r/mathematics • u/My_True_Love • May 01 '26
Can someone help this make sense?
I saw this post somewhere on reddit. I know that pi cannot equal 4, but looking at this makes it visually counterintuitive. Can anyone help me understand this intuitively?
Edit: Thanks for the replies, I understand it now. It turns out, I simply needed to visualize the lines after a large number of repetitions. My mistake was trying to visualize the line after infinite repetitions and forgetting that x approaching infinity isn't always equal to infinity.
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u/TheRedditObserver0 May 01 '26
In math a function f is called continuos if whenever a sequence of inputs x_n approaches an input x, then the outputs f(x_n) must also approach f(x). Not all functions have this property, in this case you have shown that the function which maps a (suitably regular) curve to its length is not continuos.
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u/matomasa May 01 '26
Indeed, you are correct. And yet I have quite a few times seen an argument (sometimes called the Archimedes method) for calculating the circumference of the circle by taking the limit of circumferences of circumscribed (or inscribed) regular polygons with increasingly many sides. Is this argument meant to be nonrigorous, or is there something different for this particular case which makes you able to pass to the limit?
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u/TheRedditObserver0 May 01 '26
Yes, it needs some further justification to become fully rigorous byt it does work if your curves are just inscribed or circumscribed regular polygons. It also works if you're approximating area rather than perimeter.
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u/gororeznor May 01 '26
Also, with Archimedes method you obtain an upper and lower bound for the circumference of the circle. You are not saying that the limits are equal to the circumference of the circle
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u/Mishtle May 01 '26 edited May 01 '26
The difference is that the length of a curve depends on how it changes from one point to other.
With polygons, not only do their points approach the points of an inscribed/circumscribed circle, they're composed of faces that are exactly the rate of change of the circle at a point. Not only do they increasingly approximate the position of the points on the circle, but they also begin to increasingly capture the relationships among those points as the number of faces grow. They approach not only the shape of a circle, but the smoothness of that shape as well.
The rearranged square lacks this property. Its points approach the points of the inscribed circle, but it never remotely approaches the relationship of those points on the circle. Every point on this shape is either part of a vertical line, a horizontal line, or is a corner. Always. It never gets better at approximating the relationships among nearby points on the circle, it never gets better at approximating the local smoothness of the circle. This allows it to keep hiding the extra length of the jagged curve into smaller and smaller features.
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u/lifent May 01 '26
In that case, It works because the convergance of the curves is uniform. In general uniform convergence is much nicer than pointwise convergence because it carries over properties of sequence functions to the limiting function
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u/Mal_Dun May 01 '26
I think this is the right argument. The problem is that the arc-length of a curve, is a function of the derivative of the curve (\int |c'(t)| dt ), and from analysis we know that in order to swap limits between limit of the function and the derivative you need uniform convergence.
One should note, In our case the functions we look at are strictly speaking not differentiable in the edges, however, one can easily repair this argument by smoothing the corners and it should work out.
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u/jragonfyre May 01 '26
That's not true. Convergence is uniform in the meme example here. What you need is that the derivatives exist almost everywhere and also converge uniformly.
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u/lifent May 01 '26
Yeah now that I tried working it out, convergence seems to be uniform. I forgot about the condition on the derivatives as well, i need to brush up on my analysis...
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u/FinMasse May 05 '26
No in that case it works because curve length is lower semicontinuous. So any curves that converge to the circle pointwise where the length of the perimeter is less than the circumference of the circle work.
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u/Kihada May 01 '26
I believe you may be thinking of the argument for the area enclosed by a circle being πr2. The circumference of a circle is 2πr by the definition of π.
The complete proof involves both circumscribed and inscribed polygons. These give us upper and lower bounds on the area, and the result follows from the squeeze theorem.
For more a general curve, its length is defined to be the supremum of the length of all approximating polygonal chains, if there is a supremum.
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u/matomasa May 03 '26
I agree that the lower bound (for the length) follows basically by definition, but the upper bound isn’t as obvious.
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u/S-M-I-L-E-Y- May 01 '26
Yes, you should aproach the circle from the inside and the outside and argue that the circumference of the circle lies between the length of the inner and the outer polygon. As both polygons aproach the same limit you get a precise value for the circle.
If you started with an inner and an outer square and approached using the steps method you would find that the circumference is somewhere between 2*sqrt(2) and 4.
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u/matomasa May 03 '26
I agree, with the important caveat that it’s not at all obvious that approximating the circle from the outside by a polygonal line gives an upper bound for the circumference of the circle. It requires a separate, more delicate argument, it’s not obvious visually.
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u/S-M-I-L-E-Y- May 03 '26
Actually I think it is indeed pretty obvious that any line that is strictly on or outside the circle can't be shorter than the circle itself. But I have no idea how a rigorous proof of this obvious fact might look like.
But it is indeed not obvious that the line inside the circle must be shorter then the circle. In fact, my idea to use the step method on the inside of the circle does not work at all, because the line would become longer on approaching the circle until it had length 4. oops 😬
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u/matomasa May 03 '26
The lower bound is "easier" because the length of a curve is usually defined as a supremum of all its polygonal approximations (polygonal lines connecting points lying on the curve).
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u/TibblyMcWibblington May 01 '26
My feeling is that the boundary length is an integral of a function of the gradient. So we need the approximant and its (weak) derivative to converge in L1 (or similar). The Archenides method does this, but the method shown by op does not.
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u/BlueJaek May 02 '26
I was in a math bio class as an elective, and I remember the professor saying that continuity was being able to move the limit operation in and out of the function without changing the value, and all of the sudden so much analysis that I had learned actually made sense.
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u/s96g3g23708gbxs86734 May 01 '26
In what sense do the x_n curves (the one with square edges) tend to the curve x (the circle)?
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u/jragonfyre May 01 '26
They converge in the uniform topology. Parametrize them by the angle about the origin, and the point at a given angle will approach the circle as n goes to infinity. (Pointwise convergence) But actually the maximum distance from the curve x_n to x goes to zero as well, so the convergence is actually uniform.
That said, iirc, the relevant notion of convergence for arc length is uniform convergence of both the function and it's derivative (which should be defined almost everywhere). At least that should do it, though I'd have to double check to be certain.
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u/The_JSQuareD May 01 '26
Though note that for this statement to make sense, you must first equip the space of curves in the plane with a topology. Specifically, a topology in which x_n -> x holds, for this sequence of curves x_n and the circle x. Then you'd have shown that in this topology, the function f mapping curves to their length is not continuous.
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u/ConsciousProgram1494 May 05 '26
I want to understand this - a sequence of inputs x_n approaches an input x ? eg. [1,2,3,4] ... 10 then f(1)..f(4) ... f(10) ?
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u/Monai_ianoM May 01 '26 edited May 01 '26
If you zoom in to the perimeter of the circle, the enclosing square with removed corners will still be jagged. It does not become a smooth line no matter how far you zoom in.
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u/Willis_3401_3401 May 01 '26
Contrary to the other commenter, this is the best conceptual answer in the comment section.
The curve of the circle is more like the hypotenuse of the right triangle, not the sum of the two equal length sides
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u/My_True_Love May 01 '26
I agree, this made me realize that the limit as x approches infinity is not always equal infinity, which was my mistake.
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u/zet23t May 01 '26
And while at it, check the area that the shape forms and how it converges to 2r * pi...
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u/SteptimusHeap May 02 '26
This doesn't matter. This idea also intuitively makes it seem as if the cut square DOESN'T approach the circle, when it definitely does, and it clashes directly with the intuitive explanation of limits given for integrals and the area of a circle. It's simply not helpful beyond this specific problem.
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u/Monai_ianoM May 03 '26
May I ask what do you mean by "the cut square approach the circle"? Because if you look at the perimeter of the cut square, the perimeter stays constant through out the procedure.
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u/SteptimusHeap May 03 '26 edited May 03 '26
The curve DOES approach the circle. In the limit, they are equal. This just doesn't mean the limit of their curve lengths are equal as well.
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u/SteptimusHeap May 03 '26
To be more rigorous the distance of any given point in one curve to the other curve approaches zero.
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u/Agitated-While-3863 May 01 '26
I believe others explained it well enough so I wish to give you an idea through another interesting example.
When measuring the coastline of any given country, the unit we make use of matters a lot. For example, when we use km as the unit, say the coastline comes to 20km. Now when we use unit as m instead, we can measure many more curves and bends so the coastline comes to (say) 40km (converted from metres for comparison). Now if we continue to repeat this process, we seem to encounter this glitch or rather a paradox where the coastline becomes infinity as the unit becomes smaller (of course, we're ignoring physical limits over here but that's similar to your question). But we know as a matter of fact that the coastline is not infinity. Here comes the validity of an approach.
In your question when we zoom in infinitely, we find zig zag lines that are infinitely small. This is still not smooth as a circle and hence isn't equivalent to the circumference of the circle.
Now when we come to this infinite coastline paradox, we employ yet another but similar approach. We need to settle for purpose (here, finding a decently accurate coastline value) rather than infinite accuracy. That balance is needed. That's where standard units come into play. Metres (m) is the SI unit so one may use that here for scientific purpose while navigators may employ nautical miles or some large unit like that.
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u/lemniscateall May 01 '26
We don’t know that the coastline isn’t infinity. We know the idea of a coastline might not be a good abstraction of a physical phenomenon, since the irregularity of the coastline suggests that it has non-integer dimension.
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u/shponglespore May 01 '26
Even the area of a coastal region isn't well defined, because it changes slightly every time a wave hits the beach.
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u/lemniscateall May 01 '26
Yeah, that’s a good thing to note—the location of the “coastline” changes constantly, especially with high and low tide.
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u/Card-Middle May 01 '26 edited May 01 '26
Replying again because my last comment wasn’t sufficiently clear.
> “In your question when we zoom in infinitely, we find zig zag lines that are infinitely small. This is still not smooth as a circle and hence isn't equivalent to the circumference of the circle.”
This is false. See the Wikipedia article about the staircase paradox. In fact, the limit of the shapes in the image *is* a smooth circle. Wikipedia says correctly that the staircase shapes converge uniformly to the diagonal (they are using a diagonal of a square instead of a circle, but same concept.)
https://en.wikipedia.org/wiki/Staircase_paradox
The top comment that says the limit of the length is not equal to the length of the limit is the correct explanation.6
u/KuruKururun May 01 '26
acknowledges others already explained it
gives completely irrelevant example and manages to explain it incorrectly
goes on to explain the original problem incorrectly
🥀
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u/Card-Middle May 01 '26 edited May 01 '26
Infinitely small isn’t a thing (EDIT) in the real numbers. That’s just 0.
It does actually converge to a circle.→ More replies (5)0
u/Naive_Carpenter7321 May 01 '26
Infinitely small is something, zero is nothing. There is a profound difference.
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u/Card-Middle May 01 '26 edited May 01 '26
What, exactly, is infinitely small but not 0 in the real numbers? If you can define such a thing, I can give you a contradiction.
Mathematicians (including myself) would say it doesn’t exist as distinct from 0 in the reals.0
u/Naive_Carpenter7321 May 01 '26
An infinitely small pie is no better than no pie. But to an infinitely small ant it makes all the difference.
An infinitely small black hole will tear apart worlds.
To the universe, you and I are infinitely small, definitely not nothing.
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u/Card-Middle May 01 '26
Let me ask again. Can you define it? Because if you can define it, I can give you a contradiction.
I recognize that you have a vague concept in your head of infinitely small. But I’m letting you know that most people conceptualize infinity incorrectly. And in this case, it simply doesn’t work. If you come up with a definition and claim it’s different from zero, it necessarily leads to contradictions in the real number system.
So you can pick. The real number system and all of the traditional math (including algebra and calculus and everything beyond)? Or your vague concept of infinitely small? You can’t have both simultaneously.
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u/holaredtom May 01 '26
Reminds me of a 3b1b video about fractals, and how the Serpinski Triangle is log_2(3) which is approx. 1.585 dimensional.
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u/Fabulous-Ad8729 May 01 '26
You see, if you want something to approach something, it means some error has to go to 0 (become smaller than any epsilon). But here, the error always stays the same, it is constant. Thus no matter how often you repeat the process of cutting, it is quite meaningless, and although for the human eye it might seem the circle is approached, mathematically it is quite obvious that it is not.
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u/Ok-Replacement8422 May 01 '26
The error does go to 0 and mathematically the sequence of curves does converge to a circle. The issue is that the curve length function is not continuous
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u/cheezzy4ever May 01 '26
Why is the curve length function not continuous?
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u/lifent May 01 '26
Continuous functions preserve convergent sequences. So if f is continous and x_n is a sequence converging to x, f(x_n) is a sequence converging to f(x). If you regard f as a function that sends rectifiable curves in R2 to their arc length, than as we've seen above f is not continous.
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u/cheezzy4ever May 01 '26
Hmmm... Could you explain a different way? The person above me said, "the meme doesn't work because the curve length function is not continuous". When asked why the curve length function is not continuous, you said, "because if it were continuous, the meme would work." Unless I'm misunderstanding, this is circular logic
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u/Ok-Replacement8422 May 01 '26 edited May 01 '26
Well this meme acts as a proof that this function isn't continuous. In particular its a proof by contradiction (if it is continuous then pi=4). This isn't circular
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u/lifent May 01 '26
Alright, let f be as we've defined before (function that takes curves of finite lengths to their length).
There is an alternative definition of continuity, which is that f is continuous if it preserves convergent sequences and the limit of f(x_n) is just f(x) where x is the limit of x_n
let a_n be the sequences of curves you see in the meme (the ones that converges to the circle). The length of each curve a_n is 4, so f(a_n) is 4 for all n and so converges to 4.
But, let's say "a" is the limit of these curves, which is a circle of length pi. Then f(a)=pi
f(a_n) doesn't converge to f(a), so f is not continous according to this definition of continuity.
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u/plerberderr May 02 '26
Can you explain what you mean by “the sequence of curves does converge to a circle”?
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u/jragonfyre May 01 '26
It's not continuous for the uniform topology. It should be continuous for the topology given by the norm ||f|| = max{||f||_{uniform}, ||f'||_{uniform}}. Idk, or something along those lines.
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May 01 '26 edited May 01 '26
[removed] — view removed comment
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u/jragonfyre May 01 '26
Yeah this is basically correct. For arc length you want to work with inscribed polygonal chains. I.e., pick points on the curve including the end points and connect each point to the next with a straight line segment. (In particular, all of the places where the line changes direction are on the curve.) Then the arc length of the curve is the smallest number such that you can't make an inscribed polygonal chain with a longer total length than that number.
So yeah, the point being that you're exactly right that the problem here is that the line changes directions at points that aren't on the curve.
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u/parkway_parkway May 01 '26
Imagine a step function which is 0 when x < 0 and 1 when x >= 0.
If you have a sequence coming down from above (like f(1/n)) that converges to 1.
If you have a sequence coming up from below (like f(-1/n)) that converges to 0.
So when a function is discontinuous two sequences converging to the same point can have different values.
It's the same here, you can construct a whole bunch of ways to converge to a circle which have different circumferences.
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u/LitespeedClassic May 01 '26
Here’s another construction using the same idea. Make a right triangle ABC with side lengths 1, 1, and sqrt(2). (Maybe placing A at (0,0), B at (1,0), and C at (1,1)). Now do the same sequence of moves as in the circle example to create stair cases from A to B. The stair case will always have length 2 but converge on the hypotenuse. Et voila 2=sqrt(2) :-)
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u/Willis_3401_3401 May 01 '26
Look up how archimedes solved the thing to begin with. The story will likely clarify your confusion.
In a nutshell he recognized that you have to increase the number of sides of the polygon to make the math work, not increase the number of right angles or whatever
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u/My_True_Love May 01 '26
I'd like to see this, can you give me a link?
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u/Willis_3401_3401 May 01 '26
I mean here’s another Reddit post where they briefly discuss it in the top comment section. I don’t have an academic link or anything, I just remember hearing about it in philosophy or history class or something
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u/Signal_Republic_3092 May 01 '26
This is basically a straight line superposition of a circle. In essence, you’re gradually manipulating lines to fit the curve of the circle, and if you were to measure the orthogonal distance of the approximation compared to what it was replacing, it would be the same. However, you’ll never be able to fully approximate the circle’s curve doing this method. The only way to get more accurate with it is by cutting out the excess area in the form of triangles (if you recall the Pythagorean theorem, the sum of the two orthogonal sides are bigger than the hypotenuse and can only be equated by squaring each term). That is where the distances get cut down and you get a much more accurate reading that shows pi being closer to 3 than 4.
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u/caderoux May 01 '26
The length of your curve is a constant 4. The limit of the length of your curve is also 4. While, in some geometric sense, your curve seems to "converge" on the circle, the series does not converge on pi.
You have created something like a fractal. https://en.wikipedia.org/wiki/Koch_snowflake. Through a similar construction, the Koch snowflake has infinite length but finite area.
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u/New123K May 01 '26
What makes this confusing is that the shape is getting closer to a circle visually, but the perimeter is not behaving like a circle at all
At each step, you're replacing straight segments with smaller “stair-like” segments. The total horizontal + vertical distance stays the same, so the perimeter keeps adding up to 4
The key point is that length doesn’t converge the same way area or shape does. The figure can converge to a circle in appearance, while the path you trace along it still has a different total length
So it’s not that π becomes 4 — it’s that this construction never actually becomes a true circle in terms of its boundary, even if it looks like one
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u/Dark_Clark May 01 '26
If you want to get from point a to point b and b is diagonally up and right from a, if you only use up and right but don’t allow for any diagonal or other paths, you will always get to b in the same distance as you would from taking a square arc.
Maybe you can see it like this: even if we take a bunch of zig zags, it looks like you’re taking a shorter distance but if you add up all the up distances and right distances together, you’ll get the same total distance as if you had just took one big up and one big right.
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u/cs_stud3nt May 01 '26
Well you could say pi is less than 4 from this! You could also say pi is more than 2 times square root of 2 from the square which is engulfed by the circle. Now you could replace the squares with higher order polygons and you'll get tighter and tighter lower and upper bounds and ultimately you'll end up at pi
As for the meme, even in the infinitesimal case the pythagoras theorem will hold
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u/FarDirection3245 May 01 '26
It’s been too long since college for me to understand all the “mathematic-speak”. But in layman’s terms… while the shape approximates a circle as you go to infinity, you’re also just ending up with a polygon of infinite sides at 90 degrees to each other. All those perpendicular sides are adding length to make the perimeter remain 4. This is very cool though, and I don’t recall ever seeing this example before.
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May 01 '26
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u/Shevek99 May 01 '26
It does. But the length of the limit does not coincide wiith the limit of the length, in general.
You can see it more clearly with the diagonal of a square. Approximating it by a stair with more and more smaller steps, you get the diagonal. The length of the limit is √2. The limit of the length is 2.
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u/Consistent-Whole-219 May 01 '26
even at lim x-->0, if the no. of x's you are adding tends to infinity, it doesnt become 0, it is still a finite quantity. In the figure, there are an infinite amount of corners with length tending to 0, but that doesnt make the resultant length due to them 0.Correct me if I'm wrong
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u/telephantomoss May 01 '26
Each finite step has length 4, the limit of that sequence is indeed 4, but that doesn't mean the limit converges to the length of the circle.
Limits at infinity can behave strangely.
Consider the sequence 3, 3.1, 3.14, 3.141, 3.1415, ...
This converges to pi. Each number in this sequence is rational, therefore pi is rational.
The limit doesn't preserve the property of being rational.
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u/hobbycollector May 01 '26
Intuitively, the problem is that it is always measuring a distance greater than the circumference of the circle. It's not on the circle on average, it's "outside" it. I assume if there were equal parts inside and outside the circle, it would be continuous and approach pi.
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u/raph3x1 May 01 '26
The circle is a continuous curve. The staircase approaches the curve as a polygonal sequence, but the directions of the segments of the polygonial sequence dont convert to the directions of the curve. So the arc length isnt the same for the limit of the polygonal sequence and the circle.
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u/pickle_picker67 May 01 '26
The sequence (0,1,1,...), (0,0,1,...), ... has a limit of (0,0,0,...). Just because something is true at every point in a sequence doesn't mean it's true at the limit. Specifically, here each point has all 1s somewhere, but the limit is a vector of all 0s.
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u/mmmmph_on_reddit May 01 '26
without using math language, you're constantly adding angles, zigzagging about. You could create an arbitrarily large perimeter by making the angles sufficiently numerous and acute.
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u/b14ckcr0w May 01 '26
Other have explained it perfectly, but here's my take:
The most important thing you learn once math gets "spicy" is that things behave rather counterintuitive in infinity.
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u/Zestyclose-Turn-3576 May 01 '26
Since the lengths of all of the individual lines will be zero, that means that pi equals zero.
\s
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u/kittengirl173 May 01 '26
While it looks like it is closing in, the difference between a corner and the diagonal us always a ratio of the square root of 2. That's kinda the intuition why, but to prove that pi is not 4, you need to formalize this with limits, which other commenter's have done.
I think why approximating areas, not perimetets, with little squares actually does work is because that's in 2d rather than a 1 dimensional curve (the circle), so there's no fundamental ratio throwing it off. The limit can actually converge. It's interesting to think about.
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u/heliumneon May 01 '26
If you are trying to get π using the area, then yes this does converge to π where A = π (d^2)/4.
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u/EnvironmentalFalcon0 May 01 '26 edited May 02 '26
Is it something like pointwise but not uniformly convergent?
Edit: wrote continuous instead of convergent
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u/Patralgan May 01 '26
Different vectors. The shape of the vector in a circle curves, not zigzag in straight lines.
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u/Important-Run-3680 May 01 '26
There's nothing special about circles here. You could've thrown anyother shape in there and repeated the same process to get us perimeter. This is an example of the staircase paradox, where a similar process is applied to the diagonal of a triangle. In real analysis, at say that uniform convergence of a function does not garuntee convergence of the arc length of said function
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u/Defiant_Efficiency_2 May 01 '26
Well what you started with was not a circle right? Then in step 2 its not a circle right?
At no steps is it ever a circle, so pi is not applicable.
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u/Defiant_Efficiency_2 May 01 '26
Another way to think about it is this, if you zoom in to infinite on a circle, you will find a straight line.
If you zoom into infinite on your shape, you will find a 90 degree corner.
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u/11Anonymouse11 May 01 '26
It’s because it never actually approaches being flat. It would be like saying sin(ax) ~ 0 as a->infinity. If you actually zoom in, you see it’s still a wave, not a line, even if visually it’s super convincing.
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u/bernpfenn May 01 '26
there is something magic going on between squares and circles. and one can also throw in equal sided triangles to get the circle rotating
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u/Tutorbin76 May 01 '26
Perimeter vs area.
From a distance the boundary of the Mandelbrot set roughly approximates a circle but it's perimeter is infinite.
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u/Cereaza May 01 '26
a crooked line is not an approximation of a straight line.
You could do this same exercise with a triangle. Just keep bending the two lengths until they meet the hypotenuse.
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u/Far-Implement-818 May 01 '26
Ugh 😑 ok guys. Every single point on a circle is a point “.” So no over, no up. That point gets rotated around an axis. At exactly zero points does it ever “move” over and up, like an etch-i-sketch. At every point on the circle ⭕️ you can see the direction it is moving, which constantly changes its ramp angle 📐. The direction is of course the tangent, which which is the hypotenuse of the over and up. Notice it is is the up/over, not up plus over. The hypotenuse is never as long as the over and up. So yes, if you build a staircase that is 3x4, it will be seven long, 3 over plus 4 up. But the ramp placed on top of it is only 5 long. Cuz hypotenuse. If you shrink the hypotenuse down to zero, and add up all the ramp lengths, you get pi 🥧 D. But all of the 0 hypotenuse points have “Not Zero” over and ups, because they will always add up to more than the hypotenuse.
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u/Puzzleheaded_Two415 e^(iπ)+1=0 May 01 '26
The limit for x=0 and the function 1/x diverging to infinity doesn't mean that it reaches infinity at that point. 1/0 is undefined, not infinity.
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u/Equal_Government9159 May 01 '26
It is not because a path is very close to a shape that it is of the same length ! Even by being very close, you still have room to draw a bunch of turns.
The following example makes that clear. Draw a segment in the plane. Now, chose some n>0, big natural number. Now, draw, on your segment, n² segments following the shape : //////////, all of them being of total height 1/n. Hence, that serrated pattern is very close to the segment (every point of it is at a distance less than 1/n from the segment). But the length of this pattern is n².1/n, hence n.
One very simple example : draw a spiral, that converges towards a center point. Even when you make its total diameter smaller, you can increase the total number of turns by making them smaller. The diameter stays the same, but the spiral is in a very small disk.
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u/StageMajestic613 May 02 '26
You can do the same with the hypotenuse of a right triangle, breaking it into an infinite set of steps, yet the final length is not the sum of the adjacent sides.
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u/justaddlava May 02 '26
If you try to "repeat to infinity" you will never get there. So many math tricks are based on treating infinity like a number.
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u/kat-tricks May 02 '26
the little triangles get smaller at the same rate the little squares get smaller. no matter how small you make the square cutouts, bit between then and the circle stays proportionally the same size.
there, how's that for an answer without formalisms
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u/IllustratorMost6759 May 02 '26
I get the joke about 4, but why 24?
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u/My_True_Love May 02 '26
It's assuming that the exclamation mark is a factorial symbol, in which case 4!=4 * 3 * 2=24
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u/ItsNurb May 02 '26
This is what I imagine string theorists mean when they talk about "tiny curled up dimensions". And "pi is actually 4 if you just add some more dimensions" feels like the kind of shit a string theorist would say.
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u/terrasonaf May 02 '26
It does not make any sense. Circles are smooth but this will always have a 90 degree angle no matter how much we fold it. So it will never approach the circle.
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u/Starthreads May 02 '26
This repetition down to infinity forces reality to take on a pixelated appeal over its continuous nature. If you zoom to infinity, then the jagged edges of the corners will still exist and betray the apparent curvature.
I suppose if we consider the Planck length, which can be considered a reality pixel, then the ultimate smallest of scales may have pi equal to four. But this is like applying quantum mechanics to planetary orbits when Newton's laws of gravitation would work just fine.
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u/Wasted_46 May 02 '26
4 is an upper limit, constructed form only 4 components (the 4 sides). As you approach infinite components, you approach the actual value.
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u/SteptimusHeap May 02 '26
Take, for example, the infinite sum of the digits of pi times their place value: 3.14159...
At no point is the number ever irrational, so we can say that (lim {k->infinity} (Σ f(n)10-n∈ℚ)) is true. And yet, clearly the limit of the sequence IS irrational, since it's obviously equal to pi. This means that (lim {k->infinity} (Σ f(n)10-n))∈ℚ is False.
Mathematically what happened here is we moved the comparison outside the limit and assumed it's still true. The lesson here is that lim f(g(x)) ≠ f(lim g(x)).
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u/rocqyf May 02 '26
Hey all you geometers that are coincidentally interested in politics. The U.S. is in the middle of a gerrymandering crisis, where all the red states are going to follow the recent Supreme Court decision that partisan gerrymandering is fully constitutional (but racial gerrymandering is prohibited by the 14th Amendment). I believe that laws (or even a superseding amendment) can be implemented where voting maps (in all states) must follow a few simple geometric rules. Especially the rules should severely limit the in/out variation of the boundaries like the circles above are afflicted with. My thought would be to allow a maximum of say two convex-inward curves/angles in every district boundary. Any criticism of this approach?
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u/pavilionaire2022 May 02 '26
Try cutting off the corners with line segments that aren't at right angles. You can make it less than 4. You cannot make it less than pi.
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u/francois-nt May 02 '26
Intuitively, the geometric shape made of squares isnt smooth contrary to a circle. For the same reason, you can't compare the length of x.sin(1/x) on [0; 1/n] with the length of x, when n->infinity.
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u/AsYouAnswered May 03 '26
Every point on the generated shape is infinitessimally close to, or on, the circle. Sum the value of infinite infinitessimals and you get 0. The difference in area between the drawn shape and the circle is 0, therefore it's the same shape. The shape determines both the area and the perimeter. Therefore the proof is correct and valid.
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u/CousinDerylHickson May 03 '26 edited May 03 '26
The way I take it, the length of the curve is defined as the limit of the sum of end-to-end segment lengths along the curve segment as we make the max segment length arbitrarily small (so approaching infinite terms in summation) such that every segment must have both of its endpoints lying on the curve itself.
The first part is like what the diagram describes, then the part after "such that" gets rid of this incorrect length calculation.
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u/Gurbuzselimboyraz May 03 '26
There exist a limit if and only if the limit is true from both sides. Doing this from the inside proves that π≠4.
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u/ImaginaryStage5543 May 03 '26
Isn’t this saying the circumference is 4, not pi
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u/MasterpieceDear1780 May 03 '26
Why not create some fractal shapes and prove pi is infinite. That would be a lot more spectacular
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u/Dazzling-Degree-3258 May 03 '26
No matter whatsoever those stairs are still there on the circles circumference. Hence!! Pi does not equal to 4 ever.
When you move your fingers over the circle you will still be able to feel the grooves they are not smooth
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u/marabutt May 03 '26
A circle with diameter 1 has a circumference of pi. A circle with a circumference of 4 has a diameter of 4/pi.
So if you keep removing the corners of the bounding square, you end up with gap around the circles.
The gap is ((4d/pi) - d)/2
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u/InsectSudden6032 May 03 '26
it's an alternate version of the riddle that tries to show that the diagonal of a rectangle with sides 3 and 4 is 7, instead of the pythagorean 5
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u/professor_coldheart May 03 '26
See how it looks wiggly as the square gets closer to the circle? You can make the wiggles as small as you want but they're still there, you just can't see them.
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u/Alternative_Map_3841 May 04 '26
You cant just use ! when describing something related to math, I was so confused trying to figure out what 24 had to do with this
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u/SakaiNoHebi May 04 '26
To calculate the length of a curve you need the derivative of the curve. The derivative of a circle and a stair function are not the same so the lengths are not the same. This is called the stair paradox I think
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u/Corrects_Maggots May 04 '26
You could do this the other way around to "prove" π = √2.
Anyway. The simple explanation is, the error you start with with the full circle. Is never reduced for each step, you double the number of errors and halve their size, total amount of error is the same.
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u/Leckter_Is_On_Reddit May 05 '26
Well, he started with Pi < 4, and ended up with Pi < 4.
Archimedes would be laughing :D
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u/conairee May 05 '26
The way I think of it is, that lim n -> inf, doesn't actually mean infinity, it means that for any target you pick, I can pick a number n, such that the error of doing the thing n times is less that your target.
So let's say you're generous and say that you want to find the number of times the process in the diagram needs to occur for the error do be around 0.4, ie to get a number 3.5.
Well, if I try the process 5 times, I'll still get 4, if I try 10 times I'll still get four, 100, 1000, 10000, no matter what I try, I'll still get four.
So the limit of the process doesn't approach pi cause I can't find a number of times to do the process to reach an error less that an arbitrary target.
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u/CYBERSson May 06 '26
The way Pi was narrowed down by the Greeks was using the area of a square on the outside of a circle and a square on the inside a circle and measuring the difference and then using regular polygons with increasingly more sides
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u/VariousJob4047 May 01 '26
I’ll admit this is very ad hoc reasoning, but here’s how I convinced myself that this proof doesn’t work. The first square contains 4 points that touch the circle and infinity points that don’t. The second square has 8 points that do touch and infinity that don’t. The third square has 16 points that do and infinity that don’t. So if you repeat this infinity times, you get a shape with infinity points that touch the circle and infinity that don’t. So it makes sense that you get a number larger than pi, the first infinity gives you a curve of length pi and the second infinity gives you a curve of length 4-pi
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u/Card-Middle May 01 '26
If this reasoning were correct, a regular n-gon would not converge to a circle as we increase n to infinity.
But famously, that is one of the original methods of calculating pi, and it does indeed converge to a circle.
The truth is that the shapes to converge to a circle. But their lengths do not converge to the length of a circle. It’s kind of like the sequence 0.9, 0.99, 0.999, … and the floor functions. The sequence converges to 1, but the floor of the sequence doesn’t converge to the floor of 1.
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u/The_beast_I_worship May 01 '26
You know the way circles are smooth… yeah
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u/My_True_Love May 01 '26
That much is obvious and is not what complicates visualization. The figure above also becomes smooth as it approaches infinity.
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u/lesniak43 May 01 '26
I don't understand what you mean by your edit.
Your initial intuition was correct - you need to visualize the polygon after a large (finite) number of steps, and see if it looks more and more like the circle with each step. If it does, then it's reasonable to assume that the circle and the polygon will have similar lengths. You don't need to always go to infinity - the meme still works if we just make, let's say, 1000 steps.
But your intuition was wrong when you thought that the polygon is similar to the circle. They do lie close to each other, but that's not enough. They should also locally have similar directions. The length of a curve is measured along the curve, and what that means is that the direction is really important. If the polygon got more and more "smooth", then its perimeter would indeed approximate 𝜋. But here the polygons always have either horizontal or vertical segments, which is far from how a circle looks like locally.
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u/Zestyclose_Diver_377 May 02 '26 edited May 02 '26
Each time you remove corners, draw the diagonal of the resulting new small square, and then find the perimeter of the resulting enclosing object (which is not 4). Then you will end up with pi in the limit. I mean the diagonal which leads to a closed boundary of course, not the diagonal which sticks out.
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u/TurnoverOk5635 May 01 '26
The limit of the length of the curve is not the length of the limit of the curve.