/sci/ - Science & Math

The arclength functional is not continuous.

>>7052497
in the 'repeat to infinity', i.e. after a countable number of iterations, whatever you have, you don't have a unit circle.

>>7052527
>after a countable number of iterations, whatever you have, you don't have a unit circle.

Why?

>>7052497
looks like computer science, which is not math

>>7052501
Yes it is. There are no discontinuities. It might not be smooth but it is continuous

>>7052549
Do you know what countable means?

>>7052497
That proof fails immediately after the second removal, after that you must remove asymmetrical sections in order for the corners to touch the circle because the corner snippets towards the top and side axi of the circle are smaller than the ones near the diagonal axi.

If someone wants to prove that for me you can, that's just my first thought on why this is wrong and I can't be bothered to check my work. It doesn't really matter in the end because we all know the original proof is wrong anyway.

>>7052587
Obviously OPs "proof" is wrong, but not at all for having asymmetrical sections removed. What the hell are you even trying to say?

The perimeter length doesn't change, to get a useful limit it has to shrink with n, where n describes how broken the perimeter line is.

It's wrong, because you can't do an infinite amount of work.

>>7052586

Yes but I don't understand why it's relevant here.

>>7052497
you can't have infinity iterations, and at no point during your approach to infinity are you actually measuring a circle's perimeter

>>7052587
having the removal being a square is not a requirement for the perimeter to remain the same

>>7052497
It's because... you know what? I'll let Vi Hart explain it to you: https://www.youtube.com/watch?v=D2xYjiL8yyE

It's wrong, because you don't get a circle in the limit, but an object that has an infinite amount of infinitesimal edges, while a circle is a curve defined by an algebraic expression, which is different.

>>7052497

Infinity is just an abstract concept, it doesn't actually exist. That's why the math breaks down "at infinity".

There is nothing wrong with the picture. It estimates pi from above as 4, which is accurate.

>>7052623
What a dumb thing to say
If that was true we could never talk about limits

>>7052649
> What is calc I

>>7052592
Nevermind what I said, I thought of a simpler counter-proof.

For approximating arclength you have to use a pythogorean theorem-based formula instead of this one, for the reason of that this one is fucking different. The pic produces what is called the taxicab metric for measuring lengths in the cartesian plane instead of the the euclidean geometry one.

If you disagree with the OP, then you also disagree with Archimedes method of exhaustion for computing pie. They work the same way essentially: You approximate a circle with N -> infinity straight lines

>>7052625
because there are an uncountable number of points on the unit circle

>>7052662

So why don't you just do an uncountable number of iterations, huh?

>>7052658
But that has nothing to do with OP's pic.

>>7052667
>iterations
>uncountable
Choose one

>>7052661
Except Archimedes proof didn't end with <span class="math"> \pi=4[/spoiler]

Also, YOU are missing the key difference bewteen this and Archimedes proof. With each interation of OPs thing the angle between each adjacent line segment is <span class="math"> \frac{\pi}{2} [/spoiler] while the angle between adjacent line segments using Archimedes method increases to to <span class="math"> \pi [/spoiler]

>>7052667
Again, you must not understand what uncountable means. You cannot do uncountably many iterations.

Holy shit the line is equal to the square.

>>7052497
Pretty easy, the new form does not approximate the length of the circle, but its area. Approximation means that it goes towards a value, here the value stays all the same (=4). Why? Because the process is wrong. If you actually wanted to approximate the length, then you have to use Pythagoras in order to calculate the lengths of the edges of the points and then sum them Up, which will lead to Pi

>>7052653
The only reason you think it's a dumb thing to say is because you don't know anything about mathematics, and are probably still in high school.

>>7052673
>>7052679
You can't do infinitely many, either.

>>7052655

I'm sorry but I don't believe in the ghosts of departed quantities.

>>7052698
I have a degree in mathematics. You are talking about an infinite amount of 'work' what the fuck does that even mean? Do you know how space filling curves work? This is basically the same thing except it is a curve winding onto another curve. As has been said, the shape OP describes approximates the area of a circle without approximating the circumference.

>>7052703
You can approach the limit cardinal aleph null, you cannot approach the limit cardinal aleph 1.

>>7052497
you'll never get an actual curve by doing that because all you're doing is taking straight lines and moving them around

also the shape around the circle will always have more area than the circle

>>7052709
>You are talking about an infinite amount of 'work' what the fuck does that even mean?
You can't say anything about what you get "after" an infinite process, as it is a process that has no end.

>>7052703
It's not the same thing. Even if we allow infinite iterations (and I'm perfectly fine with that) you will have a first iteration, as second iteration and so on. For every iteration you will have a NEXT iteration. This is basically the definition of countable.

>>7052716
So are you saying that <span class="math"> \lim_{x \to \infty} \frac{1}{x} != 0[/spoiler]? This is just an example
Maybe that isn't what you're saying

>>7052724
That expression doesn't mean anything as the limit does not have a rigorous definition

>>7052716
Maybe I should be careful what how I say things. I'm talking about what the shape tends towards, like how we can talk about limits. A couple examples, <span class="math"> \lim_{x \to \infty} \frac{1}{x} = 0[/spoiler]. The fraction never actually equals zero, but as x gets large without bound, the fraction approaches zero, bounded below only by zero.

>>7052637
Underrated post

>>7052728
I fixed my tex, what you are trying to reply to is here >>7052730
Are you saying that the concept of a limit doesn't have a rigorous definition or that for me to talk about a limit you want me to recite the epsilon delta definition? If it's the latter, I won't because you have fucking google

The circumference of the circle is 4? Why is this sort of reasoning wrong? All throughout mathematics we use the concept of limits to approximate with, and find convergence. If this sort of reasoning is demonstrably false then what does that say for the rest of mathematics?

>>7052737
It says that your understanding of what is happening in the OP is fallacious

>>7052497
because you can never get a shape equivalent to a circle by doing that, no matter how many times you repeat the process, if you zoom in on the image enough you'd still see pic related

>>7052742
how does a definition for a derivative work?

>>7052497
take a ball that's a foot across. now measure the circumference and tell us if it's 4 feet or not.

>>7052746
Not like that you fucking dolt.

>>7052746
There is no convergence here. Well I guess the area bound by the curve is converging to pi, but the perimeter is not.

>>7052750
that's the point though. It "converges" to 4.

>>7052754
Trivially (in that it starts at 4 and stays at 4). But it does not converge to the curve that is described by the circle.

>>7052754
No. It doesn't. Look up convergence.
From the beginning of the process the polygon's length is always 4

>>7052754
The reason it doesn't converge is there are always a slew or right angles above the circle. No matter how many times you iterate the process those angles will be present, which disallows those points to converge onto the circle.

>>7052755
because the shape isn't a curve, and never will be, only the area will approach the same value as the circle's

>>7052755
>>7052757
>>7052761

It gets better and better at describing the curve. The area difference is smaller and smaller. How is that not convergence?

>>7052763
The enclosed surface converges to a disk, but the curve itself doesn't converge to a circle.

>>7052497
Answer is easy, this is not how you find arclength. The definition of arclength is:
<div class="math">L(t)=\int{a}^t\frac{\partial P}{\partial t}dt</div>
Where P(t) is a parameterisation of the curve, L(t) is the arclength at a from the starting point P(a) until the point P(t).
And your troll picture has nothing to do with this.

It's like asking why a banana isn't a watermelon.

>>7052763
The interior angles will always be 90 and 270 when they should be converging to 180. Converging to the area has nothing to do with converging to the curve.

>>7052497
Answer is easy, this is not how you find arclength. The definition of arclength is:
<div class="math">L(t)=\int_a^t\frac{\partial P(t)}{\partial t}dt</div>
Where P(t) is a parameterisation of the curve, L(t) is the arclength at a from the starting point P(a) until the point P(t).
And your troll picture has nothing to do with this.

It's like asking why a banana isn't a watermelon.

>>7052763
The area under all the curves summed is the same from beginning to end

>>7052497
Answer is easy, this is not how you find arclength. The definition of arclength is:
<div class="math">L(t)=\int_a^t\frac{\partial P(t)}{\partial t}dt</div>
Where P(t) is a parameterisation of the curve, L(t) is the arclength from the starting point P(a) until the point P(t).
And your troll picture has nothing to do with this.

It's like asking why a banana isn't a watermelon.

>>7052779
How so? It is clearly getting smaller.

>>7052780
Arclength of a parametrised curve <span class="math">\Gamma(t)[/spoiler] with <span class="math">t \in [t_i,t_f][/spoiler] is <div class="math">L = \int_{t_i}^{t_f} \left| \frac{\partial \Gamma(t)}{\partial t} \right| \mathrm{d}t</div> the norm is very important.

This is why mankind really isn't that smart. We are baffled by things that we created ourselves (i.e: math, numbers, scientific theories) which doesn't get us anywhere but maybe money from our "jobs". Animals have no idea what we think about, yet they lead pretty great lives.

And that's when I'm going to drop in the G-bomb: God. He answers the most pondered question of mankind: the origin of the universe, which is answered through the very first statement in the bible, "In the beginning, God created the heavens and the earth". So, He says that forever means forever. It means never ending. It never ends. Ever.

>>7052799
Aman

>>7052799
science helps a lot in terms ofincreased life expectancy, infant mortality an food production.
And it has so for a very, very long time

>>7052799

>This is why mankind really isn't that smart.

No, you are baffled by things you do not understand. You can physical represent pie within a circle by its relationship between the diameter and the circumference. Get a piece of string equal to the diameter then start measuring it around a circles circumference while counting the number of times you use the string. You will be able to do this three times and a bit (.1415) depending on your measurement of the string as well as your accuracy. Therefore, a circle has been defined and anything varying from this definition is not a circle within Euclidean geometry.

>May be you should expand your reading list, particularly the section on history with specific academic domains?

>>7052842

A circle has neither a beginning nor an end. Should pay more attention to the way you formulate things when arguing.

>>7052849

So, if we assume a given point (x,y), which is also the end point, does this make a difference to my argument?

>>7052856

Sorry I meant to say:

Any given (x, y) coordinate on the circles circumference.

>>7052842

How does pi work when applied to real world objects? Is close enough simply close enough? I pick a few decimal places down the line, round up, and call it good? If I fill a cup with a liter of cola, I have a liter of cola. If I use the maths formula for volume, the description of my beverage goes on for infinity. How accurate is a number that never ends?

>>7052896
After something like 20 decimal places there is no physical difference than pi rounded to 20 decimal places and pi

>>7052742

If you do it an infinite number of times it will smooth out. Just like the limit of 1/x never actually equals 0 no matter how big x becomes, unless x is infinite.

>>7052799

God is great isn't he

>>7052640
>mfw the correct answer gets zero replies
This is why /sci/ is shit

>>7052896
The problem is that matter is discrete, you can't have a perfectly round cup just like you can't have a perfect circle on a computer screen.
>>7053193
>If you do it an infinite number of times it will smooth out.
Nope

>>7052497
It isn't actually a circle

None of the answers in this thread satisfied me, all I got is that "it's different", "you can't do it" and "fuck".

>>7053697
>None of the answers in this thread satisfied me, all I got is that "it's different"
A lot of anons even described HOW it is different. What else do you want?

The "proof" in OP's pic makes no sense. The last panel doesn't follow at all from the previous ones.
I can give you a similar "proof", try to disprove it:
Draw a circle of radius one. Draw a triangle of side one. The perimeter of the triangle is 3*side = 3. Hence <span class="math">\pi[/spoiler] = 3.

>>7053667

Hey, now. It's not the *only* reason.

yes yes...you are right..

If you put the circle at the origin and consider the first quadrant and only the horizontal parts of the approximation then you're basically approximating sqrt(0.5-x^2) by a sequence of step functions. These actually converge uniformly to the circle, but length is a function of derivatives, and convergence of functions says nothing about the convergence of their derivatives. The confusion comes from thinking otherwise.

>>7052637
>arrows on both ends of the axis

>>7052799
Hail Ahmadulla Rasillulah

When you say a sequence of curves converges to the circle, what does that mean? Convergence is a topological notion. If we're looking at the space of all curves in the plane, what topology do we put on that set?

There's actually multiple natural topologies. One of them is called the C^0 topology, which just says that all points on the sequence of curves should converge to all points of the limit curve. In this topology, the zigzag curves do converge to the circle. But arclength isn't a continuous function! The OP pic is the proof.

A stronger topology, called the C^1 topology, defines convergence of curves by requiring that all points converge, but also that all tangent lines converge. In this topology, the sequence of curves in the OP doesn't converge. There is no such thing as a "fractal limit" that looks like an infinitesimally jagged circle, it's just a divergent sequence. If instead we surround the circle with regular polygons with more and more sides, this sequence does converge to the circle in C^1. Also, arclength IS a continuous function, if we use the C^1 topology. That's why Archimedes's proof works but the OP doesn't.

>>7052799
>animals lead pretty good lives

Haaahahaha, have you seen a nature documentary? Getting throated by a lion isn't a good way to go. Starving isn't either, or any number of other ways to suffer and die or suffer and live

You guys are all fucking idiots

>>7053907
/thread

>>7053193
you can't repeat a process infinity times

>>7052748
Actually like that, you pleb.

>>7052649
4 is just an abstract concept, it doesn't exist. That's why math is unable to deal with numbers

No matter how many times we have had this thread over the years you faggots always sperg out over it.

Why is /sci/ so shit?

>>7052640
/thread

It is not a sawtoothed curve.
It is not an "infinitely jagged" sawtoothed curve.
It is not a "polygon with an infinite number of sides".
It is not a fractal.
The limit is an ordinary, perfectly smooth perfect circle.

All this proves that pi <= 4 not pi = 4.

>>7052652
so estimates is the new implying?

>>7054258
This.

>>7052497
taking that out to infinity gives you a diamond shape. its a convoluted way to rotate your square by 90 degrees.

>>7054040
lel, thats literally why limits were conceived, and why the proof is so fucking handwavy.

>>7052497

The shape defined doesn't converge pointwise to the circle. I can find a point on the edge of the circle that, no matter how many iterations you do, will never be touched by the line, which will only ever touch coordinates that are algebraic.

>>7054332
>converge pointwise
>iterations
>algebraic

Can you repeat that in English please?

>>7054653
This nigger hasn't even taken real analysis

>>7054675

All you do is hide behind deliberately made-up words to conceal that all your "theorems" are no more substantial than thin air.

>>7054788

You've just solved mathematics.

>>7054653

Certainly. A sequence of function F_n(x) converges pointwise to G(x) if, no matter what point x I pick, F_N(x) = G(x) for some N- it eventually "reaches" G at that point.

OP's construction does not satisfy that condition- at the N-th step in his process, it touches the circle only at angles that can be written as<span class="math">\theta = \frac{k}{2^N} \pi[\math]. As long as I pick a point on the circle that isn't a rational multiple of pi, then OP's construction will never reach it, and so it will never "equal" the circle.

Since his construction isn't the same as a circle, then he can't claim that the circumferecne is equal to that circle, and so <span class="math"> \pi \ne 4[/spoiler].

Make sense?[/spoiler]

>>7054806

Latex fucked up.

It touches only angles of the form T = k/2^N * pi. As long as I pick an angle that isn't a rational multiple of pi, then it'll never reach it no matter how many steps it takes. Therefore, the jagged-circle thing isn't equal to an actual circle, and so you can't say their circumferences are equal and the proof falls apart.

Make sense?

>>7052497
Most of the people in this thread are wrong; OP's construction converges perfectly to a circle, but it only gives an upper bound for <span class="math">\pi[/spoiler].
Here's a counter argument:
We assume that <span class="math">a^{2} + b^{2} = c^{2}[/spoiler] for all <span class="math">a,b,c \in \mathbb{R}[/spoiler]. This is the Pythagorean Theorem. Assume <span class="math">a,b[/spoiler] are nonzero. Assuming the OP is correct, we can show that <span class="math">a + b = c[/spoiler].
Plugging in the second equation to the first, we have that <span class="math">a^{2} + b^{2} = (a + b)^{2} = a^{2} + 2ab + b^{2}[/spoiler]. This implies that <span class="math">2ab = 0[/spoiler]. But we assumed that <span class="math">a,b[/spoiler] were nonzero. Thus there is a contradiction and both of our assumptions cannot simultaneously hold. So unless you have some qualms with Pythagoras and his Theorem, OP, you're incorrect.

>>7054865
By
>converges perfectly to a circle
I mean that, visually, the shape is a circle. I mentioned it because most people like to use some variant of
>Oh if we blow it up it's actually jagged and not round
and that same sort of argument can be used to say that the classical construction of pi is false, i.e. <span class="math">\pi \neq 3.1415...[/spoiler].
This >>7054332 is a more developed, rigorous, and correct statement.

>>7054865
>Assuming the OP is correct, we can show that a+b=c .
I don't understand that statement

>>7052497
where did the factorial come from

>>7054893
Draw a line from the point (0,0) to (a,b), call it's length c, and iterate the perimeter of the box containing the line in a manner similar to the OP for your result.

>>7054865
><span class="math"> \forall a, b, c \in \mathbb{R} a^2 + b^2 = c^2[/spoiler]
><span class="math">\forall a, b, c \in \mathbb{R}[/spoiler]

>>7054909
Haha, right. I hope it's clear that I implied c was the hypotenuse of the right triangle with base length a and height b.

>>7054865
All you've done is show a contradiction with the Pythagorean theorem. This is no different from finding a contradiction with the perimeter of a circle being 2pir.

>>7052896
>How accurate is a number that never ends?
a better question is how accurate do you really need your calculations to be in the context of beverages

>>7054958
Any way you see OP's claim disproved is going to be equivalent to that you dingus. More specifically, the way we calculate arc-length (in the xy-plane) is integration with respect to s, where <span class="math"> ds = \sqrt{dx + dy} [/spoiler]. You can't avoid the Pythagorean theorem.

>>7052497
Because pi equals 2

See proof

>>7054966
No it is not. Instead of simply contradicting the result of the construction we need to contradict the construction itself. This is easily done by invoking the mathematical definition of arc length. The length of the zig zag is NOT equal to the length of its limit. That's the flawed assumption here.

>>7054992
Did you read my post? I agree with you. Again: The mathematical definition of arc-length uses the Pythagorean theorem.
In any case, if we find a contradiction with the result of the construction, we have found a contradiction with the construction (since we know that the Pythagorean theorem holds, lest you abandon the definition of arc-length). This is a simple counterexample man.

>>7055013
You found a contradiction in result rather than the construction. A contradiction in result is ambiguous. Again, you might as well just say "but pi is less than 4" if that's your reasoning. That doesn't actually reveal anything about why the false conclusion was reached.

>>7055024
>Assume OP construction
>Assume Pythagorean theorem (and a,b,c nonzero)
>Contradiction implies thing1
>Pythagorean theorem implies thing2
>thing1 and thing2 cannot both be true
>thing2 was implied by Pythagorean theorem
>Pythagorean theorem is true (from some previous proof), so thing2 is till true
>Therefore thing1 is false
>Therefore OP construction is false
These are the logical steps I took. I fail to see any missteps. Please point out the exact misstep.

>>7055051
Again, I don't know why you continue to ignore the simple sentences I posted. Some idiot could argue that the Pythagorean theorem is wrong because it contradicts this proof. It's fucking ambiguous. What you should be doing is finding the flaw, not saying "it's wrong because it's wrong".

>>7055051
Read the OP again:

>Why isn't pi equal to 4 /sci/? What's wrong with this proof?

>>7055081
but we have valid proofs of the pythagorean theorem so unless you deny proof by contradiction, the argument holds

>>7055086
This proof is valid therefore the Pythagorean theorem is wrong by contradiction. Until you can show how the proof is invalid, you are using an ambiguous argument.

>>7055086
Also, if we go back to your original post. it's just mistaken:

>>7054865
>OP's construction converges perfectly to a circle
true

>but it only gives an upper bound for pi.
false. It gives a true estimate of pi if you evaluate the limit. OP did not evaluate the limit, he merely claimed it was 4.

>>7055085
I showed what was wrong with that method of proof. As for why pi does not equal 4, we can find a simple counterexample invoking the proper definition of arc-length; i.e., we say 2*pi*r = circumference and show that 2*4*r d is not the circumference.

>>7055103
We can form a Cauchy sequence of the length of the box surrounding the circle. It looks something like this:
{4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4....}
In R, all Cauchy sequences converge. The sequence above obviously converges to 4. This is not pi.
>OP did not evaluate the limit, he merely claimed it was 4
Yeah, I think he was justified in doing so.

>>7055092
I am convinced you are baiting now. My stomach is quite full, so I'm going to leave this pond and go whine about shitty anime characters on /a/.

>>7052896
when you pour a liter of cola you'll still be off by a small amount, you're just working with a simple value of "1" that you're trying to get as close as possible to

>>7055117
>I showed what was wrong with that method of proof.
Where did you do that? You merely stated it was wrong because the conclusion was wrong. Nowhere did you show the false assumption being made. All you did to "clarify" the method was write something false about it, that it was finding an upper bound.

And for the fifth time, a counterexample does not disprove anything. It merely shows that the proof and counterproof cannot be right at the same time. It does not tell us which is wrong. To do that we must find invalid reasoning in the proof itself.

>In R, all Cauchy sequences converge. The sequence above obviously converges to 4. This is not pi.
WRONG. It is not necessary for the limit of a sequence to have anything to do with the members of the sequence. Please learn some real analysis before you talk out of your ass.

http://en.wikipedia.org/wiki/Uniform_convergence

>>7055117
It's not a Cauchy sequence though. You are begging the question by assuming it is. Now you're being just as illogical as the troll proof.

>>7055121
I'm not baiting, you're just too stupid to understand why you're wrong, even though I've explained it in very simple terms.

>>7055135
>And for the fifth time, a counterexample does not disprove anything. It merely shows that the proof and counterproof cannot be right at the same time. It does not tell us which is wrong. To do that we must find invalid reasoning in the proof itself.

But we know the counterclaim must be wrong because we have previously established the veracity of the pythagorean theorem.

>>7054865
>OP's construction converges perfectly to a circle

This is incorrect, see >>7054806 for a demonstration that it doesn't converge pointwise, which is what's needed for the "proof" to work.

>>7055156
No one established the veracity of the Pythagorean theorem. Since the veracity of the proof that pi=4 is established (since I presented the argument and no flaw was shown), the Pythagorean theorem must be wrong.

>>7053860
yeah, that's how axes work. what's your point?

>>7055162
That doesn't demonstrate that doesn't converge pointwise. He gets the definition of pointwise convergence wrong. It's the limit of f_n(x) = f(x), not f_n(x) = f(x). So yes, it does in fact converge perfectly to a circle.

The shape defined by removing the boxes from a square is a circle when you use the taxicab metric, and in that metric, pi does equal 4.

I'm drunk right now, but I think the flaw has to do with that. You're not getting a euclidean circle when you take away the squares.

>>7055162
The convergence IS pointwise, it's just not uniform. The person you quoted has no idea what he's talking about.

>>7055164
>Pythagorean theorem must be wrong
ebbbbiiinnn xD

>>7055182
*According to your logic

>>7055180
You are getting a Euclidean circle. The flaw is in the assumption that the convergence is uniform and the perimeter of the limit is 4. In fact it's pi.

>>7055187
The convergence is uniform though.

The problem is that the arclength functional is not continuous, anon got it right in the first post in the thread.

>>7055187
The convergence of what specific sequence though? The perimeter at every iteration of your procedure? So it goes 4,4,4,4,4,4, ... , pi ?

>>7055196
The limit is not a member of the sequence.

>>7055196
sorry to have you spell it out, but how does 4,4,4,4,4, ... not converge uniformly? This is probably wrong, but is seems like the sequence is given by <span class="math">f(x) = 4[/spoiler] and hence <span class="math">\vert f(x)_n - f(x) \vert = 0 < \epsilon[/spoiler] for all x and for any positive epsilon

>>7055196
<span class="math"> \lim_{n \to \infty} \int f_n = 4 \neq \pi = \int \lim_{n \to \infty} f_n[/spoiler]

The proof uses the assumption that
lim pathlength(curve) = pathlength(lim curve). But you can only move the limit outside if the function is continous.

>>7055228
Drunkposter here. I finally get it.

please enjoy this amusing picture in return.

>>7052799
In Ye Olde Days, science was considered a Holy act, a means of understanding Him through study of His work.

What happened to those days /sci/?

>>7052497
I hate using greentext but in this case
>Implies you can count to infinity

>>7055205
>>7055228
I'm ashamed it took you this long /sci/.

>>7055181
>>7055162
You're both wrong. OP's construction DOES converge both uniformly and pointwise to a circle you idiots. The derivative needs to converge uniformly for the proof to work, see
>>7052794
But it doesnt. Therefore the proof is invalid

And who the fuck was saying the sequence (4,4,4,4,4,4,4,4....) doesnt converge or doesnt converge uniformly? Jesus

>>7053756
but it is and any other explanation is simply an algebraic representation of that original expression... even if they originate in totally unrelated 'math fields'

>>7052640
>>7055453
Its actually 100% wrong. You do get a circle in the limit. Look up what a limit is.

>Infinitesimal edges
u wot m8. This is the cartesian plane not some hyperreal plane or something

>>7054790
no you fucking ingrate
i hate you so much
I can't even understand what's being said but I am cataloguing it and trying, but a fucking ape like you just yells at your betters. Math is the most rigorous subject on the planet, where people have converged to the same realizations since the invention of language - how can you be so disrespectful of all of human achievement???

>>7055471
you clearly have a lesser understanding of the limit

>>7052799
you mean what you believe you comprehend from words invented by humans, and clearly transmitted through such non-god like language that they can't transcend the difference between us and it. So, no. you're wrong and everything you assume to be correct is simply not rigourous - try harder

>>7055471
Different guy, but this problem always makes me think of Karl Weierstrass, sorta kinda proved that you can have infinitesimal edges using real numbers - really he proved you could have continuous everywhere but differentiable nowhere functions.

I always think of Weierstrass functions as being infinitely jagged and I have constructed one before by starting with the absolute value function and doing pretty much what the troll is doing, forcing it to converge to some smooth function (i think I chose a parabola). I think of that assignment every time I see these threads...

>>7055545
Nowhere differentiable functions have to be carefully constructed though. Often they are constructed through limits or series but in the case of OPs picture there is a sequence of curves, each with finitely and infact growing numbers of points where the curve isn't differentiable, that converges to a smooth circle that is differentiable everywhere.

The sequence of jagged curves doesn't converge to some "infinitly" jagged curve, or nowhere differentiable curve that sits on the circle or something

>>7052497
OP, you aren't actually wrong here, you're just mixing into some non-euclidean bullshit.

In what's known as taxicab space, the value of pi is, in fact, 4.

By calculating distance using jaggies, you're computing the taxicab distance.
By ignoring the jaggies as "infinitesimally small," however, you're trying to apply taxicab space to our euclidean plane. This is where it falls apart. You're saying "In this hypothetical geometry, Rule A is true. Therefore, in euclidean space, Rule A is also true." even though you can't do that.

>>7052640
>It's wrong, because you don't get a circle in the limit,

Just like how you don't get 1 at the limit of:
.9
.99
.999
.9999
Ect.

>>7056681
that's not a limit retard

>>7055580
>that converges to a smooth circle that is differentiable everywhere.
Except it's not differentiable everywhere you fucktard

>>7056858
>(r cos(t), r sin(t)) is not differentiable
You must have been educated in America.

>>7056871
I meant the converging curve you moron.

A sequence of non-differentiable curves can't converge towards a differentiable curve you inbred moron

>>7052497
Excellent post. I used the same reasoning to disprove a^2+b^2=c^2.

That's the squariest circle I've ever seen.

>>7054865
The very same principle from the picture applies to the PT. By letting the length of the steps converge to zero, we obtain a+b=c.

>>7056970
I mean, we don't, because the proof is flawed, but if you wanted to explain to someone why it is flawed, do not invoke a theorem that is susceptible to the same flawed proof.

>>7056917
>A sequence of non-differentiable curves can't converge towards a differentiable curve

>>7055228
Which function is not continuous? Prove it.

>>7055228
It is continuous. It might have 90 degree angles but it is continuous

>>7057052
>>7057088
He talking about the function that gives the pathlength of a curve (basically the integral along the path) for the topolgy of curves over the plane (I'm guessing this is the topology of simple convergence).

There are probably simple counter-examples to prove that it indeed isn't continuous (OP is one, but probably even simpler stuff in the similar vein where a family of curves is made to converge to a single point).

>>7056917
>being this stupid

So could someone demonstrate how we know that this sequence converges to a circle?

>>7057883
Let S_{n} denote the nth iteration of OP's construction.
Give me a point p on the circle and I can give you an N such that n >= N implies p is on S_{n}.

>>7052799
>drop the G-bomb

Praise te Lort.

>>7056993
>>7057844
Care to give an example of a non-differentiable curve which converges towards a differentiable curve? I'll be anxiously waiting.

>>7057983
<span class="math">f_n(x) = \frac{|x|}{n}[/spoiler]

172 replies and still no definite answer. Why doesn't OP's proof work?

>>7059225
The autists have cleverly hidden, amongst impenetrable gobblydygook, the simple explanation: "The extra perimeter is hidden in the small ridges. As you make more ridges, you still have the same perimeter, just hidden in more and smaller ridges".

>>7052497
It never becomes a circle.

>>7059225
you can't get 4! from a perimeter of 4

>>7052497
it works for surface. not for circumference.
of course, the surface gets smaller with each iteration, eventually converging towards pi*r^2

>>7055142
>>7055142
I'm not the guy you were baiting, but I thought I would add that its probably for the best if you embrace the idea that it's just bait.

Your failure to follow simple formal logic is showing. Do you have any training with logic for proofs at least? I would recommend it, it's good for the mind and it can keep you from embarrassing yourself like this.

Just sayin

>>7052497
Do any of you bitches know what a motherfucking triangle is?

If you remove only squares you get a visible gap within 2 iterations. Dumbest shit I've seen

>>7059840
You're an idiot.

<span class="math"> \displaystyle
\lim_{n \to \infty} n \cdot \frac{4}{n} = \lim_{n \to \infty} 4 = 4
[/spoiler]

<span class="math"> \displaystyle
\Rightarrow \pi \leq 4
[/spoiler]

mystery solved

>>7059845
You mad bro?

Why don't you do this a few times and find out that the difference in perimeter is on the edges of the triangles that are formed, and those approach a fixed size as the area removed decreases.

If you are in 5th grade like me this shit would be so easy

>>7059855
Gr8 b8 m8

>>7059855
First of all, do you not realize that this method cuts out rectangles from each peak and not just the ones on the sides? And notice that they are rectangles, not squares after the first iteration. You can't create a square there that touches the circle with its corner. Lastly, you aren't creating any triangles, because triangles do not have curved sides. The limit actually does create a circle with no space left between it and the original. The problem is that has nothing to do with the perimeter.

>>7052497

here's the best way i have for explaining what's wrong with this proof

unroll the circle and the square around it, the circle is now a line, and the square is now 4 weird triangles with the line as the longest side of each triangle

after one iteration of the proof, it is now 8 weird triangles attached to the same line

after two iterations, it is 16 weird triangles

no matter how many times you do the iterations, you still have weird triangles lined up end to end, and since the shortest way between any two points is a straight line, the straight line is always going to be shorter than a path following the legs of the triangles

and the triangles are split in such away that with each iteration there are twice as many triangles, but the length of the legs is half, so the length of the path tracing the triangle legs is always the same length: 4

I have another stupid challenge for /sci/ !!

prove that 0 divided by 0 cannot equal 0 without appealing to "you cant divide by 0"

>>7059865
If you admit they are rectangles then they don't to 4, simple

all these shapes have the same perimeter

>>7059884
Why not? Trolling or retarded?

>>7059875
Let 0/0 = 0
Well 0*1 = 0
Therefore 1 = 0/0
Contradiction! Therefore 0/0 cannot be 0

>>7052794
>the norm is very important.
Indeed, thanks for correcting.

>>7052497
the circle is inscribed to the square, so the square define just an upper bound. you need to define a lower bound and make it growing up to fit the circle. if the lower bound goes to 4 you can say pi=4 (pro tip: you can't)

now you can only say this >>7059854

Why do we have two of these threads? Anyway:
>>7060505

>>7052637
Shit explanation. The limit is a perfecly fine circle, but because it is 2dimensional only the area is guaranteed to converge correctly.
Take this example: a rectangle of length <span class="math">L[/spoiler] and height <span class="math">\varepsilon[/spoiler]. Note <span class="math">A[/spoiler] its area. Now increase <span class="math">L[/spoiler] and decrease <span class="math">\varepsilon[/spoiler] in such a way that <span class="math">A[/spoiler] remains constant. Your limit is a line of area 0 (this time though the perimeter diverged correcly as a line is of dimension 1).