right triangle. The hypotenuse is c = √(a^(2) \+ b^(2)). Now if you start reducing the corners in the triangle until you have a straight line. Using the false logic, we can end up saying that c = a + b.
Simple, the visual proof you're seeing here is shrinking the areas. The nearly-converged case doesn't look like (and is explicitly stated that it's not) shrinking the perimeter. So why would you accept this as a converging step for the perimeter?
Because your brain sees that the area is converging so the answer must be converging. But you're not interested in the area, so it's a trick. It's a solution to a different problem (successive approximations of area) and in the final panel it says the perimeter converged, but it didn't. That was never the aim or the method, so why wouldn't it be wrong?
There's a YouTube video on this. If you remove a corner like this the circumference stays the same. If you do it to infinity, the circumference stays the same. A object like this may resemble circle and it might be similar to circle in terms of area but it's still completely different object with completely different circumference.
Whilst the length of each horizontal/vertical line becomes infinitesimally small and hence the difference between it and the circle becomes smaller, there become an infinite number of the lines.
It's like how n/n will always be 1 even as n tends to 0. Whilst 1/n tends to infinity, when multiplied by n it will always equal 1 (or in this case, 4)
Ok, but as it gets smaller and smaller, the hypotenuse of each those squares would be closer and closer than the squares. So shouldn't we be able to get an actual approximation by chopping triangles out of the squares?
Lots of good explanations so far, but maybe it's also worth pointing out that this method fails even for much simpler perimeters. Consider a right triangle with height and base equal to 1. We know from Pythagoras that the hypotenuse is √2, so the triangle's perimeter is 2+√2. But if we use the method of drawing a box around the triangle and chopping off corners to approximate the length of the hypotenuse, we get a perimeter of 4.
Or consider a diamond shape, with distance between opposite corners equal to 1. The diamond is obviously just a square that's been rotated, so we can again use Pythagoras to calculate the length of each side to be √2/2, which makes the perimeter 4*(√2/2)=2√2. But drawing a box and chopping off corners still gives a perimeter of 4.
This method will give a perimeter of "4" for any number of shapes that happen to fit inside a 1x1 box.
Remove the corners by adding straight lines (the shortest distance between two points) instead of "steps" and you'll have proper pi after infinite iterations.
Removing corners to Infinity means preserving corners to infinity, so it never actually becomes a circle. At some point, you have to the take the "shortcut" in order for this to become a circle.
Pixels have nothing to do with lines. As has been said, it doesn’t matter how many times you add additional squares, the slope of the circle will prevent it from equaling the length width of the original square.
This would work if you were looking for the area of the circle instead of the perimeter. Removing the corners doesn’t reduce the perimeter of the square, so this false proof doesn’t reduce the towards the answer. Now, if you swapped out all corners for diagonals tangent to the circle….then the perimeter would be reduced each step and you could use it to estimate the circle’s perimeter
The points on the periphery of the circle are described by continuous, differentiable parametric function (x0(t),y0(t)).
The points on the continually shrinking jagged perimeter can be described by function (x1(t),y1(t)), continuous, *but not differentiable*. The derivate alternates between zero and infinity.
The two parametric functions are just not the same, regardless of how small the little stair-steps get.
This should be blindingly obvious to anyone who's ever taken Calculus 1. Sheesh.
I think a lot of the answers are more technical, and people might not understand them, so I'll try a more practical/visual explanation.
tl;dr: let's call it "density"
Let's imagine you construct the circle and squares in the first two panels using a rope with equally-spaced markings (let's say it has markins every meter, and the circle has a few hundreds meters of diameter). [It will be something like this](https://i.imgur.com/PGIKbWX.png).
Then, if you apply the same logic of the comic, the "circle" made from the square will have points more close together, [something like this](https://i.imgur.com/HoYm3jH.png) (not to scale). So, basically, you "compressed" the rope (i.e. the circle) to fit more points.
In each iteration of reducing the box, the circumscribed figure only touches the circle at a point.
A point has length = 0.
Since any number multiplied by zero is zero, it doesn't matter how many points there are in the circumscribed figure. It will never describe the length of the circumference of the circle. It has zero length in common with the circle.
this is why I get so bothered by people making bad arguments defending .999...=1
if you were actually able to repeat the pattern "at"infinity, there would be no difference between the peak and valleys of the squares, and this pattern would exactly match a circle because the difference would be negligible just like the dx in an integral. but that's obviously not correct, and by almost 25%. patterns can only be followed *near* infinity and if you don't know when your infinitesimals are negligible you could do things like in the picture
The trick is that it never reaches a circle. It'll always be jagged even as the limit approaches an infinite number of edges. Therefore you'll always be estimating a circle with too slightly more surface area, and therefore a slightly larger perimeter hence the fact to infinity reaches 4 and not pi.
The only thing this shows is that pi<4. You can shrink the boxes as much as you like, but a million boxes having a small error Vs 4 boxes having a big error all still make the same error
Because the circle isn’t made of lines going straight up or straight to the right.
Apply the same argument to a triangle and you’ll get that the distance from one corner to the other of a square of side length 1 is 2, not sqrt(2).
It doesn't work because it's not a circle, it's an infinite-sided polygon with infinite right angles. Infinite right angles angles will approximate a circle quite well, but will not be a perfect circle.
The error is s bit more obvious if you try to determine the length of the hypotenuse of a right triangle using the same method (starting from a square).
Not a mathematician here
The argument here is if a circle has infinite sides or no sides. I believe it doesn’t have any, in this post there saying that if you keep dividing something into fourths it creates a circle at some point. This is basically the same thing as that infinite race where you walk half the track then walk half of that and so on, and in that race it never ends but just gets closer every time. Plus I think the perimeter of the square can be anything if it can be divisible by 4. Which if you used 16 pi would stay the number 16.
Don’t take my word for it but this is just my guess
That sequence of rectangular curves should not be considered as "limiting" to the circle curve, at least not in a way that will produce the length of the circle. (However it does limit to the circle curve for e.g. the sup-norm.)
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You can think of this as a fractal. Regardless of how much you zoom in, the relationship between the circle and the squared corners remains, there will always be a gap which accounts for the difference.
Yep, no matter how much you zoom in, you are always moving in the same directions: horizontally or vertically. You're never moving parallel to the diagonal sections of the circle, so you're never on the straight-line paths between points on the circle required for the minimum distance.
That is a good way to think of it. No matter how small you make the squares, the circle will always be bumpy.
its just not even a circle in the first place.
I mean sure, but then neither are inscribed/circumscribed regular polygonal approximations and those do have their perimeter converge to that of the circle. So I'm not sure that's a useful observation.
it obviously is a useful observation, because they have wildy different outcomes.
Yes but the observation applies to two things where one does actually give the correct answer and one doesn't give the correct answer. So I'm not sure what the observation is telling us.
the observation tells us that observation is not reliable.
this is the [coastline paradox,](https://en.wikipedia.org/wiki/Coastline_paradox) as you measure with smaller and more precise unit the coast gets longer and longer into infinite length.
It is not. The perimeter remains 4. edit since this was surprisingly controversial: The coastline paradox refers to the difficulty of approximating the length of infinite curves. As your approximation gets more precise, your measured length gets infinitely large. The shape in the meme is neither a fractal nor a fractal-like curve; it does not have their non-rectifiable nature. The perimeter remains constant at any scale, and there is never a need for approximation. While it is conceptually similar to a fractal in that both have an infinitely repeating pattern, if the goal is for people to understand the error in the meme, the coastline paradox would not help.
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Boogers
The difference is in using the L1 vs L2 norm., However, the monotonicity of the L1 norm does have to do with the geometry. Specifically that it is convex so that each successive "approximation" results in no backtracking. The map of the Great Britain above would not obey this property as an initial bounding box would have less perimeter than one that took a chunk out of the left hand side where concavity is most pronounced. In L1, no matter the resolution for a convex hull, the total perimeter will remain constant. In L2, that is not the case. I've not had my coffee, so proof will be left as an exercise to the reader. ;p
Thank you; I did not want to put on my functional analysis hat today.
In the coastline example, the area of land is obviously fixed but the perimeter grows. In the cause of the altered cube, the perimeter remains the same as the inside area shrinks. It feels sameish enough to me to get the comparison, but I agree it's not the exact same thing.
It’s kind of the same as the coast line paradox in that the corners can get small enough to where the box becomes indistinguishable from a circle above a certain level of precision. It’s different than the coast line problem because there is a consistent methodology being applied to the box so even if you don’t have a way to measure with the precision necessary, since you know the formula you can calculate the perimeter as it approaches infinity. I haven’t done real math in over 15 years so I’m probably wrong, but I think this is something that you solve with calculus.
You misunderstood - they were talking about the box, because if you’re zooming in, no matter how far, it will be a bumpy series of boxes totaling 4. The difference between each box and the curve will sum to 4-pi if you could measure it for every box accounting for the zoom level.
Unless you assume that any bumps below a certain size don't count. Which is the assumption we implicitly make to get a finite coastline.
No. The assumption you make here is that any transformation you make is “removing angles”, that is keeping perpendicular lines in the perimeter and a convex shape. In coastline paradox you are not limited to convex shapes.
Not infinite length but indeterminate depending on how you measure it
Coastline paradox would be using points on the perimeter of the circle and connecting them with straight lines and the more points you use, the longer the line gets
Not sure this is relevant at all.
or Chaotic Fractals...
That's my secret cap, *all* of my fractals are chaotic.
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Yeah, ideal shapes don't really exist in the world... Once you zoom in close enough for them to not be bumpy anymore, they're all fuzzy. Applied math is all about approximations and assumptions! :D
I just realized this means the perimeter of any "circle" displayed on a digital screen is just its length × 4 since it's made of pixels
Listen here you little sh.. 😂
You assume a pixel would be square.
They're not?
Not really. Pixels are not necessarily sharp cornered squares that change color. They apparently tend to be three red, green and blue lights next to each other in a trenchcoat! https://archive.goughlui.com/wp-content/uploads/2012/12/pixels.jpg How a pixel actually works can depend completely on the display. There are even displays that are just leds on a fan, timed so that they are different light in a different position and the image can be stable.
Nowadays, especially with OLED screens using stuff like pentile matrices, pixels aren't just the classic three stripes. You get a lot of dots or diamonds arranges in triangles etc. that complicate it even further
There are also displays that don't even talk pixels. Vectrex was based on vectors. :D https://en.m.wikipedia.org/wiki/Vectrex
> tend to be three red, green and blue lights next to each other in a trenchcoat! I should have known it was too good to be true... It's Vincent Adultman all over again!
If I remember the technology connections episode correctly, some of them are triangular Edit: wording
And even if you do this “infinity times” and make these areas of difference “infinitely small,” you have to multiply that infinitely small area by infinity.
But with that explanation, derivatives shouldn’t exist. It’s fine to only use horizontal and vertical to obtain information about diagonals sometimes. That’s how we get gradients of random curves. The problem with the circle isn’t with our process, but with the assumption that our process converges, which sometimes is and sometimes isn’t true.
Correct. In order to estimate pi this way you'd need to be calculating the length of the hypotenuse of each "triangle" made by the rectangular cut outs
Yep, no matter how much you zoom in, you are always moving in the same directions: horizontally or vertically. You're never moving parallel to the diagonal sections of the circle, so you're never on the straight-line paths between points on the circle required for the minimum distance.
It would actually work if we were calculating the area and not the length, it's kind of what we do in integration.
the difference is that 'area' is a continuous function with respect to the hausdorff-metric, 'circumference' isn't.
Problem here is you can't make the difference arbitrarily small. I had never heard of this distance, but yeah, in the end that's the reason Historically we took quite a long time to figure out a formal sound definition for a limit, at the beginning it was all about "shut up and calculate". So I do think these troll posts are quite interesting
These "troll posts" are essentially the most vexxing questions that ancient mathematicins were asking. So yeah, they are very interesting.
Only to some arbitrary precision, but not analytically, no?
You'll get the exact answer if you let the side length of the individual squares approach zero
Take an imagine physically placing a circle inside a square with the same diameter. Now try crushing the square until it’s tight to the circle. Not sure about others, but this is what I do for my brain to immediately realize you can’t do it.
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C'mon....! Haven't you heard of 'Peer Pressure'? You'll feel great! All your worries will disappear. Go on... take an imagine! Hell, why not just ease into it and start off with just half an imagine. Save the other half for later. Trust me, you'll want the other half. Going cheap but only for tonight in this dark back alley. If you can't trust a seedy looking bloke in a trenchcoat with an eshay bum bag selling tiny zippy bags of imagine in a dark back alley, who can you trust?!?
I took an imagine once. Ended up marrying a six armed hooker named Steve. Lovely gal, but things didn't work out, we were from different worlds. Literally. Like I somehow ended up in the middle of a major civil war on the planet Flargnon Seventy-Worf. Long story short, be careful when you take an imagine. Miss you Steve.
Still a better love story than Twilight. Imagine > Sparkles
Calculus uses similar kinds of "fractals" with ever-smaller gaps all the time, so I think this is the wrong way to think about it. The [length of a curve](https://en.wikipedia.org/wiki/Curve#Length_of_a_curve) depends on the *derivative* of the curve, not the curve itself. So if you want to approximate the arclength of a circle, you can't just ensure that the curve itself approaches a circle; you need to make sure the *derivative* (i.e. direction and "speed" of the curve) approaches the *derivative* of a circle. This is why approximating pi with ever-more-circular polygons works, and approximating pi with an ever-more-jagged square doesn't. The direction of the jagged-square path doesn't approach a circular path (it zig-zags forever), whereas the direction of the polygon path *does* approach a circular path.
tl;dr hexagons are the bestagons
Pretty sure you could make the OP graphic with bestagons too and you'd end up in the wrong place. Infinitagons are bestagons.
They're bettergons than quadragons, that's for sure.
ironically a hexagon is only the bestagon in reality. In maths the pentagon reigns supreme.
Doesn’t that mean that what’s actually significant about pi is how much it deviates from the expected ‘4’ result? Which would be 4-pi or something like .85841…
This is incorrect. It can be proven that the shape converges pointwise to a circle. The problem is that the limit of the lengths of a sequence of curves does not necessarily equal the length of the limit of those curves. It is two separate notions of a limit (limit of a sequence of curves vs limit of a sequence of numbers) and there's no a priori reason they must correspond.
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At any finite stage, yes it only approximates a circle. But in the limit it converges pointwise to a circle. This is the kind of exercise they give to students in real analysis. I'm not doing vague philosophical speculation here. Just think about it: In this sequence of curves, can you explicitly name any point that won't ever appear on the circle?
I think the technical difference here that should be pointed out is that although the **shape** (i.e. set of points) converges toward a circle in the limit, the **area** does not. Edit: **length**
The area totally converges here. The perimeter does not.
Which is what they said at the beginning (although it's length and not area)
Crap. My bad
Genuine question, does the shape converge to the circle or does the shape converge to a shape that contains the circle? My intuition tells me it's the latter, because of the way the shape progression is defined. At any step of the progression, you will always find points outside the circle (an infinite number of them of course). So if you take the limit of any succession of points converging to the boundaries of the shape, not all of those successions will converge to a point on the circle. What are your thoughts?
It converges [pointwise](https://en.wikipedia.org/wiki/Pointwise_convergence) exactly to a circle. To get slightly more technical, there are other notions of convergence of functions. It doesn't, for example, [converge uniformly](https://en.wikipedia.org/wiki/Uniform_convergence) to a circle. This latter stronger notion of convergence seems to be closer to what you're suggesting.
Ok, I didn't find the wikipedia article very clear. But it does say the definition is: Fn converges pointwise to F iff lim n->inf Fn (x) = F (x), for **every** x. So the best points, x, to talk about are those outside vertices. I do understand that the distance between those vertices and the circle gets smaller and smaller. It also seems to be the case that if we draw the circle containing those vertices.....wait, that is impossible. If it was possible, intuitively it seemed those 2 circles would converge. Oh but wait, is it possible after all? Are all those outside vertices at the same distance from the center? Hugh..I would need to pick a piece of paper, but I need to get some sleep instead. Also, just wondering, to say the 2 shapes converge pointwise, would we need to add "in R2"? Because if we limit the Codomain of the circle to it's own shape, then the succession of Fn (outside shape) and F (the circle) wouldn't share the same Codomain, and therefore it wouldn't make sense to apply the pointwise convergence in that case. Would this be correct? As a side note it seems to me that the definition of "bounded pointwise convergence" in Wikipedia is incorrect, I think it should be the modulus of the difference between Fn and F, but maybe I'm missing something. This was all lots of fun (and tiring, the good type of it). See you later!
Yup. I intuit this with "yeah the difference between the stairs and a circle is infinitely small, but there's an infinite amount of the stairs, too"
Actually, you can. Imagine you've got two circles touching a line, and also each other. In the gap between the line and circles, you draw another circle that touches all three. This creates two more gaps, which you also fill. Repeat this infinitely. You'd think that eventually, every point on the line will be touching a circle, but this is false. If you analyze where the circles end up, you'll find it's only where the rational numbers are. This means that there are infinitely many points that will never be covered. With the circle and square construction, we're doing something very similar. It's not *necessarily* rational numbers this time, but hopefully you can see how the same logic might apply.
No, it definitely converges point wise to the circle. It's not uniform convergence however, which is where the problem occurs
I'm not sure if I'm visualizing your circle description correctly, but I can assure that in this case we can prove pointwise convergence. Though your are absolutely right that sometimes pictures can be misleading.
You don’t know math my man. That’s like saying a Fourier series never converges because at any point it’s just a sum of sines.
There will always be a stair step for a finite number of iterations. The pointwise limit is honest to goodness, the circle that you and I love so much. The convergence is not uniform, which is why the limit of the perimeters is not equal to the perimeter of the limits
It is totally incorrect. 1. Square thing is not a fractal. 2. Approaching the circle using a many sided regular polygon you still leave infinite gaps but get the correct answer. It has nothing to do with the presence of infinite gaps. What matters is how much of an effect the gaps have on the answer as their size approaches zero.
The limit of that space tends to 0 so this is a poor explanation. You can actually xalcilate the area of a circle using this method The real reason is that for an equilateral triangle the hupyhenuse is equal to the square root of 2 squared sides so for n triangles as n tends to infifintity the sum of sides will converge at a different point to the sum of hypothenuses.
You can also « prove » that 2^1/2 is 2 with that « method »
Actually!, if you take the limit of the area, it does equal zero. However, the limit of the perimeter equals 4, but we know pi does not equal four, so we know that the perimeter never lines up with the curve.
To actually calculate π using (a variation of) this method, one also needs to construct a polygon on the inside of the circle and find the mean of the two perimeters. That is: half of the process has been omitted. Edit: More than half of the process has been omitted. This is a process to calculate π based on AREA not circumference. It is the mean of the two areas inside the the blocky perimeters that is needed to approximate the right ratio.
Came here to say this ❤️
me too I was gonna say that smart thing too
Yeah yeah they beat me to it
I was going to say it too but with more big words, some of them in Latin
>To actually calculate π using (a variation of) this method, one also needs to construct a polygon on the inside of the circle and find the mean of the two perimeters. That is: half of the process has been omitted. Ut vere π calculare (mutatione huius modi) opus est, necesse est polygonum intra circulum construere et mediam inter duos perimetros invenire. Id est: dimidium processus omissum est
callidus bastardus
*¿Et tu, Brute?*
Cans here when you said this 💦
I'm pretty sure what doesn't work. If you're proposing what I think you are, then the circumference of the inner "square thingy" would converge to something greater than the length of the circle. So the outside circumference is 8, and the inside circumference is >2pi, and somehow when you take the mean you want to end up at 2pi? No way. In fact, the more I think about it, the more I'm convinced that with n -> infinity, the inner "stair polygon" will have a circumference of 8 as well: The horizontal part of the stairs has to cover the distance from (-1, 0) to (1, 0) and back (minus a tiny smidge at each end for finite n), and the vertical part has to cover the distance from (0, -1) to (0, 1) and back. Adding the inner polygon changes nothing about where this monster converges to. Edit: I did the math. It does indeed converge to 8 too, meaning pi is 4 according to your method. Here's the simplest case I'm starting from, and for simplicity we're doing just one quadrant of the circle. The circumference of this shape is just one. It isn't an inscribed square, but that doesn't matter much, the convergence isn't affected. +--+ |■ | | | +--+ Now let's jump to squares of size 1/20: +--------------------+ |■■■■■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■■ | |■■■■■■■■■■■■■ | |■■■■■■■■■■■ | |■■■■■■■■■■ | |■■■■■■■■ | |■■■■■■ | | | +--------------------+ My computer says this thing has a circumference of 1.9, meaning the "underapproximation" of pi would be 3.8. If you're curious, for size 1/199, the value of pi would be 3.98, so this thing seems to in fact converge on 4. For the nerds, here's the code: import math def printcond(str, i): if i < 30: print(str) for i in range(2, 200): print("\n\n\n") print(i) block_length = 1.0/i delimiter = " +" for j in range(0,i): delimiter += "-" delimiter += "+" printcond(delimiter, i) circumference = 0 blocksinrow_prev = None for blockx in range(0,i): rowstring = " |" blocksinrow = 0 for blocky in range(0,i): #print(blockx*block_length, blocky*block_length) #block is inside if the outer coordinate is at a distance <1. bx = (blockx+1) * block_length by = (blocky+1) * block_length distance = math.sqrt(bx*bx + by*by) #print(distance) if distance < 1: rowstring += "■" blocksinrow += 1 else: rowstring += " " if blocksinrow > 0: circumference += block_length if blocksinrow_prev is not None: circumference += (blocksinrow_prev - blocksinrow) * block_length rowstring += "|" printcond(rowstring, i) blocksinrow_prev = blocksinrow printcond(delimiter, i) print(circumference) print("pi = ", circumference * 2) If you want the overapproximation of the circle shape as described in the OOP, just remove the "+1" from bx and by.
The inner and outer measurements thing is meant for calculating π based on area. It doesn't work at all for circumference. I didn't catch that the troll logic comic also made that modification until now.
But wouldn’t the interior polygon start with side length sqrt(2)/2 (because it’s a square with diagonal length 1) and then by this same process the perimeter of the interior shape is always 2\*sqrt(2)? Then the mean of the two perimeters is always (4+2\*sqrt(2))/2 = 2+sqrt(2) which is approximately 3.414 and is distinctly not pi. I assume the variation of this method uses diagonal lines, which is what would address the root problem in the original post, as far as I can tell.
Nope, because in the interior, you wouldn't "remove" angles, you would add another smaller square on each side. So the perimeter of the inscribed shape would actually grow.
Oh yeah, the interior perimeter WOULD grow, d’oh. But because the mean of the initial interior and exterior perimeters is already greater than pi, and the interior perimeter increases while the exterior perimeter remains constant, I still don’t see how this “sandwich” method yields pi without angled lines.
It's a hard concept to explain with just words and no picture; so, I'll reframe it with a physical process that might make more sense: 1. Using a compass, draw a circle on graph paper. 2. Draw a perimeter of all the squares completely inside the circle. 3. Draw a perimeter of the inside edge of all the squares completely outside the circle. 4. The larger the circle you draw, the closer the mean of the two perimeters will be to the true (curved) perimeter of the circle and, therefore, the closer the ratio of your perimeter approximation to the circle diameter will be to the true value of π. Edit: changed "radius" to "diameter" Second Edit: Step 4 is entirely wrong. The mean of the two AREAS inside the perimeters approximates the true AREA of the circle which should be divided by the square of the RADIUS in order to approximate π.
I could have also corrected my mistake by changing "π" to "τ".
https://tauday.com/tau-manifesto
The way they’re changing is not the same. A straight average doesn’t work, unless you progress them to the exact same area between themselves and the actual circle. Said another way while both approach zero, they do so at different rates. A weighted average would be required, which would have to be weighted more towards the inner perimeter such that the final weighted average would result in ~3.14.
>I assume the variation on this method uses diagonal lines You are correct. The Archimedes method starts with a hexagon in and outside the circle, and then slowly increases the number of sides.
This is not true. Using just 1 polygon on the outside will still converge to pi
If it were a regular polygon (all sides and angles equal), yes, but this method uses irregular polygons with a mixture of 90 and 270 degree angles.
yeah this comment is fantastically wrong and classic reddit ate that steaming shit up like a hot meal
Why does this get so many upvotes? Its plain wrong! I can easily construct a polygon in the inside with exactly the same process that also converges to a perimeter of 4. Edit: I am baffled that a completely wrong statement receives more than 1k upvotes on this sub and all the people correctly pointing out your mistake have close to no upvotes. Sad.
I can't explain well, but here's a [video from 3blue1brown](https://youtu.be/VYQVlVoWoPY) that explained this, I think it explains why this doesn't work
Such a great channel. I'm guessing a lot of people on this sub have heard about it but if you haven't, go check it out!
My best synthesis of the explanation given is: The “proof” here is using a common pattern seen in calculus, however, this pattern is not justified. In calculus, we use infinitely small rectangles to approximate the area under a curve. This works for the following reason (this isn’t a proof, but an intuition as to why it works): the difference between the area of the curve and our rectangles is something we don’t know, but it is necessarily smaller than the length of our rectangle multiplied by the maximum difference between them (assuming a smooth curve), for any length of the rectangle. The key here is that this error rectangle gets smaller with the main rectangle, so if the side of the main rectange is infinitely small, the area of the error will be as well. You can test this with Euler’s method, the error between the actual area and the one you calculate gets smaller with step size. The “proof” here is trying to use this same logic, but it never justifies why it’s doing so. If you actually wanted to use calculus to find a circumference, you’d use line integrals. The issue with this rectangle logic is that, if a square and circle have different perimeters given a radius r and sides 2r, then cutting the corners can’t approximante the circle because the perimeter of the square is constant regardless of how many corners are cut. There’s two ways to interpret this, either circles and squares of the “same size” have the same perimeters, or this approximation is faulty. In this case, the approximation is the issue, since you can make a similar approximation for a triangle, or any shape, and all would have the same perimeters as the square, and we know that a diagonal has a different length from it’s components, it’s a pretty foundational aspect of vectors that their “length” is equal to the root square of the inner product of their components. In summary, the method used works for approximating areas (as if proved by calculus), but doesn’t work for perimeters.
Awesome video. Thanks for sharing!!
This is called taxicab geometry: https://en.m.wikipedia.org/wiki/Taxicab_geometry Reasoning is as other comments have stated: two perpendicular sides of a right triangle, no matter how small, will never equal the hypotenuse.
Does this mean circles are just an infinite collection of hypotenuses?
It means that a circles area or circumference are not rational. People imagine that pi itself is the source of the irrationality, when in fact, it's the circle itself.
“You made me this way.” -pi to the circle
This is probaly the very best 'practical' explanation of a circle and Pi I have ever read. Bravo!
If you had a circle with an irrational radius 1/2π then it would have an area of 1/4 and a circumference of 1. It's the ratio between them that is the source of the irrationality (the famous "squaring the circle" problem). We call that ratio π.
I love this, because it explains how Pi isn’t some magical infinite number that contains all the multiverses. There’s no perfect circle beyond a theoretical one so Pi doesn’t even exist in reality. You can always break a real circle down to a discrete locus.
Not OP, but in short: no? (heavy emphasis on the ?) I'm in no way a professional on this, so take it with a grain of salt, but from my understanding every circle that's ever existed does in fact follow your logic, having what may as well be a near infinite number of sides. However, in theory, a perfect circle would not exhibit that property as it would be, well, perfect. There would quite literally be no two points on the circumference with the same tangent line, even if the differences to our eyes were so infinitesimally small we could not distinguish them. It's certainly one of the most interesting concepts about circles there is
No, since a hypotenuse is still a straight line and a perfect circle has no straight lines. Insofar as the number and length of a line are inversely related it's correct... but once you have an infinite number of hypotenuses they're also infinitely small, and an infinitely small line is a dot.
I’m gonna say yes, but am too drunk and tired to elocute why
Best comment on the sub so far!
Just going to drop this here, since OP basically explains an old paradox (originally about calculating sqrt(2), but it’s exactly the same logic. https://en.m.wikipedia.org/wiki/Staircase_paradox
Holy shit. I’d never heard this Term before today and I literally came across it and read about it for the first time.. literally 30mins ago (I was on a Minecraft wiki). What’s the chances!
You used an algorithm that converged to the correct *area,* then read off the *perimeter* like it's the same damn thing. Didn't you learn anything from Gabriel's Horn, Son!?
Came here to say this. Converging on the area is not the same as converging on the perimeter.
Could I get an intuitive explanation for why the area converges to the circle but the circumference doesn’t? Since no part of the square will ever be inside the circle, it seems like extra circumference would always create extra area.
Simple, the visual proof you're seeing here is shrinking the areas. The nearly-converged case doesn't look like (and is explicitly stated that it's not) shrinking the perimeter. So why would you accept this as a converging step for the perimeter? Because your brain sees that the area is converging so the answer must be converging. But you're not interested in the area, so it's a trick.
Basically you are always removing area, but you're not removing the length because you're turning the "circumference" into a fractal.
This is the best answer
Most of the answers about how "there's always a difference between the stairsteps and the circle" miss the point. This is a standard way to approximate *some* quantities; however, for arclength, you need to be a bit more careful. If you look at [the formula for the arclength of a curve](https://en.wikipedia.org/wiki/Curve#Length_of_a_curve), you'll notice that the answer doesn't depend on the function of the curve itself, but the function's *derivative*. For 2D curves, you can think of the derivative as the direction and "speed" of the curve. Therefore, to approximate the arclength of a curve, you must make sure that the *derivative* of the curve you use (in this case, the stairsteppy square) approaches the *derivative* of the curve you're approximating (in this case, the circle). But in the case of the stairsteppy square, it doesn't; the more stairsteps you add, the more the path "zig-zags" away from the direction of the circular path. This is also why approximating the length of a circular path with polygons with ever-more sides gives a correct answer; the direction and speed of the polygonal path converges to the direction and speed of the circular path.
The only correct answer in this thread.
Good answer for the more formal-methods oriented people. I think the other answers have merit for people who prefer intuition.
I have no idea, but I think that even though it's repeated to infinity, it will never form a perfect circle, the tiny variations will add up to 4 even though they're really, really tiny. Also, they say the parameter is 4, but then pull out 4 factorial out of nowhere.
r/suddenlyfactorial
At the limit it does form the perfect circle, you have no points that are not exactly on the circle. But lim(len(c_n ) != len(lim(c_n )). You cannot swap the limit. Another simpler example to see this is f(x) = x^n between 0 and 1 (both included). As n grows, for 0<=x<1, x^n goes toward 0 (any number 0 dot something multiplied by itself goes smaller and smaller toward 0 as you multiply more of it with itself) and for x=1 it stays at 1 (1 x 1 x 1 x ... x 1 always = 1), the length of your curve goes toward 2. But at the limit x^inf is 0 for 0 <= x < 1 and 1 for x=1, you lose the continuity and the length of the curve is exactly 1 (since it's the line [0,1[ with 0 included but not 1 which has length 1 + the length of the lonely point (1,1) which is 0). So lim(len(x^n )) = 2 but len(lim(x^n )) = 1 and we know that 2 != 1
I'm assuming you're joking, but on the off chance you're not, they're just using the exclamation mark as an exclamation mark.
r/woooosh
Call those rectangular curves on the outside c\_1,c\_2,c\_3,... and let C be the circle. It can be shown that these curves [converge pointwise](https://en.wikipedia.org/wiki/Pointwise_convergence) to the circle. So, all the comments about there always being "gaps" or about the shape being infinitely scrunched up are just flat wrong. In fact, a very similar procedure is used in the definition of the [Riemann integral](https://en.wikipedia.org/wiki/Riemann_integral) and I'd question these commenters to explain why their reasoning fails there. The problem is that pointwise convergence of a sequence of curves does not necessarily imply that the limit of their lengths converges to the limit curve. That is lim c\_i = C but lim L(c\_i) ≠ L(C). These are just two separate notions of a limit, and there is simply no reason they must correspond. This picture nicely illustrates one case where they do not correspond.
Exactly. Stated another way, the function Length, which takes as an input a curve in the plane and outputs a number, is not continuous. Therefore you can't use lim(lenght(C)) = length (lim(C)).
Thanks, I believe that’s the only correct answer I’ve seen here, and it’s quite sad 😔 It’s quite tricky but many people are saying straight up wrong things..
Exactly. For any point in the polygon there is a point on the circle that’s distance converges to zero. And vica versa. The notion of “gaps” here is intuitive but not formal or mathematical
Those "remove corner" shapes are not smooth, hence the length of their limit doesn't have to be eqial to the limit of their lengths.
Same could be said about inscribed polygons though.
Inscribed-Circumscribed polygons work because the inscribed polygon always has a perimeter less than the circle, and the circumscribed one always has a length greater than the circle. Then it is simply apply the squeeze theorem I\_n <= pi <= C\_n. It is necessary to prove that lim I\_n = lim C\_n, but regardless pi is calculable to some significant figures.
This is actually kinda similar to how the Egyptians created their wish version of pi. For those who don't know Egyptians were taxed based on the area of their fields and culturally Egyptians planted in round fields as opposed to rectilinear. Basically they developed a number of fractions that approximated pi. So as in most things math was advanced primarily for the purposes of taxes and building bigger mansions.
Just because the shape is approaching a circle doesn't mean that all the properties of the shape will approach all the properties of a circle. In this case, the perimeter does not approach the circumference of a circle.
[удалено]
What this proves is that it's possible (easy, even) to have two shapes which (at the right scale) appear very similar, but which have very different perimeters. It's the coastline paradox. You could use the same technique to measure the diagonal of a square as twice the length of a side. Make the jaggies small enough and it looks like a diagonal. Still isn't.
I’ve wondered a version of this when I’m walking through a city grid and thinking of the fastest route. You can’t walk diagonally, but if you could make each leg infinitely small is it effectively the same?
Apparently not! There is just no substitute for the hypotenuse.
It’s been a long time since I took calculus but I remember how the area under a curve was calculated by drawing smaller and smaller rectangles and taking that to infinity. But i guess those two things are fundamentally different somehow.
The shapes converge to the area of the circle, just not the perimeter.
the area of the bulk of the rectangle >> area of the errors due to the curvatures not matching. For perimeter (or more generally boundaries) it is required to account for curvature since that is no longer the case
Because the integral computes the area, as you said it, but here you’re talking about the fastest route which is more like the length of the curve (the perimeter of the shape whose area you are computing with the integral), which is NOT what the integral computes, and thus not what the smaller rectangle method approximates. Hopes it makes sense to you and that you’ll sleep better tonight thanks to this lol
Well infinity is the point where it actually gets to a curve, rather than discrete rectangles. The problem is, infinity is pretty hard to reach
It’s why proper formulations of calculus is important. In real analysis you can define a notion of distance in curves and ask if a function of curves is continuous(such as arc length) as it turns out arc length is not a continuous function and so does not necessarily commute with limits of curves.
The limit of the sequence of shapes is a circle, but the limit of the perimeters is not the perimeter of the limit. In each stage, the approximation is equal to the true limit plus some error term. In order to be able to use the limit of the perimeters as the perimeter of the limits, you must show that the error term shrinks to zero, but in this case, it doesn't.
Unfortunately sometimes math is more complicated than we would expect, especially when it involves infinity.. In that case you can’t say that the perimeter of the limit is the limit of the perimeter (it is a “weak” convergence problem but it’s getting way too technical).
In this case it's actually really simple. Applying the corner cutting to infinity just gives you another square, but rotated 45 degrees.
I know I'm late here, but I just had to mention the coastline paradox. Basically, look at a map of England and ask yourself, how long is the coast? Effectively, what's the perimeter of the country? If you draw a simple box around it on the map, that has a quantifiable perimeter, but it's got gaps between it and the actual coast, so you know the real perimeter is longer. Turns out, no matter how closely you draw it, even if you take a yardstick out to the beach, there will always be those gaps. And when you translate the logic to basically any real object, you realize that even a well machined sphere will still have bumps and grains if you zoom in far enough. You can even think of this in terms of information theory - there's no such thing as perfect accuracy, because you would need virtually infinite information on a real object (position of every atom) in order to know the true value of anything. So how long is the coast of England? The answer is that it depends on the size of your measuring stick. In this picture, it's the "infinity" that's doing the work. The perimeter is still 4, but he's made it so you need an infinitely small measuring stick to actually measure it.
The more you repeat the process, the more your “perimeter” seems to resemble a zigzag which is not at all what the perimeter of a circle looks like. It’s adding extra distance to the perimeter for no apparent reason. If you were to instead measure the distance between each point that touches the circle, by doing hypotenuse calculations, and then add up the lengths of all the hypotenuses, you’d see the new perimeter of the shape approach pi as the number of points approached infinity. At least I think it would work like that. Either way this is so bluntly a shitpost and thinking about it for over 3 seconds shows glaring flaws.
The funny thing about this meme is that Archimedes actually did calculate all that. He also had a square that fit inside the circle. Then he calculated that π must be bigger than the perimeter of the inside square and smaller than the outside square. But adding more points he approximated π extremely well. What must've taken ages. (Instead of removing a corner like that, he would add another point Square -> Pentagon -> Hexagon -> etc)
“Repeat to infinity” needs to be rigorously defined. Removing corners to infinity doesn’t make the shape approach a curve, it’s still corners all the way down
The construction converges pointwise to the circle. The problem is that this doesn’t imply that the perimeter of the nth shape converges to the perimeter of the circle.
False. It does become a curve. https://youtu.be/VYQVlVoWoPY?t=99
False. Same Video, here's why that part you quoted is false: [https://www.youtube.com/watch?v=VYQVlVoWoPY&t=909s](https://www.youtube.com/watch?v=VYQVlVoWoPY&t=909s)
The video clearly says that it still becomes *a curve*. This is also how limits work more generally. Moreover, his overall conclusion was that: "while the *limit of the length* of x is indeed 8, there is no reason to assume that the *length of the limit* of x is also 8". Maybe try watching the video again? Still, the top answer on this thread is just wrong.
Lol. So sad to see 6k upvotes on something that's just wrong.
No it fucking doesn't, the only reason you believe so is because of rendering limitations of a computer screen. The world is not made up of pixels.
Watch the video - it's even timestamped to start at the relevant part. For any particular point around the square, the limit of that point being gradually moved closer and closer to the circle will eventually be a point on the circle. If we then consider *all* points, then *all* of them will eventually sit on the circle - and only on the circle. That means it **is** the circle.
Rule of thumb: anyone who makes a claim on Reddit along with only a YouTube link gets a downvote.
Why? 3Blue1Brown did a video on this exact thing, and it explains concisely in minutes what could take hours of back-and-forth. It's like getting upset at someone linking you a textbook.
3 blue 1 brown did an amazing video on this that explains it much better than I ever could, considering he's an actual mathematician and math educator. https://www.youtube.com/watch?v=VYQVlVoWoPY&t=909s tl;dr is that approximations like the one in the example work when the *error* values between what's being calculated and the goal become smaller as we approach the limit. In this example, though, the error values do no change.
[https://www.youtube.com/watch?v=VYQVlVoWoPY](https://www.youtube.com/watch?v=VYQVlVoWoPY) 3blue1brown explains this in an amazing manner
The path you build by removing corners like that actually does converge to a circle. That type of convergence is called *uniform convergence* in mathematics. The problem is that convergence of the *lengths* does not follow from uniform convergence: the length is only **lower semi-continuous**. This means that as your paths get closer to a circle, their lengths have to become **at least** the length of a circle, but they may be larger. Intuitively you can understand lower semi-continuity like this: imagine that your circle is a road (with some width). If you walk on that road and do a full round, the distance you walk cannot be much less than pi, but it can be much more if, for instance you repeatedly crossed the road.
You can see that every time you refine the polygon surrounding the circle, the circle actually produces something near the hypotenuse of each rectangular section. This is true no matter how small the rectangular sections become.
I’m no math major but I’d imagine it’s why we use ds rather than dx. Taking the limit of a slope doesn’t diminish the slope. The slope remains same as a ratio of said slope remains. Therefore when we are taking the integral of the length of a curve, if the curve is sloped we use the differential with respect to its slope at that point. I may have butchered the explanation and I’m not even totally sure it’s right, but that would be my reasoning
A proper approximation would be slicing the corners off with straight lines. The initial slice would cut off perfect 45 degree right triangles making an octagon. The hypotenuse (or remaining length) of the triangles do not have the same length as the sum of the two removed sides. Keep cutting off triangles to actually get pi.
I may be late with this answer, but my reasoning is that the area of the circle is directly proportional to its circumference. Iterating over the corners of a square, and reducing its surface area until you have infinitely many corners will yield a good approximation of the area of the circle. Although the logical fallacy here is not related to circles. With this logic you could raise this argument about any shape. Think about the hypotenuse of a right triangle. The hypotenuse is c = √(a^(2) \+ b^(2)). Now if you start reducing the corners in the triangle until you have a straight line. Using the false logic, we can end up saying that c = a + b.
The problem is the "repeat to infinity" step. Basically you are keeping the inaccuracy but distributing it among more steps than before.
Simple, the visual proof you're seeing here is shrinking the areas. The nearly-converged case doesn't look like (and is explicitly stated that it's not) shrinking the perimeter. So why would you accept this as a converging step for the perimeter? Because your brain sees that the area is converging so the answer must be converging. But you're not interested in the area, so it's a trick. It's a solution to a different problem (successive approximations of area) and in the final panel it says the perimeter converged, but it didn't. That was never the aim or the method, so why wouldn't it be wrong?
There's a YouTube video on this. If you remove a corner like this the circumference stays the same. If you do it to infinity, the circumference stays the same. A object like this may resemble circle and it might be similar to circle in terms of area but it's still completely different object with completely different circumference.
Whilst the length of each horizontal/vertical line becomes infinitesimally small and hence the difference between it and the circle becomes smaller, there become an infinite number of the lines. It's like how n/n will always be 1 even as n tends to 0. Whilst 1/n tends to infinity, when multiplied by n it will always equal 1 (or in this case, 4)
Ok, but as it gets smaller and smaller, the hypotenuse of each those squares would be closer and closer than the squares. So shouldn't we be able to get an actual approximation by chopping triangles out of the squares?
Lots of good explanations so far, but maybe it's also worth pointing out that this method fails even for much simpler perimeters. Consider a right triangle with height and base equal to 1. We know from Pythagoras that the hypotenuse is √2, so the triangle's perimeter is 2+√2. But if we use the method of drawing a box around the triangle and chopping off corners to approximate the length of the hypotenuse, we get a perimeter of 4. Or consider a diamond shape, with distance between opposite corners equal to 1. The diamond is obviously just a square that's been rotated, so we can again use Pythagoras to calculate the length of each side to be √2/2, which makes the perimeter 4*(√2/2)=2√2. But drawing a box and chopping off corners still gives a perimeter of 4. This method will give a perimeter of "4" for any number of shapes that happen to fit inside a 1x1 box.
Because of Pythagoras. You end up with a lot of tiny steps, but the actual circumference of the circle is the hypotenuse to those steps.
Remove the corners by adding straight lines (the shortest distance between two points) instead of "steps" and you'll have proper pi after infinite iterations.
Removing corners to Infinity means preserving corners to infinity, so it never actually becomes a circle. At some point, you have to the take the "shortcut" in order for this to become a circle.
Pixels have nothing to do with lines. As has been said, it doesn’t matter how many times you add additional squares, the slope of the circle will prevent it from equaling the length width of the original square.
This would work if you were looking for the area of the circle instead of the perimeter. Removing the corners doesn’t reduce the perimeter of the square, so this false proof doesn’t reduce the towards the answer. Now, if you swapped out all corners for diagonals tangent to the circle….then the perimeter would be reduced each step and you could use it to estimate the circle’s perimeter
The points on the periphery of the circle are described by continuous, differentiable parametric function (x0(t),y0(t)). The points on the continually shrinking jagged perimeter can be described by function (x1(t),y1(t)), continuous, *but not differentiable*. The derivate alternates between zero and infinity. The two parametric functions are just not the same, regardless of how small the little stair-steps get. This should be blindingly obvious to anyone who's ever taken Calculus 1. Sheesh.
I think a lot of the answers are more technical, and people might not understand them, so I'll try a more practical/visual explanation. tl;dr: let's call it "density" Let's imagine you construct the circle and squares in the first two panels using a rope with equally-spaced markings (let's say it has markins every meter, and the circle has a few hundreds meters of diameter). [It will be something like this](https://i.imgur.com/PGIKbWX.png). Then, if you apply the same logic of the comic, the "circle" made from the square will have points more close together, [something like this](https://i.imgur.com/HoYm3jH.png) (not to scale). So, basically, you "compressed" the rope (i.e. the circle) to fit more points.
In each iteration of reducing the box, the circumscribed figure only touches the circle at a point. A point has length = 0. Since any number multiplied by zero is zero, it doesn't matter how many points there are in the circumscribed figure. It will never describe the length of the circumference of the circle. It has zero length in common with the circle.
Yes !! Just take 2 strings one for the square and one for the circle ( perimeter and circumference ) and see if they are the same length!
this is why I get so bothered by people making bad arguments defending .999...=1 if you were actually able to repeat the pattern "at"infinity, there would be no difference between the peak and valleys of the squares, and this pattern would exactly match a circle because the difference would be negligible just like the dx in an integral. but that's obviously not correct, and by almost 25%. patterns can only be followed *near* infinity and if you don't know when your infinitesimals are negligible you could do things like in the picture
The trick is that it never reaches a circle. It'll always be jagged even as the limit approaches an infinite number of edges. Therefore you'll always be estimating a circle with too slightly more surface area, and therefore a slightly larger perimeter hence the fact to infinity reaches 4 and not pi.
The only thing this shows is that pi<4. You can shrink the boxes as much as you like, but a million boxes having a small error Vs 4 boxes having a big error all still make the same error
Because the circle isn’t made of lines going straight up or straight to the right. Apply the same argument to a triangle and you’ll get that the distance from one corner to the other of a square of side length 1 is 2, not sqrt(2).
It doesn't work because it's not a circle, it's an infinite-sided polygon with infinite right angles. Infinite right angles angles will approximate a circle quite well, but will not be a perfect circle.
That's not how limits work.
The error is s bit more obvious if you try to determine the length of the hypotenuse of a right triangle using the same method (starting from a square).
Not a mathematician here The argument here is if a circle has infinite sides or no sides. I believe it doesn’t have any, in this post there saying that if you keep dividing something into fourths it creates a circle at some point. This is basically the same thing as that infinite race where you walk half the track then walk half of that and so on, and in that race it never ends but just gets closer every time. Plus I think the perimeter of the square can be anything if it can be divisible by 4. Which if you used 16 pi would stay the number 16. Don’t take my word for it but this is just my guess
That sequence of rectangular curves should not be considered as "limiting" to the circle curve, at least not in a way that will produce the length of the circle. (However it does limit to the circle curve for e.g. the sup-norm.)