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/sci/ - Science & Math

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11171853 No.11171853 [Reply] [Original] [archived.moe]

What is wrong with this reasoning?

>> No.11171870

Look closely in the 4th image, the boxes are rectangles, not squares, slightly longer on the x axis. Using this method to account for that gets you pi.

>> No.11171886

Why would that matter? The perimeter is still 4.

>> No.11171902


>> No.11172060

uniform convergence of piece-wise smooth (or even linear in this case) curves doesn't imply convergence of the arc-length

>> No.11172080
File: 19 KB, 831x486, TIMESAND___makeway.png [View same] [iqdb] [saucenao] [google] [report]

The limit has infinity corners.
A circle has zero corners.

>> No.11172220

To have even a hope of reaching a proper smallest upper limit, the sum should change along the way.
In the prompt it's just a constant.

>> No.11172230

No it fucking wouldn't

>> No.11172236
File: 207 KB, 1110x1600, 1570155246832.png [View same] [iqdb] [saucenao] [google] [report]

This has nothing to do with circles.

>> No.11172262

Remove the geometric bullshit in the interpretation and the image doesn't say anything. Geometric interpretations of elementary calculus are for the weak, and physics majors.

>> No.11172398

Wut, it just becomes the same rotated square, not a circle

>> No.11172414

How many times do you need to remove corners before the perimeter becomes less than 4? Like what number value?
This isn't a limit problem or fundamental calc theorem problem btw. The top horizontal distance, for instance, does not get closer and closer to some different number each time you remove corners and add up the horizontal lengths. It stays at length of 1. There is no rate of change.

>> No.11172711

The "limit" doesn't really apply here.
The problem is that OP's picture never becomes a circle.

>> No.11172721

This image is the reason why analysis is important to study.

>> No.11172751

The transformed square always has infinite points that are not on the circle. Clearly the two shapes are not equal.

>> No.11172767

holy shit this board is so fucking low iq it hurts

>> No.11172768
File: 4 KB, 400x400, image059[1].gif [View same] [iqdb] [saucenao] [google] [report]

>The problem is that OP's picture never becomes a circle.
why does it not? Serious. A Riemann integral of non regular intervals that tend to zero should arrive at the correct area of the circle when they become infinitely small, so why does an almost identical concept not arrive at the correct arc length? Or even approach it at any rate at all?

>> No.11172770

because it does not converge in perimeter, only in area. You can make up a fractals, for example, that have an infinite perimeter but bounded area.

For an intuitive understanding, look up the coastline paradox

>> No.11172839

uniform convergence of piece-wise smooth (or even linear in this case) curves doesn't imply convergence of the arc-length

>> No.11172847

Intuitively, imagine “inflating” the square.

>> No.11173060
File: 1.96 MB, 1440x2984, troll.png [View same] [iqdb] [saucenao] [google] [report]

It's true with respect to the taxicab metric d((x1, y1), (x2, y2))=|x1 - x2|+ |y1 - y2|
pi is defined by approximating it with figures that have finitely many corners. Your reasoning for why OPs pic is wrong is wrong.
Take the straight line and approximate it with straight lines. The sum doesn't change, but you still reach the smallest upper limit.
More like, why definitions are important. Defining the length of a curve wrt. the taxicab metric gives you the result pi=4 (with pi here being the perimeter of the radius 0.5 euclidean circle wrt taxicab metric, I'm not claiming 4=3.14...).
Because area and length are two fundamentally different concepts, as OPs pic demonstrates. You can't do what you do in measuring area to measure length - it's not well-defined.
>because it does not converge in perimeter, only in area
It does converge in perimeter to 4, clearly. It doesn't converge to 3.14..., but you'd have to ask why would you want it to converge to that number in the first place, how do you know the length is actually 3.14..? You need a definition of length.
What do you mean?

>> No.11173229

>but you still reach the smallest upper limit.
yeah because pi=4

>> No.11173252

You have no idea what you're talking about.

>> No.11173264

it's true that the polygonal curves do converge uniformly to the circle, and it's true that their length remains constant. but this doesn't imply anything. there's no theorem in math which would say that if a sequence of curves converge, then the sequence of lengths must also converge to the length of this limit curve. such theorem doesn't exist, because it's simply not true and this picture is actually a counter example. it does work for areas though, that's why it seems weird at first.

>> No.11173274

Exactly. OP might as well posted
>1/2 = 1/(1+1) = 1/1 + 1/1 = 1+1=2???? PROOF THAT 2=1/2!!!!

>> No.11173293

>dude just use the dominated convergence theorem, its obviously bounded by the shape

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