Thread 16827327

Prove that this is wrong.

>>16827327 (OP)
Take a 4 inch string and wrap it around a 1 inch diameter pipe. If there's overlap then pi is less than 4.

>>16827327 (OP)
Slide thread, but it isn't wrong for a pixelated circle. It is wrong for a polygonized one or one constructed using a compass. Perimeter describes the distance required to optimally traverse a shape using a series of infinitesimal steps. In the example given, you're unnecessarily constraining yourself to movements of +dx, -dx, +dy, -dy, while the space of 2d shapes in general allows traversal using line segments that can be oriented at an angle. Physically, the stroke of a compass is perpendicular to the line between the needle and the pencil, so having that angular degree of freedom is necessary to match those of the tools used to construct circles geometrically.

This is a classic paradox you'll learn in post-multi var calculus (or you can find yourself if you're trying to understand lenght and approx. of geometrical shapes), the the function "lenght of a curve" isn't always continuous when extend your iterations to infinite.
The TL;DR is related to the derivative, to not 'break' the lenght function the derivative of your curve should still exists when you extrapolate to inf iterations (that example works with derivable functions like a sinusoidal curve).

This is a classic paradox you'll learn in post-multi var calculus (or you can find yourself if you're trying to understand length and approx. of geometrical shapes), the the function "length of a curve" isn't always continuous when extend your iterations to infinite.
The TL;DR is related to the derivative, to not 'break' the length function the derivative of your curve should still exists when you extrapolate to inf iterations (that example works with derivable functions like a sinusoidal curve).

>>16827330
I shan't be doing that. Fuck off idiot

>>16827330
>>16827364
it works in base 11

>>16827330
The virgin numerical approximation vs the chad physical approximation

>>16827336
>post-multi var calculus
When was that developed?
Are you saying mathematicians prior to its development would have seen the proof in OP or something like it and concluded pi=4 etc? (maybe not for this particular example, because you can physically measure a real-world circle with some string and get a better value, but for some other shape or related concept)
Because the line gets closer and closer to the circle, just like mathematicians do with polygons in circles to find the true value of Pi, or with integration for the area under a curve.
In fact, this looks a lot like the area under a curve, except that it is length rather than area.
Are there examples of simple "proofs" that were widely held, later to be shown false?
And is the "post-multi var calculus" explanation applied "organically", as in it would crop up as a natural explanation for why OP's proof does not work, or is it applicable because we know OP is wrong ex post facto?

>>16827419
Or maybe not conclude that pi=4 (or similar), but conclude that the concept of a proof is dead, because apparently sound reasoning can be used to show pi=4, which it is not.

>>16827327 (OP)
Circles have no corners. You can't make corners go away by making them infinite.

>>16827419
>the line gets closer and closer to the circle, just like mathematicians do with polygons in circles
See >>16827425
The polygon cheat actually makes the corners go away.

>>16827425
I would actually take on corners like this.

- Let's say we on any iteration of an OP proof.
- Now cut any number of corners, without touching the circle.
- Perimeter now is < 4, but the resulting shape closer to the circle. This means the approximation doe not converge to the circle but to a different made-up shape (? need help here)

>>16827450
>the resulting shape closer to the circle. This means the approximation doe not converge to the circle
At most, that would prove that one approximation converges faster than the other. But the simple fact of the matter is that a circle has no corners.

>>16827327 (OP)
Because circles are defined by points, not sides. This shape would have infinite sides and a circle has none, just points from it's radius.

Whoever made it tried to be clever but you would have to redefine what a circle is in order for this to work. It's like... 3rd grade math.

>>16827459
>Because circles are defined by points
But all the points in OP's pic do end up at the same distance from the center.

>>16827459
Reading this thread is making me incredibly sad. People saying this is a paradox and a bunch of other retarded shit don't even know what a circle is. Like, this guy >>16827336 He clearly knows a bunch of math shit but not what a point is.

>>16827462
He's right and you're a brainlet.

>>16827456
>But the simple fact of the matter is that a circle has no corners
But fitting polygons with more and more... corners
Just werks?

>>16827461
literally impossible if there are sides.
>>16827463
Look up the definition of a circle you retard. They aren't defined by sides. There is no paradox. How are you people so fucking stupid that you don't even know what a fucking circle is?

>>16827466
>>16827467
>80 IQ and brown

>>16827468
t. 105 IQ thinks he is 120 IQ

>>16827470
>80 IQ
>brown
>seething
Name a more iconic trio.

>>16827468
>>16827470
Are you guys trolling or are you genuinely retarded? Do you even know how you get pi? Do you know what a point is? Circumference?

I'm surrounded by idiots. This world is over.

>>16827472
>t. tar-black gorilla nigger

>>16827471
>>16827470
>>16827468
>>16827468
>A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the center.
>all points

>>16827474
>all points
Yeah, all the points end up the same distance from the circle at the limit. Imagine being so brown you get filtered by the premise of the meme and never even get to a position from where you can start analyzing what actually went wrong.

>>16827476
ok you are genuinely retarded.

>>16827477
You will never increase your fluid intelligence but you can at least get your mental illness treated.

>>16827479
What's your goal here chief?

>>16827327 (OP)
Easy, (cos(1)/2, sin(1)/2) is a point on that circle but no corner can ever map to it because they always map to algebraic coordinates. Therefore your map is bullshit.

>>16827480
To convince you to take your anti-psychotic meds so that you snap out of your delusional episode making you think you belong on a "Science & Math" board.

>>16827483
I honestly just feel bad for you people. You have no idea the hurt that is coming your way. You've been given every opportunity to turn things around but you just choose not to.

I just don't understand how you got this way. What happened in your life to make you like this? Like, you're clearly not ok but why do you let greed control your life? What makes you want to hurt people? Do you get sexual gratification from this or what? Did they promise you a mail ordered bride? Does loneliness cause people to do things this desperate?

I really need to understand people like you. Just... there is no humanity in you. There is no intelligence. Your eyes are hollow and empty. There's nothing going on in there. You just think of pussy and money and that's it.

Also before you tards go off about the mental illness thing... who do you think is reading this? We all know you tards are retarded here. No one in my "real life" is going to be reading anything I post online. There is no goal here, you're not achieving anything but this is how you choose to spend your time. There are no consequences here but you still put in the effort.

You are following the orders of some incredibly stupid people. It's all just really sad.

>>16827327 (OP)
when you have a lim f(x) while x is approaching something, the f(x) has to change too
all that pic "proves" is that p =< 4

>>16827487
I'm not reading this. Take your actual, prescribed medications.

>>16827462
It's not that complex, some curves will not approximate others doesn't matter how many iterations you make to get close to "infinite", if you try to extrapolate the length of your approximations to infinite (the exact geometric curve) then the result will be different, fractals have the same "problem", you would be wasting time if you expect a good behavior by using segments to measure them.

Even if "visually" (for your eyes) the approximation looks similar to a circle it will never be a circle, even if it tends to a circle. You just have to compare the tangents of the approximation vs the circle and it'll be obvious there's a problem, the tangents of your approximation never tends to something circle-like (very smooth).

>>16827523
>Even if "visually" (for your eyes) the approximation looks similar to a circle it will never be a circle
>t. 80 IQ and brown
The limit of what OP depicts IS the circle.

>>16827526
NTA but (cos(1)/2, sin(1)/2) is a point on that circle and is found nowhere in the limit of what OP depicts.

>>16827540
>(cos(1)/2, sin(1)/2) is a point on that circle and is found nowhere in the limit of what OP depicts.
Prove it.

>>16827526
>muh IQ
That pic shows an approximation of a curve with another one, approximation that fails to have a continuous length function because of the way it is built, the example is right before the last 2 rows, the last one is wrong because it assumes the length function will be continuous even if you use a curve that breaks in the "step" to infinite.

>>16827547
That approximation is the circle at the limit. What doesn't follow is that the length you associate with that approximation will approach the perimeter of a circle.

>>16827544
>>16827482

>>16827584
Do the points that touch the circle always map to algebraic coordinates?

>>16827425
isn't that the whole idea of limits and integrals?

>>16827555
Basically that.

>>16827588
Not sure what you mean.

>>16827330
Me with my 3 inch string. );

>>16827327 (OP)
you are implicitly assuming that your error approaches zero, this is false.
What I think is happening is as the length of your mistake approaches zero, the number of times you make that mistake approaches infinite, and it cancels out.

>>16827334
>Perimeter describes the distance required to optimally traverse a shape using a series of infinitesimal steps.
That is not how teacher defined perimeter in school.

>>16827613
>Not sure what you mean.
Look at OP's pic. The approximation touches the circle at some points. Then think about my question.

>>16827627
That's literally what >>16827482 says. Your question seems to be a precursor, not a follow up, which is why it doesn't make sense. Can you clarify what you mean or are you just confused somehow?

>>16827327 (OP)
Take a square of perimeter 4. The top left corner. The perimeter is still 4. Remove additional corners. Perimeter is still four. In this limit you end up with something that looks like a triangle, but with a perimeter of 4 instead of the expected perimeter of a right triangle with sides unity of 2 + sqrt(2), and it's trivial to see why - drawing a step function between two points and drawing a straight line between two points is not equivalent.

>>16827644
Can one of the points that touch the circle be (cos(1)/2, sin(1)/2)?

>>16827655
No, that's literally what >>16827482 (You) says. Literally. You're replying to a post by asking a question that the post is literally an answer to. Like wtf lmao.

>>16827659
My bad. I was too charitable in my interpretation of your post. Your statement is simply wrong.

>>16827677
More like you're too charitable in your assessment of your own intelligence.

>>16827685
>Science & Math
>states his "math" without proof
Brainlet.

Actually pi=3sqrt(3)

>>16827701
>Doesn't realize his post is largely self-referential.
Sub-brainlet.

>>16827729
I didn't make any mathematical statements without proof ITT. Looks like you're hallucinating.

>>16827745
>Your statement is simply wrong.
This is well below retarded. The only form of reply you deserve is insult and mockery. So is >Looks like you're hallucinating.
You're just a total clown with no redeeming qualities at all lol.

>>16827749
Whom are you quoting, brainlet?

>>16827751
Total clown show you are
>>16827677
>Your statement is simply wrong.
>>16827745
>Looks like you're hallucinating.

>>16827752
Did you forget your meds today?

>>16827761
>Self-inserts in anonymous dialogue as anon
>"Did you forget your meds" subhuman retard quip
Total fucking clown lol.

>>16827765
I hinted at your error multiple times but you just kept doubling down due to your mental illness.

>>16827768
This isn't a dating app, you irrepressible faggot. I don't give a fuck about whatever flirty "hints" (lmao) you think you're dropping

>>16827770
See >>16827701

>>16827771
Not typesetting your homework for you. If you can't figure out what points the first three iterations of corners map to and induce your own proof, that's all on you.

>>16827772
See >>16827701

>>16827773
>>16827773

>>16827775
>retard seethes so hard it starts spazzing out

>pi is an algebraic number, i know this because i did 3 iterations of [insert pi approximation]
the absolute state of /soi/

>>16827777
>>16827777

>>16827779
Non sequitur using "Indian reply syntax" I believe lol.

>>16827779
It's 80 IQ browns from /pol/.

>>16827796
>New brahmabot update installed

>>16827801
Anally abscessed.

>>16827802
Sorry to hear it

>>16827487
>CTRL-C + CTRL + V in google
>it's not a copypasta
holy shit did I just witness the birth of a new pasta?

>>16827482
Any interval, no matter how small, contains infinitely many non-algebraic numbers. Pick a point on the circle where the tangent is 45 degrees. Zoom in on its vicinity until it's arbitrarily close to a 45 degree straight line. The jaggies are recursively bisecting it. Focus on one coordinate. The jaggies are recursively bisecting that, too, covering every real number on that interval.

>>16827838
Works for a line segment, fails the diagonal argument for coordinate tuples: | {x, y}^N | < | N |

>>16827840
*Flip the inequality, you get it.

>>16827840
Do it on the diagonal where x=y. "It's a line segment" obviously doesn't help your case. Bisecting the line segment bisects the intervals for both coordinates.

>>16827843
You're making a Cantor crank 101 mistake. You can't cover an interval with a countably many steps.

>>16827468
>80 IQ and brown
Yes, let's explain your concept as we all are 80 IQ, brown and 7 year old.

Well, circle has no corners by definition, all points are on the same distance from the center. To make a corner you have to put at least one point further than another.

>>16827847
Every step doubles the number of points.

>>16827854
Yeah, that's what makes it uncountable. It's literally one of the most common Cantor crank arguments.

>>16827857
So what's the problem? It's a countable infinity of recursive bisections producing uncountably many points to cover the interval.

>>16827858
Equivalent to saying we can use countably many nodes in a binary tree to count R.

>>16827862
How many nodes in a full binary tree with infinite levels? :^)

>>16827864
How many bisections to cover the circle?

>>16827868
I dunno but unless you can name a specific flaw in my argument (besides >Cantor bad) your claim is refuted:

>>16827482
>they always map to algebraic coordinates
The recursive bisection will inevitably cover non-algebraic coordinates.

>>16827869
>Cantor bad
Your argument fails due to Cantor's argument. Not because Cantor's argument is "bad" lol what the fuck how did you even come up with that?
>The recursive bisection will inevitably cover non-algebraic coordinates.
Nope, see >>16827862

>>16827327 (OP)
The perimeter would be spiked instead of smooth which isn't a circle

>>16827880
So what's the cardinality of the set of points you get from infinite recursive bisection?

>>16827887
Power set of N. What did you think it was?

>>16827891
https://en.wikipedia.org/wiki/Power_set
>In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers

>>16827896
Doesn't mean it covers the circle lol. Simple counterexample.
>https://en.wikipedia.org/wiki/Cantor_set
It's your lucky day, you learned something on 4chan. Rare event.

>>16827899
I'm getting pretty bored of your incoherent thrashing. Should've heeded your interactions with other posters and realized you're a retard.

>>16827904
>I'm getting bored of being wrong
Bad habit, leads to failure.

>angles
>still beleiving in determinism
seriously?

>needs help with religion
Now isn't that who we really are? As scientists y'know... dimming the sideofyeh phasallity has some pixious verbat set siyndrom in sameness is only inevitable as psychologies


Anyone up fore developing the beanstalk with the castle at the top?

... fuqn miracle gro everywhere....
fqqqnnnnnn 3G testicalliformia
Glue bell warming... ahsighaights~- <@#>

>>16827910
You made a series of false claims, then settled on a "counter-example" to a claim I didn't make. You can get arbitrarily close to any number on an interval using repeated bisections. Since we're talking about every possible infinite sequence of bisections, that indeed covers every point on the interval. Not wasting any more time on obvious retards.

>>16827913
Nope. Good luck in your fantasy world I guess.

>Retard having a meltdown because his meme flopped
Just ban all non-metric countries.

>>16827623
I'm surprised this one got as much attention as it did. If your shape is composed of line segments you can just add up their lengths, which is the high school definition of perimeter. You do need to modify the definition a bit when you don't have line segments.

You can make an analog to this slide thread bait question that isn't complicated by curves by rotating the square in the OP 45 degrees and doing a similar construction by inscribing it inside of a larger square of side length sqrt(2). The perimeter of a square with side length 1 obviously isn't 4*sqrt(2) by the typical definition of perimeter, so the construction is problematic in the exact same manner.

>>16827838
based
>>16827840
cringe

>>16827950
jeet

>Linearity
>1 Dimensional
How many positions? Each step of towards an equation of dimension itself... the serious not only does it exponentiates it makes fractions of beyond inside itself.... 'nows' like.. bullshit. call that shit elementary... fucking hyper suspension

...Go love that shit you jeez...

#binaryheartsmithszzz's

>>16827327 (OP)
>Prove that this is wrong
Just drill a hole that way.

>Prove that this is wrong.
It's not. You just need to do it with a regular polygon with side count approaching infinity. The answer shrinks with the count increasing.

>>16827327 (OP)
>Prove that this is wrong.
This construction is only a circle at the limit, but then the side lengths are 0, so it gives you no actual way to calculate the perimeter.

>>16827482
Good post, simplest disproof itt.
OP construction will never be able to construct a non-algebraic coordinate.

>>16829370
Extremely obvious samefagging.

>>16829731
>Extremely obvious seething
Nope, >>16829370 isn't me. You're just mad you still don't understand why 2^N points almost never covers an interval lol.

>>16829735
>same misunderstanding about how to use a coma
>same insertion of newlines where they don't belong
>same use of made up terms
>same nonsensical "argument" repeated
>updooting a post that literally no one cares about
Learn to samefag correctly.

>>16829742
comma*

>>16829742
You seem to be wrong about everything. Sucks to be you I guess

>>16829743
No, coma works better for you lmao. Dumb mf.

>>16829745
Learn to samefag correctly.

>>16829747
>01:38
>02:40
Kek. Learn to samefag correctly. Or better yet, just go back to the upvote arrows website.

>>16829748
>If I say it enough I'll come out of my coma
Doubt

>>16829750
Those are both me you absolute ass. Fucking lol.

>>16829731
Not samefagging lol, you're bitter

Me
>>16829752
Not me
>>16829757

Faggot
>>16829750

>>16829752
>Those are both me
Clearly. Also see >>16829742. Closing this retard-tier thread now, but your mental illness will force you to shit out another post that no one will ever read.

>>16829759
>Punch bug no return
Indian debate syntax.

>>16827391
its base 8 baby in base 11 you cant justify any fractions at all and its not 12 because 12 is already more than 10 and theres no reason anyway 5 is upside down 2 because 10/2 is 5 and 10/5 is 2 which indicates that god would like you to remove numbers 3 and 7 or add numbers before and after 5 and or remove 5 when you do this

Guys, these sorts of arguements never accomplish anything, what we clearly need to do is build an arena.
Then we gather up all they people who think pi = 4 into one team, and all the people who think pi is approximated by 3.141... into another team.
We equip them with weapons like swords and spears and make the two teams fight against each other to the death. Which ever team wins is clearly right.
Then we do the same again but for 0.999... ≠ 1 and 0.999... = 1, then again for flat earthers and globers. Then for moon hoaxers and moon landers, atheists and believers, and so on for everything until there is no one left alive who wants to argue about anything and so we can all live peacefully jerking off to AI porn.

>>16827327 (OP)
you can clearly see the error in step 4 which makes step 5 ridiculous.

>>16827327 (OP)
Induce a geometry by changing the definition of the norm. Pi can be anything, except variable.

>>16830059
I disagree.

>>16830059
>we wuz decide with stick
I will repeat myself: it's time to ban all american posters on /sci/

>>16830590
>"arguements"
>american
Your blind rage and jealousy has induced you to seethe eurocringe. Many such cases.

>>16830606
>this lack of self-awareness
Notable.

>>16830623
>takes notes about euros being cringe and Americans being masters of the universe
Good start.

C = circumference = π*d = π*1 = π
P = perimeter = 4*s = 4*1 = 4

C <= P
π <= 4

>>16830705
>C <= P
the circumference of a circle is its perimeter, so:
C = P, and
π = 4.

>>16830076
I thought I was going insane scrolling through 100+ posts without anyone pointing this out.

>repeat to infinity
this is where you fucked up. you can't take the limit to infinity here because it never converges. the whole process is equivalent to just zooming into a fractal with endless self similarity where a corner is made up of more corners. it can't resolve to anything and has an indeterminate form.

>>16831307
Is it easy to shot that it converges and the limit is indeed the circle, but that isn't the problem, the lenght of curves isn't something as intuitive as it seems.

>>16831307
Is it easy to show that it converges and the limit is indeed the circle (ie. using the epsilon-delta definition), but that isn't the problem, the lenght of curves isn't something as intuitive as it seems.

>>16831382
Nope.

>Manhattan distance between two points does not depend on the number of turns.
Great find op.

>>16827330
Someone like Parmenides would just say that proves that the string is an illusion.

>>16830076
There's no error there though. The error is in assuming that the length of the limiting curve has to be the limit of the lengths of the pre-curves.

>>16831463
Not an argument, let me guess, you don't even know the epsilon-delta definition for limit.
It's trivial to show using an epsilon-delta as base for a construction.

the correct answer is that you can only approximate a curve by a finite number of straight segments if each of the endpoints of the segments are on the curve. in fact, that is part of the definition of arc (curved) length (the supremum of the sum of the length of straight segments between the points of a partition of the curve). if any of the points is not on the curve, then the sum of the segment between and after it might be greater than the length of the curve itself (if all points are on the curve, since the shortest path between two points is a straight segment, the length of the curve will always be either greater than or equal to the approximation by finite segments, thus the supremum in the definition). approximating the length of a curve by segments with points not on said curve renders nonsensical results like pi = 4 and a^2 + b^2 = a + b. sorry for bad english, not my first language.

the correct answer is that you can only approximate a curve by a finite number of straight segments if each of the endpoints of the segments are on the curve. in fact, that is part of the definition of arc (curved) length (the supremum of the sum of the length of straight segments between the points of a partition of the curve). if any of the points is not on the curve, then the sum of the segment between and after it might be greater than the length of the curve itself (if all points are on the curve, since the shortest path between two points is a straight segment, the length of the curve will always be either greater than or equal to the approximation by finite segments, thus the supremum in the definition). approximating the length of a curve by segments with points not on said curve renders nonsensical results like pi = 4 and sqrt(a^2 + b^2) = a + b. sorry for bad english, not my first language.

>>16831904
If the derivative of the approximation curve tends to the tangent of the circle (or other curve) then the length should converge to the circumference, your construction forces that situation. If you try to approximate a curve with another curve of fixed length and longer than the 'target' curve and the derivative will get "pathological".
ie. replacing the circle with a simple, you could replace the straight lines with a sinusoidal of fixed lenght and variable amplitude×frequency. The end result will be the same, the approximation fails when n inf and the derivative function actually isn't defined for the "limit of the approximation" that in principle is the same segment points-wise.

>>16831918
srry for having deleted and reposted.

>If the derivative of the approximation curve tends to the tangent of the circle (or other curve) then the length should converge to the circumference
i dont see why you say that. the approximation curve isnt even differentiable on all of its points, and even in the op's pic the derivative of the curve (if you mean the tangent vector) certainly does not have the same direction as the tangent to the circle on all points. theres no need of a derivative in the definition.

>your construction
its the definition of arc length youll find in any differential geometry book. surely you can define whatever you want however you want, but its the definition people usually mean when they say "curve length" (even if theyre not aware of the formal definition).

alternatively, you could define it as the integral of the euclidean norm of the tangent vector, but that is equivalent to the definition i presented (you can prove it using the mean value inequality for vector functions)

i see no point of using derivatives in the problem.

>>16831382
>Is it easy to show that it converges and the limit is indeed the circle (ie. using the epsilon-delta definition)
no because it's straight up impossible. again, your so called perimeter is not definable using limits. you aren't converging to a smooth circular shape it's just undefinable numerous corners. that doesn't just "become" a circle because you said so with words. it doesn't at all. it's simply undefined at infinity. that's the nature of indeterminate forms.

now if you respond to me with "it's obvious that it works you're just an imbecile" i'll know that you were simply trolling with absolute clarity. feel free to suggest otherwise.

90% of this shithole is bait

>>16831577
What do you get out of posting here? I mean it's not even a thoughtful satire or a funny schizo post. You simply have no idea what you're talking about. Yet you voluntarily came to a math and science board to express your ignorance in the most boring way imaginable. And you presumably completed a captcha to do so. Why?

>>16832041
>Can't answer but must attack
A lot of characters to avoid saying "I don't know ε-δ". I you think that will induce some emotion you're wasting time.

>>16832053
I was actually curious. But not surprised that you completed yet another captcha just to post a very boring nonanswer.

if that is true then this is true

>>16827327 (OP)
if that is true then this is true

All the retards claiming the curves don't converge to a circle need to immediately stop posting on this board and not come back until they've read a book on calculus. It is easy to see (though difficult to actually prove because this process is difficult to formalize) that as a sequence of parametrized curves R->R^2, the curves described here converge (even uniformly) to one whose image is a circle. The issue here is that arc-length is defined based on derivatives, but uniform convergence of a sequence of functions is not enough to allow you to interchange a limit with a derivative operator. A sufficient condition to make that swap is uniform convergence of the sequence of derivatives plus convergence at at least one point of the sequence of functions. Unsurprisingly, that isn't even working because the sequence of derivative here does not even converge pointwise a.e..

By contrast, the classical Greek method using regular polygons works because the sequence of derivatives converges uniformly. Well, the corners are a little bit of a wrinkle, but it's converging in L^\infty which is enough for our purposes.

>>16832320
yeah you double the surface area with those corners pi is actually 2*(4/2)

>>16832801
Idk if youre trolling but if you arent youre really overthinking.

The (differential geometry) definition of arc length uses points on the curve. The pic includes points not on the curve. Thats why its wrong (The set of "approximated" lengths using curves with points not on the curve has no supremum, thus if you wanted you could say The circle has a perimeter of any real number greater than pi; if you wanted, you could say this circle has a perimetrr of 999999999999999. Which is absurd).

>>16832816
Are you serious? The set of approximated lengths does have an upper bound, it's 4. Limit and arc-length do not commute due to the previously mentioned issue with derivatives.

To demonstrate, imagine that the curves were parametrized f_n(t) = (x_n(t), y_n(t)) for t in [0, 2pi), such that f_n(0) = (1, 0), and f_n'(t) -> (-sin(t), cos(t)) uniformly. Denote by L_n the arc-length of f_n, then we have

lim_{n->inf} L_n = int_0^2pi sqrt((lim_{n->inf} x_n'(y))^2 + sqrt(lim_{n->inf} y_n'(t))^2) dt = int_0^2pi sqrt((-sin(t))^2 + (cos(t))^2)dt = 2pi

i.e., it works perfectly. The only step that fails in OP's version is commuting the limit into x_n' and y_n'.

>>16827327 (OP)
Just because the sequence is {4,4,4,4,4,4,4.....} does not imply the limit is 4

>>16834595
...yes it does?

>>16834604
Why would it?

>>16834614
the "distance" between any member of the sequence and 4 is 0, which is smaller than any positive real number

>>16827327 (OP)
>Wrap a fractal around the circle
>Pi is now infinity
Checkmate

You can remove aleph-gorillion corners and the perimeter will always be greater than the actual circumference

OP is a G I G A F A G G O T

>>16831072
Yeah, but I also wrote: circumference = π*d
And: perimeter = 4*s
Which clearly show, that I was referring to the circumference of the circle and the perimeter of the square.
(The perimeter of every other shape, or geometrical entity, is also 4*s.)

What's the point of TT anyway?
To waste the time of fools?

>>16827327 (OP)
What does it mean for a sequence of shapes to approach a shape? Pic related is how the ancients approximated pi. Someone tell me the difference between that and OP. I've read every explanation in this thread so far, and they're all lacking. Except for >>16827330

>>16834629
Let 1/Z be the set of fractions of the form 1/n, where n is a natural number. Let f(x) be a function that is equal to 1 if x is positive, and equal to 0 if x is zero. the limit of {1, 1/2, 1/3, 1/4...} is zero.
but {f(1), f(1/2) f(1/3)...} = {1,1,1,1...}, while
f(lim {1,1/2,1/3...}) is 0

>>16835335
>hold this Z for me nigga

>>16835288
It depends on what are you approximating.
If your approximation only considers: (A)
Area(Curve - Approximation)
Distance(Curve - Approximation)
It will work with a lot of 'bad' curves, you can create a Series of curves that approximates the circle with each new step, if your approximation also considers: (B)
Tangent(Curve - Approximation)
Length(Curve - Approximation)
the OP example will fail but a classic approximation using polygons with edges restricted to the circle will work.

Both (A) and (B) have the same curve as limit, the circle but (B) is stricter than (A).

>>16827849
>circle has no corners by definition
Don't really do much maths but doesn't this line of reasoning also dispute the foundations of calculus? For the curve has no points (excepts minima and maxima) yet the gradient is reached by treating them like they are infinitesimal points- is not that exactly what OP's pic does?

I beg can someone please coherently explain why the pic is wrong

>>16835444
Because π = 3.14... uses √(x^2 + y^2) for distance not x + y. Diagonals not corners.

>>16835624
The problem isn't related to the corners/diagonals but the fact that you can't 'remap' and maintain length if a curve has a restricted length that is different than the circle.
Picrel, you could ignore the 'loops' and imagine a loop created with a sinusoidal wave contained in the yellow area and of constant length but of variable frequency and amplitude

>>16835288
the derivatives of this one converge in L^\infty to the derivatives of a parametrization of a circle so the arc-length limit argument is valid.

Note that if you instead look at the sequence of areas and use A = \pi r^2 to calculate pi, both approaches work because regularity is not required

>>16835659
The problem and solution are exactly what I explained. The limiting sequence in L^2 goes 2√2, 4√(2 - √2), ... π. The limiting sequence in L^1 goes 4, 4, ... 4.

>>16835670
>L^1
My example is unrelated to the taxicab distance, you can use an arbitrary curve to approximate a shorter curve, forcing a polygon as only alternative to the OP's example is wrong because it isn't the only alternative.

>>16835659
depending on how you precisely set this up, it's either converging to something that isn't injective (in which case the usual arc-length formula does not apply to the limit curve) or derivative values are getting arbitrarily large which prevents you from saying that the limit of the arc-lengths equals the arc-length of the limit

>>16835666
Why should I use derivative nonsense when it comes to measuring length? I should be able to rely on a theory of length and area to explain why one sequence of shapes gives the right answer, and the other doesn't.

>>16835685
>or derivative values are getting arbitrarily large which prevents you from saying that the limit of the arc-lengths equals the arc-length of the limit
That's the idea (a derivable curve without corners that fails at the limit itself), the approximation will "behave badly" in any (or both) of the ways you mentioned.

>>16835687
Because arc-length of a curve is defined based on its derivative. Pass calculus 1 and this will not be a mystery. Technically you are still relying on a theory of length and area here since you are integrating the l2 norm of the derivative over the whole length of the curve.

>>16835681
I don't care about your example. When you connect the points on the circle, you get L^2. When you go out to the corners, you get L^1. Nothing else matters to "explain why the pic is wrong" (>>16835444).

>>16835695
>Arc length of a curve is defined by its derivative
Do better

>>16835713
Boo hoo faggot, if you disagree about how length of a parametrized curve works then I suggest you figure out for yourself why pi isn't equal to 4. You can report back here when you're done. Or maybe with your definition of length it is 4?

Pi = the total amount of base Plank level building block magnetized RGB cubes that ride upon the grid lattice track.
A large, yet rational #

YOU
HAVE
BEEN
MISLED

space and time are not real. we are all one consciousness.

>>16835735
>>16835740
>>16835742
>>16835746
>>16835748
Counterpoint: you are contained in a diamond cat.

>>16827327 (OP)
We observe that pi is not four, so the perimeter function cannot be continuous under any topology on curves in which this sequence converges to a circle. In other words, the assumption that the perimeter of the limit is necessarily the same as the limit of the perimeters must be wrong, simple as that.

A
C
T
I
V
A
T
E

I
T

>>16827327 (OP)
Pressure will buckle achirally brainlet

what if you found out that your entire life you've actually just been talking to yourself and that there is nobody else?

well guess what? that is literally what is happening.

>>16835288
in order for a sequence to converge to a value, the difference between the sequence and the value has to get arbitrarily small (i.e. no matter how small a number you pick, eventually the difference will become smaller than that number).
technically your pic isnt approximating pi, its approximating the area of the circle, which can then be used to approximate pi. notice how each time a side is added to the (outer) polygon, the new area is slightly smaller than the previous polygon's, but still strictly larger than the circle's. if you can show that the difference in these areas get's arbitrarily small, then you can say that the polygon's area converges to the area of the circle. the ancients used two polygons because they wanted both an upper and lower bound.
OP doesnt work because the difference between the square's perimeter and the circle's perimeter is constant, unchanging. if you agree that the circle and square have different perimeters in the 2nd panel, then you must agree that they still have different perimeters in the 5th panel. neither of the perimeters changed, so how could they suddenly be equal?

>>16827327 (OP)
It's not a circle, it's a fractal.
Only midpoints ever touch the inscribed circle.

>>16835779
That's genius.

>>16831496
Parmenides was a retard

>>16836413
>a retard
I know you are, but what am I?

n < c retards