/sci/ - Science & Math

It does not work that way. It will not be a real circle; a real circle is smooth
It will be a spiky shape that is almost a circle, but has a larger perimeter.

>>4703150


nothing is smooth unless is 2d*

perimeter is not necessarily conserved by limiting

>>4703150
harriet cant into limits lel

>>4703743
she's pretty much right, just said it in a simplified way.
it doesnt converge to a circle.

also
<<<

OP you've just given me an idea.
I'll try to find the area of the shape at each step. Knowing the area converges towards the area of the circle, I will be able to find a sequence of limit Pi

>>4703751
it DOES converge to a circle, you faggot. the perimeter is the problem, dumbass

>>4703754
pi = (sin60)/e

>>4703755
that pic is full of derp

>>4703759
no you fucking moron it DOESNT converge to a circle, because if it did, the perimeter would be pi, not 4!
dumb fucking retard!!

>>4703769
did u read le filename?

>>4703775
>HUUUUUUUUUUURRRRRRRRRR

ur too dumb. you cant into limits and convergence. shit converges to a circle. perimeter is not conserved. deal with it and learn basic math.

>>4703775
nope

the limit (curves' length) =/= length(limit of curves)

even though it limits to a circle

>>4703778
yes. assumed it was a reply to the troll, rather than more troll. akin to the 0.999... = 1 pic

>>4703775
>>4703782
oh hai guize.
Is someone talking about uniform convergence?

>>4703769
Where ?

>>4703778
It is called it Troll_math because it is in response to a troll, not because the picture itself is a math troll.

>>4703775
He is right, for all points on the circle you can find a number N of iterations such as the point on the polygon with the same polar angle is as close as you want from the point on the circle.
What is wrong is stating that the the perimeter of the polygon will converge to the perimeter of the circle.

>>4703787
it'd still always be spiky, a real circle is smooth.
if its spiky, it aint a real circle, and thats why the perimeter isnt the same as a circle.

>>4703792
You're such an insane mad idiot shit eater.

Just more proof you aren't even in university.

>>4703792
fuck you and you're stupidity. learn what a fucking limit is or get the fuck out of my SCIENCE board if your to stupid for simple calculus

>>4703794
>still think it is not a circle when n is "infinite" well you'd be right

>>4703794
>with the same polar angle
actually it could be the same x coordinate if you only work with the upper part of the circle.

>>4703798
fuck you retard. a circle has perimeter D*pi
anything else, and it AINT A FUCKING CIRCLE!

>>4703792
nope

in the limit it is "smooth". it is a circle

>>4703805
i bet you dont even know how to calculate the lenght of a curve

>>4703803
Heh, guess I forgot to correct that part.

>>4703806
>>4703798
ITT: Tripfags who can't into fractals.

>>4703805
don't you realise how a limit of a sequence

4, 4, 4, 4, ...

has no relationship to the perimeter of the limit of a set of points.

1/n doesn't converge to 0 as n goes to infinity because 1/n is always positive but 0 is not.

>>4703806
it aint smooth
if you zoom in to infinity (if it were actually possible) the shape is still only made up of vertical and horizontal lines. (an infinite amount of them)
whereas a real circle is just 1 smooth line all the way around.

>>4703811
>implying fractals

>>4703814
lel

>>4703815
>if it where actually possible

there's your problem

the limit is a circle. deal with it.

i suppose if you could zoom into 0.999... you'd realise it wasn't actually 1 hurr durr

>>4703815
>mfw EK fails at understanding convergence

How's highschool going for you?

>>4703814
exactly

deal with that EK

>>4703827
I like your tripcode.

...and tripfags have officially ruined /sci/ beyond redemption. All of you are retarded.

>>4703826
i do fucking understand convergence
my pic does converge to a circle, OP's doesnt

deal with it

>>4703817
>Implying not fractals.
Here, something for your level of intellect: http://www.youtube.com/watch?v=D2xYjiL8yyE&list=UUOGeU-1Fig3rrDjhm9Zs_wg&index=2&feature
=plcp

>>4703843
Then show me one point where the limit of OP's sequence of curves differs from a circle.

>>4703850
Like all of her videos this one is just another insult to math.

>>4703852
>converging to ininity
>'show me a point'
dumass
<<<

>>4703856
YOU are an insult to science... in general.

>>4703859
Do you agree that the squence converges? Then there has to be a unique limit curve. Please show me how you think it looks like.

>>4703864
Why?

>>4703872
Beacause you are obssesed with metaphysical nonsense. Also that video, is pretty decent even though it lacks mathematical formality, just like the image.

>>4703868
oh sure, it'l LOOK like a circle, because we obviously cant 'zoom in to infinity' but the fact remains that its made ENTIRELY out of just horizontal and vertical lines, and a real circle isnt.
thats why the perimeter is fucked up, while the area does converge.

its like fractals, with a finite area and an infinite perimeter.
shit just doesnt work in the real world if you try pulling shit like 'repeating to infinity'

>>4703875
I am not "obsessed with metaphysical nonsense". I am aware of it, nothing more. And the lack of rigor is exactly what makes that video bad. A mathematical question has to be answered with mathematics, i.e. rigor.

>>4703859
yes EK, show me a point of the limit curve that is greater than 1/2 from the centre? or less than 1/2 if you like?

remember, a curve is just a set of point. the points determine "spikyness" or what have you. not the curves prior to limiting.

>>4703876
Answer my question: How does the limit look like?

>>4703876
OK, if you think there are vertical lines, find two points in the same quadrant, where 1 is vertically above the other.

>>4703876
This is fucking hilarious

You don't even know calculus

AMAZING

>>4703879
For it to be answered to mathematical rigor, it should be asked with mathematical rigor.

>>4703883
>oh sure, it'l LOOK like a circle...

>>4703893
protip

weird shit happens in limits

>>4703890

He's right though.

She always just makes baseless statements with no real proof, and approaches problems in a very sloppy way.

>>4703893
So you're saying that a shape that equals a circle in every fucking point is "not" a circle?

>>4703890
If OP is not sufficiently educated to ask the question in a rigorous way, that's no excuse not to teach him mathematical rigor by showing him how it's done.

>>4703887
you already know i cant find a particular point when it zooms to infinity
but look at OP's pic, it starts with vertical and horizontal lines. each time you remove the corners its still made of vertical and horizontal lines. and you do this at each stage, and its STILL vertical and horizontal lines after any particular N number of steps.
there is no particular point where this pattern suddenly breaks. its always made of vertical and horizontal lines, its just that when you 'go to infinity' it LOOKS like a smooth circle, but it isnt really, and the perimeter is 4.

>>4703907
like how 1/n is positive for every n, but it's pattern suddenly breaks at infinity and it's not positive

>>4703907
You can't find a single point where they differ? So you agree that both the circle and the limit are exactly the same in every point. How do you want to distinguish them?

>>4703898
But it's not directed to mathematicians, so for non-mathematicians who are interested in mathematics like highschoolers they are really good.
And anyways this discussion should be have formally, otherwise it makes no sense.

>>4703900
you cant have an infinite number of points.
and yeh, a circle has perimeter D*pi. anything else isnt a circle.
its just as if you had a quadrilateral and converged one of the sides to almost zero and said its now a triangle
it isnt though, a real triangle has 3 sides by definition. if you have a 4th side that's infinitely small because you cut it in half an infinite number of times, then it still isnt a triangle because by definition a triangle HAS to have 3 sides, just like a circle HAS to have perimeter pi*D.

>>4703919
math breaks down if you 'go to infinity' just like if you divide by zero.

>>4703929
so why don't you accept that the 4 breaks down to pi?

>>4703926
>you can't have an infinite number of points

HAHAHA

just fuck off

>>4703926
>you cant have an infinite number of points.
>>4703929
>math breaks down if you 'go to infinity' just like if you divide by zero.

Fucking screencapped. The undeniable proof of EK not even knowing limits (highschool math).

>>4703936
it cant, its perimeter 4 at each N number of steps
its always 4.
except when you try to converge to infinity, and then it breaks down simply because infinity isnt a number.

>>4703939
there is some non-rigorous merit to the second statement

>>4703937
alright, i believe you
just draw me a circle and list an infinite number of points that apply to it
...i'll wait

>>4703943
>except when you try to converge to infinity
so you're saying it does "break down" at "infinity"

>>4703952
yeh, it'd be true for any actual real number of points.
if you go to infinity, then its basically just a circle, and perimeter must be pi.

>>4703949
Could you please post the mathematical definition of a circle and the definition of convergence? Because obviously your definition differs from what is taught in math lectures.

>>4703949
here's my list (it's infinite naturally)

(cos1, sin1), (cos2, sin2), (cos3, sin3), (cos4, sin4), .....

all on the unit circle, all unique (sin cos take radians)

>>4703949
Are you saying a circle consists of only a finite number of points?

>>4703957
so you've completely changed you position?

>>4703966
>it's infinite
Prove it.

>>4703975
if finite, pi would be rational, or N would be finite

>>4703966
wouldnt that stop at cos360 sin360?
after that you're just repeating yourself.

>>4703969
nah, infinite.

>>4703970
nope, i still disagree with OP's pic.
if you accept the pic, then you're admitting that you think pi = 4.

>>4703983
Prove pi is irrational.

>>4703988
>stop at 360
learn to radians

>still disagree with OP's pic
not what you were derping about. at great length

>>4703988
Could you please answer >>4703964 ?

>>4703992
high school calc level proof exists

learn to google

>>4703994
alright, i think we can agree on this:

for any real number of points, it's jagged/spiky, and the perimeter is 4

for an infinite number of points (not a real number)
the pattern breaks down, and the perimeter is actually pi. and it isnt jagged/spiky anymore, because the vertical/horizontal lines that are in the pattern when it's real numbers, now have length 0.

is that pretty much on the money?
i'm not a mathfag btw

>>4704001
no, he can fuck off and use dictionary.com or some shit. i aint his fucking dictionary.

>>4704011
>i'm not a mathfag btw

so perhaps less vehemence over what you believe to be true is in order?

>>4704012
I know the definitions. It's you who doesn't know them. Please post yours so we can see what you understood wrong.

>>4704016
i still KNOW a circle with D = 1, has perimeter pi.
anything that says it 4 or anything else i already know is wrong, even if i cant explain it.

>>4704012
See http://nobrain.dk/ you stupid fuck....

>>4704019
i do know them.

>>4704027
/thread

>>4704030
The please post them. Because right now it seems you either don't know or don't understand them.

>>4704045
*Then

>>4703150
>>4703751
>>4703752
>>4703775
>>4703792
>>4703805
>>4703815
>>4703843
>>4703859
>>4703876
>>4703893
>>4703907
>>4703926
>>4703929
>>4703943
>>4703949
>>4703957
hey you may have drop highschool for a long time now, so here is a reminder of the definition of convergence <span class="math">U_k \to \l \Rightarrow \forall \epsilon >0 \ \ \ \exists N \in \mathbb{N} , \ \forall n>N \ \ d(U_n,l)<\epsilon [/spoiler] so if you are telling us this does not converge to a circle , could you please show us a point of the limit that is not on the circle?
If it's too hard for you with a circle you can try with the example in my pic and show us the limit isn't a "smooth" line .

Oh you can't? So shut the fuck up and get the fuck out of this board if you don't even know how limits work.

>>4704086
trying again the definition of convergence <span class="math"> U_k \to l \Rightarrow \forall \epsilon >0 \ \ \ \exists N \in \mathbb{N} , \ \forall n>N \ \ d(U_n,l)<\epsilon [/spoiler]

>>4703926
>you cant have an infinite number of points.
>you cant have an infinite number of points.
>you cant have an infinite number of points.
>you cant have an infinite number of points.
>you cant have an infinite number of points.
>people say he is in high school
no, even a fucking high schooler knows that there are an infinite amount of points.
EK is in fucking middle school/elementary school.
fuck i dont ever even post, but god damn EK is fucking dumb
>inb4 hur dur ey trol u

Is EK a woman? Only explanation..

>>4703926
>you cant have an infinite number of points.

>>4703793
don't know what you are up to , but the sequence in op pic DOES uniformly converge to a circle. I'm actually trying to find out all the necessary conditions so that the limit of a the perimeter is the perimeter of the limit of a sequence of curves.
uniform convergence is obviously necessary but not sufficient as shown in OP pic

actually a sequence of continous curve cannot converge to another continous curve without being uniformly convergent, so uniform convergence is not needed, you have it by default

>>4704250
my hunch is piecewise smoothness, with the number of pieces being bounded

>>4704250
Ok i found out , the condition needed is that the derivative of the curves converge to the derivative of the circle.
why? because the definition of the length of a curve is <span class="math"> \int_{a}^{b} \left \| f'(x)) \right \| dx[/spoiler] where your curve is parameterized by f on the interval [a,b].

it's not the case in OP pic, it's the case (by luck?) in archimede method

I'm not a mathematician, but isn't it obvious what went wrong here?
There are only countably many integers under N. There are more than countably many points on a circle. We've mapped a countable number of points of the constructed shape to the circle. For each pair of corners on the circle, there must be a point on the circle that has no corner on it (basic definition of denseness). Hence, the two shapes are different.

>>4704331
That would also break the good version of this process >>4703752

>>4704331
I thought of something like that too, but that's wrong :
-firstly because archimedes method works and use the same king of approaches
-secondly because at the limit there is no more corners , no infinite corners, no coutable infinite corner just pure smoothness.

this guy has it right.

>>4704317
I can't believe it was actually trivial as fuck. just needed to go back to the very definition of length...

>>4704340
No, it would not.
In OP's version, after one corner has been fixed, the location of a sufficiently close point never changes, as the direction of the line segment will never change. In the correct version, such a point will slowly converge towards the circle too, as the slope of the line segment tends to the tangent of the circle.

>>4704373
but the fact is, it does converge to a perfect circle. it is extremly easy to see in the sqrt(2)=2 pic since you can say the sup of the distances between the 2 curves is halved at each step , hence converging to 0.
so there is uniform convergence wich is far stronger than just convergence.

>>4704385
Yes there is uniform convergence? What's your point?
My statement is merely the geometrical interpretation of the post that was incidentally posted at the same time as mine, namely that it's about the derivative.
Saying it's about the tangent is essentially the same as saying it's about the derivative.

>>4704373
i cannot really follow you, but you seem to have changed from your countable argument

>>4704404
No I have not.
In both versions, you only fix a countable number of points. But in the correct version, that is sufficient to put the rest of the points on the circle as well. In OP's version, only the points that are fixed are on the circle, not the other points.
The intuitive reason for this is that in Archimedes' version, the slope of the sections tend toward the tangent. Hence, take a point that has not been fixed by the construction sequence (in Arccimedes' case, a point that does not make a rational angle w.r.t. the horizontal line), and calculate it's position, it WILL be on the circle. In OP's case it will not be on the circle.

>>4704399
my point is the sequence converge to a circle means the limit curve is a circle. argument to which you disagree on >>4704331

>>4704418
By rational angle, I mean expressed in turns (or in degrees or gradians), not in radians. In radians, it's a rational multiple of pi.

>>4704418
in both curves there are only a finite number of points in the circle for all n, and all in the circle in the limit.

which is why i can't follow you.

>>4704430
>finite number of points on the circle
I never claimed that, and if you think I did, you did not understand my argument.
Also, there are an infinite number of points on the circle in both constructions. My claim was about whether it was countably infinite, or uncountably infinite (and in fact all).
>>4704419
How? The perimeter of circle isn't 4.

>>4704418
a limit doesn't need to be part of a sequence.
It's not because a point will never be in the circle that the limit is not in the circle.

as you can see in this pic, the upper corner will never be on the "goal" line, because his coordonates on the x axis is <span class="math">\frac{1}{2^{n}} [/spoiler] and will never reach 0, which is the x coordinate of the upper point in the "goal" line. BUT the limit is 0, so the limit if a fucking perfectly smooth straight line.

>>4704456
nope

in both constructions only a finite number of points of the curve are actually on a circle for all n

>>4704456
god damnit! I'm sick of this argument. the limit of the perimeter is not the perimeter of the limit untill proved otherwise. just because the limit of the perimeters is 4 doesn't mean the limit of the curves isn't a circle!!

>>4704459
Why do you keep coming with your diagonal line. It is exactly the same principle as OP's drawing. Just like I disagreed the limit of OP's construction is a circle, I disagree that your staircase is a straight line in the limit.
What is the definition of limits of sets?

>>4704469
>for all n
I was talking about the limit of n...

>>4704477
I agree that all points contained (as in area enclosed by it) by your curve are all contained by a circle, and vice versa. But that doesn't prove that they are the same shape does it?

>>4704488
for the limits. both constructions are a circle. for all uncountable infinity of points

>>4704500
yes it does

two sets are the same if they have the same elements

you'll end up with a rhombus not a circle...

>>4704482
are you retarded?

then you are a moron in middle school if you can't see that the blue lines are converging to the black one. And every fucking point of the blue lines is farther away from the black line than the "corner curves" is .

if you drew the smallest circle containing the curves in OP pic , you 'll have circles with a diameter closer and closer to 1. but every point of those circles will still be farther away from the circle of diameter 1 than the "corner curves" are.

then why the perimeter of this circles converge to pi and not the "corner curves" because >>4704317 the notion of length of a curve is tightly linked to the derivative

>>4704503
Let's try to come up with a more formal definition of OP's procedure.
Start with a set of points S = {(x,y) | -1 <= x,y <= 1, x,y in Real} and a list of lines L = ["vertical line through 0,0", "horizontal line through 0,0"].
For each line in L, calculate both intersections with the circle. For both intersections remove from S all elements in the quadrant furthest away from (0,0).
Take the bisection of each consecutive pair of lines in L (including the bisection of the first and last), and add them to L.
Repeat this procedure countably infinitely many times.
Now, if OP's method is correct, then the entire circle must be contained by S, and every point in S must either be on the circle, or it should be closer to the origin then a point on the circle.
Take any two arbitrary points (x0,y0) and (x1,y1) that have a rational turns (that is, there is a step N, s.t. for all n>N, the point is on a line in L). Without loss of generality, we assume x0<x1 and y0>y1 (say we are in the upper right quadrant, and (x1,y1) is further clockwise than (x0,y0)). Then there is an irrational turn between these two points on the circle, on the line with that turn, there is a point (x1,y0) that was never removed from S. However, (x1,y0) should not be in S, since (x1,y0) is neither in the circle, nor contained by it. Hence we reach contradiction. Thus we know that S, the upperbound of the set removal, does not equal the circle together with the points closer to the origin.

>>4704629
I can see tons of flaws in this argument the most obvious one is <span class="math"> \bigcup_{n=0}^{+ \infty}A_n \neq ( \bigcup_{n=0}^{+ \infty}A_n )\cup \lim_{n \to \infty }A_n <span class="math"> for example 0\notin \bigcup_{n=1}^{+ \infty}\left \{ \frac{1}{n} \right \}
so showing that the union of the intersection is not equals to the circle don't tell you shit about the limit[/spoiler][/spoiler]

>>4704629
I try again:
I can see tons of flaws in this argument the most obvious one is <span class="math"> \bigcup_{n=0}^{+ \infty}A_n \neq ( \bigcup_{n=0}^{+ \infty}A_n )\cup \lim_{n \to \infty }A_n [/spoiler] for example <span class="math"> 0\notin \bigcup_{n=1}^{+ \infty}\left \{ \frac{1}{n} \right \} [/spoiler]
so showing that the union of the intersection is not equals to the circle don't tell you shit about the limit

>>4704684
It's not about equality, but about being a subset, but ok.
I understand what you're trying to say; I know the epsilon delta definition of limits.
My point is that there is a difference between looking at each point seperately and applying the definitions on it, and at looking at the entire curve, and applying the definition at that.
That was the main point of the post.

The thing that S converges to is a shape close (human language) to a circle, however, the thing it converges to is some kind of fractal structure that has inner corners at every rational angle.
"For all e>0, there is a d>0, s.t. for all n (emptyset \subseteq n - N \subseteq d => |S_n - C| < e)"
C is the set of points in the circle, S_n is the set of points in the structure after n iterations. S_n - C denotes set difference.
This should be the correct definition, and it doesn't hold for OP's construction, but it does for Archimedes'.

You were only talking about single points on the segment. That's not correct. That was why I was rambling informally about sets.

i wish moot would axe this board

>>4704794
>emptyset \subseteq n-N \subseteq d
That should read:
>0 < n-N < d
I was thinking about adapting it for sets, but decided against it, because it'd be too informal, and didn't add anything. Obviously I was too hurried to check whether I properly undone it.

>open up this thread
>read Harriett's reasoning
>think to myself, that sounds right, huh interesting
>read everyone trashing on that idea
>read EK's shit tier posts

>the more I read her trying to reason out the "still a polygon" idea, the more I realize how wrong she is
>and I ONLY read her posts (and the guys who just say "l2limits")

>tl;dr EK trying to prove one method only makes it easier for me to see how wrong she is

>>4703147
>repeat until infinity
there's your problem, it does not create a circle

>>4704938
lel

>>4704938

>/sci/ - unable to into limits since 2007

Boys, this is a well-known concept with a well-known name.

https://en.wikipedia.org/wiki/Taxicab_geometry

>>4705150

Contradiction: "Circles" (defined as the points equidistant from a shared center) in taxicab geometry are squares, so either that isn't a circle, and those formulas are wrong, or this isn't taxicab geometry, and you are a faggot.

OP's pic is simple.
Although it seems to converge to the form of the circle, it doesn't really become the same circle.
As we can see in OP's pic, there will always be corners outside the circle which is basically the same amount as the corners touching it. Therefore, when reaching infinite corners touching the circle we will have infinite corners outside the circle which create the extra perimeter. In a sense the polygon converges into a diamond with rounded edges rather than a circle.

An infinigon is not a circle deal with it

An infinite amount of infinitesimal quantities make a finite error.

>>4706478
An example:
http://www.mathacademy.com/pr/prime/articles/cantset/