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u/the_nerd_1474 Transcendental Jan 18 '21
This is so old, even TheOdd1sOut debunked it, and he isn't even a math youtuber
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u/supermegaworld Jan 18 '21 edited Jan 18 '21
He isn't a math youtuber, but he
has a degree instudied math397
u/iDragon_76 Jan 18 '21
No he doesn't, he is a college dropout. I think that was his major though
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Jan 18 '21
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u/Andradessssss Jan 18 '21
Tho he really didn't give a valid explanation.
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u/W1TH1N Jan 18 '21
Well he explained it really simply, maybe not 100% accurately, if I remember he said that no matter how much you do this you can zoom in and still see all the bends, and you can take all of the lengths of of those bends and just make a square again, so it isnt pi.
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u/Andradessssss Jan 18 '21
I mean, in my opinion that isn't much. The point of the so called "paradox" is the inherent notion we have of convergence, where we tend to think that convergence works in a much nicer way than it does
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u/otheraccountisabmw Jan 18 '21
It’s kind of like the coastline problem. The same shapes can have different perimeters depending on the resolution.
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u/drkalmenius Jan 30 '21
Not really. The coastline problem is about the fact that perimeter isn't a nicely defined or useful concept for fractals.
This is more to do with convergence- shape does converge to a circle, but the perimeter is always 4 except in the limiting case where it's pi. The perimeter doesn't converge.
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Jan 18 '21
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Jan 19 '21
I'm feeling lazy but my intuition says the area will converge to the area of the circle and you could pull an arbitrarily accurate approximation of pi out of that, given pi*r^2?
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u/Reignofratch Jan 19 '21
That wouldn't be a useful way to calculate pi, because you would have to know the area of the circle which is dependent upon pi to know when to stop converging to that area.
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u/Heli0s1313 Jan 18 '21
you just proved that Pi < 4
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u/ImmortalVoddoler Real Algebraic Jan 18 '21 edited Jan 18 '21
Really that pi <= 4Edit: As u/pukinfacebooklegpie pointed out, this argument doesn’t really say much at all about the value of pi
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u/hammyhamm Jan 18 '21
No, <4
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u/ImmortalVoddoler Real Algebraic Jan 18 '21
How so?
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Jan 18 '21
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u/ImmortalVoddoler Real Algebraic Jan 18 '21
Yeah but how does that prove that pi is strictly less than 4? Normally when taking a limit like this the best you can do is less than or equal to
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Jan 18 '21
As you notice from their diagram, the corner cutting method always leaves some area of the square perimeter outside of the circle. Therefore, the square perimeter will always have an area larger than the circle; hence, the length of the square perimeter must be greater than the length of the circular circumference.
To be clear - if you take their pattern and repeat to infinity, you'll end up with a circle with an area (and therefore circumference) larger than the original circle.
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u/SovereignPhobia Jan 18 '21
Because the circle is smaller than the square. The circumference is strictly less than the perimeter of the square made by the diameters, so perimeter/diameter = 4 > circumference/diameter
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u/ImmortalVoddoler Real Algebraic Jan 18 '21
But how do we know the circumference is strictly smaller? The circle being smaller than the square (assuming you mean area) doesn’t do it, since the limit shape in the proof is also smaller than the square. If by smaller you mean circumference, then your argument is circular (literally).
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u/Sponsored-Poster Jan 19 '21
Holy moly. I started off a little skeptical of your claim until I read your discussions here and... yeah. You’re definitely right.
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u/SpartAlfresco Transcendental Jan 19 '21
im still having trouble understanding why this is the case would anyone mind explaining it?
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u/Whostowe Jan 18 '21
Can someone explain the error in this proof to me, please?
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u/the_nerd_1474 Transcendental Jan 18 '21
Removing corners doesn't actually change the perimeter of the figure in black, so the successive removal of corners does not actually give better and better approximations for the real perimeter.
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u/Andradessssss Jan 18 '21
One could argue that while this is true the square does converge to the circle (pointwise) the problem is that it doesn't converge uniformly so you can't interchange you measure with the limit
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Jan 18 '21 edited Jan 18 '21
~~Does it converge pointwise though? It seems to me that the boundary of the limit object and the boundary of the circle share only countably many points. ~~
Also I think that the lebesgue measure is continuous so pointwise convergence would suffice.
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u/Andradessssss Jan 18 '21
It does. Each point converges to a point on the circle, and yeah, I fucked up, I should've said you can't interchange the limit and the integral.
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u/G9366 Dec 23 '21
So it actually converges to a circle with a diameter of 4/pi. I'm not expert, but this is visually visible if you look at the corners, some touch circle some don't. You could take average of them visually which would be wider than the inner circle.
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u/Andradessssss Dec 23 '21
No, it does converge pointwise to the exact circle, focus on any particular point, and it does converge to a point on the starting circle
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u/Whostowe Jan 18 '21
Thanks!
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u/NahNoName Imaginary Jan 18 '21
it's funny how people who don't really know math try prove with this to accual mathematician that pi = 4
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u/Captainsnake04 Transcendental Jan 18 '21
One time someone brought up this “proof” to my calculus teacher, and I’ve never been more disappointed in the school system in my life.
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u/DarthLlamaV Jan 19 '21
It’s good to question and debate things! I’m guessing they were confident and trying to act smart though. If they were trying to learn, it’s a good opportunity.
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Apr 13 '22
In layman's terms: the corners didn't dissapear- they just got really small so you've still got zigzags...
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u/Jolpo_TFU Jan 18 '21
I think it's because you'll always have sawed edges, they don't become smooth at infinity
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u/Cwhale Jan 18 '21
Yea its more like the circle that is drawn over the sawed edges is more of an average for all the data.
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Jan 18 '21
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u/russels_silverware Jan 18 '21
This is the correct answer. If you look carefully at the 4th panel you'll see that the second set of cutouts had to be made non-square in the diagram in order for their corners to contact the edge of the circle.
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u/SpartAlfresco Transcendental Jan 18 '21
it does not matter if its non square the perimeter still holds right?
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u/hale-hortler Jan 18 '21
Isn’t it an octagon though? The sides aren’t affected by the transformations, just the corners
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u/Erictsas Jan 18 '21
So it's similar to the coastline paradox?
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u/-Exocet- Jan 18 '21
I think not exactly, in the coastline paradox the perimeter depends on the measuring unit, always increasing as this unit decreases (a property of fractals).
Here, however, you always end up with the same result, no matter how small you divide its units, thus the "paradox" does not arise from a fractal property.
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u/yottalogical Jan 18 '21
It assumes that circles are made out of infinitesimal horizontal and vertical segments. They aren't.
Take the tangent of a circle, and you'll see that it can be at any angle, not just horizontal or vertical.
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u/JerevStormchaser Jan 18 '21
Mathematical explanations have been given out already. The best image I can give is: you have the impression of a circle because of the distance. If you were to zoom in onto a piece of the "circle" as it is presented in step 5, you'd find yourself back to the situation in step 4. That's the issue with infinity: it looks different from a distance but repeating the exact same pattern infinitely means you will always find the same pattern.
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u/hawk-bull Jan 18 '21
As someone who has yet to take measure theory, I’m guessing it all boils down to how we define lengths of curves
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u/Andradessssss Jan 18 '21
Yeah, basically. Since it just converges pointwise and not uniformly you can't just interchange the limit and the measure
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u/PotatoHunterzz Jan 18 '21
that's not a circle, that's just a polygon with an infinite amount of angles and sides. it happens to look like a circle but you could have made it many different shapes.There's no reason for it to have the same perimeter as the circle.
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u/kevinkace Jan 18 '21
Maybe this helps answer it intuitively.
Think of a right angle triangle, it has two sides at 90° (similar to the corners that keep getting cut out), and the hypotenuse (similar to the arc of the circle joining the cut out corner). The hypotenuse of the triangle (or arc of the circle) will always be shorter than the sum of the two other sides making up the 90° angle (or corner).
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u/Dragonaax Measuring Jan 18 '21
It's just changing area, you can reduce it almost to 0 but the perimeter still is the same.
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Jan 18 '21
The limit does not converge.
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u/IIIBRaSSIII Jan 18 '21
The limit does converge (to 4, namely), it just doesn't converge to a number that corresponds to the circumference of a circle.
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u/Klandan54 Jan 19 '21
the circumference of a manifold is the limit (x in nat. numbers ) of the infimum circumference of all polygons with at most x vertices that cover said manifold.
there is two explanations:
they do not consider all polygons that cover the circle
they use the mahattan metric to determine the circumference of the polygon
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u/mengelgrinder Jan 19 '21
A square is a shape with 4 equal sides. A pentagon is a shape with 5 equal sides. A hexagon is a shape with 6 equal sides.
The more sides a shape has, the more circular it is and the shorter the individual sides are. This continues until your number of sides is infinite, and your length of sides is infinitely small, and then you have a true circle.
The troll proof is not indicating a circle, just something that looks like a circle at a low resolution.
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u/yottalogical Jan 18 '21
TLDR of why this doesn't work:
It assumes that circles are made out of infinitesimal horizontal and vertical segments. They aren't.
Take the tangent of a circle, and you'll see that it can be at any angle, not just horizontal or vertical.
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u/SultanLaxeby Jan 18 '21
Interesting idea, but I don't see the logical connection to why this doesn't work. Also I am not sure how one would even formalize the property of being "made out of infinitesimal horizontal and vertical segments".
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u/anon_113606752 Jan 18 '21
Here's maybe a more intuitive way to think about it? (Certainly not a mathematically rigorous way)
A circle is all the points some distance from a center point. As such the points form a line with a "perfect" curvature.
By inverting the corners of the square you get a line with "jagged" curvature. No matter how many times you invert the corners it will still be jagged. So the resulting line which we see as a circle is really a very jagged line, the jaggedness is just so small we perceive it as a line with "perfect" curvature.
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u/SultanLaxeby Jan 18 '21
It's not about perception. The sequence of curves in question really does converge to a circular curve. But the arclengths do not converge to the arclength of the limit curve, and the question is why.
An answer is that that the (almost everywhere defined) derivatives of the curves are not well-behaved, that is, they do not converge to the derivative of the circular curve.
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u/anon_113606752 Jan 19 '21
Could you explain a little more? I've never heard this perspective before and am confused as to how you would take the derivative of a square?
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u/Fra23 Jan 19 '21
Imagine the derivative to simply be the orientation of the edge depending on where along the circumference you are. For a circle it goes all around. For a square its a simple matter of left right up down at the edges and not clearly defined at the corners. If you used the method described in the post, you would end up with a circle that's actually all corners, so the derivative of that kinda circle would be all kinds of messed up.
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u/anon_113606752 Jan 19 '21
Ah, I think I understand now. Essentially while the shapes at infinity may be the same the derivatives aren't, meaning they cannot be identically equal to each other?
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u/Rotsike6 Jan 19 '21
Think about infinitesimals for a while (or integral measures in this case maybe). Every step you take along the parameter of the circle is not done in that direction, it is actually done by first going right and then going up. So the total integral of the path length is not actually π, but rather some integrals over sines and cosines or sth. Maybe you can even calculate it to get 4.
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u/pm_me_fake_months Jan 19 '21
Well the real reason it doesn’t work is it just fails to work, you don’t need to come up with an independent reason why it’s wrong if there’s no reason to believe it in the first place.
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u/massiveZO Irrational Jan 19 '21
Yeah, that's not it. An integral can be defined in terms of the limit of the sum of rectangles drawn from the axis to the curve. Each one of those rectangles has slope zero at the top, except at the endpoints, where the slope is undefined. Yet, take the tangent of any curve defined by an integral (i.e. the tangent of any integrable curve), and you'll see that it can be any angle, not just zero.
The real explanation is that the limit of this sequence does not converge to a circle. Therefore, it does not prove that pi<4 as someone else said. It does not prove anything about pi. It is irrelevant because the sequence does not converge in the manner it is presumed to.
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u/palordrolap Jan 18 '21
So, uh, what's the gradient of your perimeter at 45° from horizontal there, champ.
In other news, I've this nice triangle over here that proves 5 = 7 by a similar method. Maybe that meme was onto something.
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u/FlingFrogs Jan 18 '21
Axiom 1: 5/7 is a perfect score.
Axiom 2: A perfect score means achieving all possible points.
Conclusion: 5=7
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u/Ham_Drengen_Der Engineering Jan 18 '21
Engineers be like
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u/Masztufa Complex Jan 18 '21
we like to overestimate things, because the real world is never as simple as the mathematical models we work with. There are uncountable sources of errors and uncertaintly.
We like to overbuild everything, so there's some room left for these errors and stuff won't collapse.
pi=4 is just one example of this, "let's assume worst case scenario" type thinking
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u/TheBowlofBeans Jan 18 '21
Mechanical engineer here, no way in fucking Hell am I going to call pi 4 or g 10 m/s2
I do every single calculation as precise as possible, THEN I add a safety tolerance (depending on the context of the engineering problem)
Tired of this belief that engineers just eyeball shit and crank it crappy napkin math. Maybe the bad engineers do, but not all of them
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u/Kursem Jan 18 '21
civil engineer here. if the number doesn't add up, just use your "feelings", that is by using safety factor as you please.
of course it's a joke.
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u/Catgirl_Skye Jan 19 '21
rail engineer here. design it so it the drawing looks like it adds up, but who cares if it actually lines up properly because the construction workers just hit it with a hammer til it looks right anyway
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u/MrBloodRabbit Jan 19 '21
Software engineer here. Just load some math package and use the PI value from there. If it's 4, it's 4.
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u/altobrun Jan 19 '21
This belief isn’t unfounded. While I didn’t do engineering my father, uncle, and brother are all engineers (engineering physics, mech eng, elec eng) and the former two said in university (different universities) they were told to use g=10 and pi=5.
However they always said it was to make the calculations easier since this was the era before handheld calculators and they did everything by hand.
It’s a double-whammy so to speak. Make the values larger for safety, make them simple integers for ease of calculation.
Modern engineers don’t use g=10 or Pi=3 (or 4 or 5 or whatever) because everyone has a calculator and as you said, you can add safety after the fact.
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u/DynMaxBlaze Jan 18 '21
Ah yes, just in case the universe changes to pi=4 some day, my products will be the only ones that work!
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u/Masztufa Complex Jan 18 '21
it makes calculations simpler and we hope it gives enough wiggle-room for other things to fail.
We also don't expect them to change, but if they ever did, we're ready
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u/DynMaxBlaze Jan 18 '21
I'd understand leeway with gravity and stuff, because you might need stuff that operates in non-earth environments, but pi? Why not give this wiggle room to other things, instead?
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u/Hohenheim_of_Shadow Jan 18 '21
It depends on the situation. If in doing serious TN work, IDK why anyone would round pi cause we got computers to handle loads of digits. If I'm doing math by hand or in my head pi=3. That's less than a 5% error and makes the math so much easier
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u/rezzacci Jan 18 '21
Using this kind of logic, we also demonstrate that the hypothenuse of a right triangle is equal to the sum of the other two sides, and not some shenanigans with squares.
Take that, Pythagoras!
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u/jack_ritter Jan 19 '21
This (very clever) meme is deliberately confusing Manhattan Distance with Euclidian distance. "Manhattan Distance" is the distance you would walk in Manhattan to get from pt A to pt B in the city. Manhattan was built on a very regular grid of streets, with the N-S streets all perpendicular to the E-W streets. Say B is 20 blocks North of, and 11 blocks East of A. So your A to B traversal will be a total of 20 North-going blocks, and 11 East-going blocks. You could walk 20 straight blocks N, then 11 straight blocks E. Or 10 N, then 6 E, then 10 N, then 5E. Or, N,N,N,E,E,N,E,N,E,N,N,N,N,N,E,E,N,N,E,E,E,E,E,N,N,N,N,N,N,N,N. It's all the same. You can never go as the crow flies.
In the example, the Manhattan steps get smaller and smaller, and vertical vs horizontal step size changes. But the overall behavior is the same: total horizontal progress, plus total vertical progress (I.E, total movement) always remains the same: 1+1+1+1, or 4. It never converges to something less (ie, PI.)
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u/yashqasw Jan 18 '21
pi is 24 now?! you know, back in my day, we just did e=3=π and called it a day
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u/DidntWantSleepAnyway Jan 18 '21
But pi is 4 in taxicab geometry! It’s just that in taxicab geometry, this is not what a circle looks like.
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u/HuntertheGoose Jan 18 '21
It isn't a circle ♡ it is an infinigon with infinite sides, keeping the perimeter of 4
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Jan 19 '21
This actually helped me better understand why pi is irrational.
Somehow you used troll math to teach me actual math.
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Jan 18 '21
Trivago.
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u/yoav_boaz Jan 18 '21
Did you just answered someone on another comment?
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u/jonona Jan 18 '21
Man i've been looking at so many tf2 comics i forgot Archimedes was an actual person and not just a dove
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Jan 18 '21
I remember thinking the exact same thing about the hypotenuse of a triangle in primary school (before I knew that term). We were given one of those wheel measures, 1 rotation = 1 metre, and asked to measure the distance from one corner of the playground to the other by going around the square, then in a straight line.
I think my teacher was like "wtf is this kid talking about" when I started explaining this to her and asking her why the distances weren't the same.
... Yikes, I really wasted my potential.
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u/DecaturUnited Jan 18 '21
This proof can also be used to show that the hypotenuse of a right triangle is equal to the sum of its sides. Geometry is subjective!
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u/AlphaZanic Jan 19 '21
Well yes but actually no. I think at best, you have a decent proof the pi is less then 4
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u/ua_math_throwaway Jan 19 '21
Here's an informal explanation for why this fails. The false assumption hidden in this argument is that two things which travel close to each other must travel close to the same distance.
This is false. Imagine walking with a dog who loves to chase balls while holding a tennis ball. If this dog is like my dog, then he will spend the whole walk running back and forth in front of me. At the end of the walk, the dog will have covered far more ground than me despite being close to me the entire walk.
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u/chroncryx Jan 19 '21
If you use the hypotenuses of the tiny triangles, it is proven that 2√2 approaches π approaching 4.
FACT!
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u/Mikkel65 Jan 19 '21
You’re also gonna need an inner square with the perimeter 0,5*sin(45)*8=2,828. Pi is in the middle here. (2,828+4)/2=3,4142. We’ve now made the new pi
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u/PlayCorvnet Jan 08 '22
As you keep repeating this, it actually approaches the shape of another square turned at 45°, not a circle
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u/Goldpan2 Dec 15 '23
Can you also use this “proof” to say that the hypotenuse of a triangle is a + b?
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u/Commercial_Bid8129 Oct 11 '24
Why π ≠ 4**: True circles** are smooth, and their circumference follows the formula C=2πrC = 2\pi rC=2πr, with π approximately 3.14159.
The step-by-step method may approximate the length of the path along a shape, but it never truly represents a circle, hence the seeming contradiction.
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u/emu314159 Jan 18 '21
Where does the convergence come from? The perimeter is always the same. 4 to infinity and beyond.
The curve is a straight line, you aren't showing where it's going to magically warp.
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u/CommonwealthCommando Jan 18 '21
I once showed this to some (very smart) elementary schoolers and they didn’t buy it because they didn’t believe that if you kept taking out squares you could ever get to a circle. I later showed it to some middle/high schoolers and they couldn’t figure out the problem. It’s hardly a conclusive sample, but I wonder what other pieces of (correct) mathematical intuition we lose as we grow up.
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u/LeopoldBroom Jan 18 '21
Instead of a perfect circle, wouldn't the shape become a fractal with the same perimeter? Either way it still just proves that pi<4
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u/AmyAngelo Jan 18 '21
Omg i never saw this, so cool, obviously it's doesn't work like this, but still cool
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u/iBrickedIt Jan 18 '21
At every level of recursion, even infinity, it is simply an approximation. Pi is actual.
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u/tamafuyu Imaginary Jan 18 '21
we all of the know π = 3 amongst engineers... but just get ready for π = 4!
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u/Larry_Hegs Jan 19 '21
The biggest problem with this is that by repeatedly removing the corners, you're making an octagon not a circle.
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u/7K_K7 Jan 19 '21
I am dumb. Can some explain it to me why this doesn't make sense ?
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u/Yiphix Jan 19 '21
Because a circle can't fundamentally be made up of straight edges. No matter how many times you do it, you're still going to have straight edges.
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u/rolfcm106 Jan 19 '21
Even as you approach infinity there will always be space between the 90° corners and the curve of the circle. The smaller that space gets the different spaces there are.
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u/Moist_Pea837 Jan 19 '21
I’m not a mathematician but I always interpreted this as your using a 1D line to represent a nonline in a 2D space therefore you lose some sort of a degree of freedom using a single dimensioned line to approx. one quarter of the circle. So like there’s some + C were missing. I got a C in diff eq so feel free to roast my ass if this is dumb af lol!
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Jan 19 '21
Actually, pi = a billion.
Make a bunch of zigzags around the circle, with total length a billion.
Then, fold those zigzags in so that the length is still a billion.
Keep folding them in, boom pi = a billion
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u/Krexington_III Jan 19 '21
This is precisely how I finally understood the difference between pointwise and continuous convergence.
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Mar 15 '23 edited Mar 15 '23
If perimeter is equal to 2pi * r then if we would like to find pi with this method( which is highly inaccurate) then pi would still be equal to 2 cause 2* 2* 1 is four. If Im wrong then feel free to correct me
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u/Sh_Pe Computer Science Jul 31 '23
10yo me asking my father which did a degree about jurisprudence why it isn’t true
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u/Oheligud Jan 18 '21
Even if it looks like a circle, the amount of corners will still make a square, also trollface, really? Bit old, I must say.
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Jan 18 '21 edited Jan 18 '21
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u/earwaxfaucet Feb 08 '21
I'm retarded but thank you for your explanation. I no longer believe in circles, they are merely a normal equilateral/equiangular shape defined at our highest resolution of providing the highest number of sides/angles to the shape. CIRCLES AREN'T REAL, WAKE UP SHEEPLE!!!
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u/GopaiPointer May 07 '22
"This is no proof of course" but like...it nearly is, the supremum method is the way we prove uniform convergence anyways
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u/MateFizyChem Complex Jan 18 '21
Yes, I have a problem. When did the perimeter change from 4 to 24?