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/sci/ - Science & Math

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11029809 No.11029809 [Reply] [Original] [archived.moe]

My brain aches

>> No.11029819
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c is shortest distance

a's and b's add up to 0.8584073464102067615373566167205...

>> No.11029822

Based and Koch Triangle pilled

>> No.11029826

the reason it doesn't converge to a circle is because the tangents to circles aren't either vertical or horizontal. they can be other angles. in a nutshell

>> No.11029835

No matter how small you make it, it's still not a circle.

>> No.11029873
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>> No.11029917

similarity is by homeomorphism, not diffeo.

>> No.11030101


What you have computed is an outer bound for Pi (4), not Pi itself.

>> No.11030127
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>> No.11030136

>[math]\pi = 4! = 12 [/math]

>> No.11030145

The panel ordering makes my brain ache.
Anyhow, zoom in ad infinitum. For Real.

>> No.11030152 [DELETED] 

IF you crush a beer can, does the can disappear because the volume goes to zero? So why would the perimeter of 4 reduce to 2pi?

This is why people vote republican....

>> No.11030153

4! = 24 = 4*3*2*1

>> No.11030156

it's all in that last 1, innit?

>> No.11030157

If you crush a beer can, does the can disappear because the volume goes to zero? Then why would the perimeter of 4 reduce to pi?

>> No.11030163

Crushing a beer can reduces its volume, not its surface area.

This is why people vote democrat....

>> No.11030169

>The panel ordering makes my brain ache.
it's pretty standard wtf

>> No.11030223

>on an anime/manga website
>doesn't know they're panelled right to left

>> No.11030231

found the autistic weebs

>> No.11030270

This is simply a counter example to the claim that the mapping from graphs (under some reasonable topology) to arc length is continuous.

So it isn’t because pi isn’t 4.

>> No.11030490

Pie = 24?
My brain really hurts trying to understand this

>> No.11030495
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>On 4chan(nel) in 2019
Anime website. Go back to plebbit or whatever hole you crawled out of.

>> No.11030802

>Anime website
lol epic xd

>> No.11030847
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>> No.11030874

A circle is defined as a shape where all points of the boundry are equally distant to the center. In your illustration the shape merely approaches the shape of the circle, with the edges multiplying in number and approaching the same radius, they will never be exactly the same distance, thus regardless of how many steps you take, you won't come any closer to having a circle.

>> No.11030876

Yeah, in taxicab metic. We use Eucidian.

>> No.11030879

>post elementary geometry paradox in trollthread
>come back to find /sci/ getting baited by the anime girls

No, zooming in doesn't work since you'll still see the bumps "fractally", so doing it "ad infinitum" can't change anything.
The correct resolution is to abandon our intuitive notion of a length measured from the "extrinsic observer's perspective", in favor of the proper notion of arclength (alluded to by >>11030270) which is essentially "distance measured intrinsically, from the traveller's perspective".
The paradox highlights that our usual intuition for limiting arguments can fail unless the intrinsic path is known to be sufficiently smooth.

>> No.11030893

the transformation of the curve DOES NOT converge to the circumference
therefore those are two different things

>> No.11030895
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Guys, I'm actually freaking out about this.

If we can just arbitrarily say that a series of small steps doesn't approach a curve, how can we say that a series of small angled lines approaches the circumference of a circle? What if we've been wrong about PI this entire time?

>> No.11030897

Because their nature are different, as two different infinite sums approach different values.

>> No.11030903

Look up rectifiable curves.

>> No.11030906

That's not what's going on. Convergence of curves as you talk is not even defined. Learn some maths bro.

>> No.11030911

>top right to bottom right
>top left to bottom left
>pretty standard

>> No.11030921

Only good explanation. Thanks.

>> No.11030924

Pi was historically defined not as the ratio of two lengths (circumference over diameter), but by the "method of exhaustion", which involves squeezing the area in between an inscribed and exscribed polygon. Unlike length, the measure of area is well-defined (in 2D Euclidean space at least), so such issues don't arise.

By the way, the OP's use of a curved shape (circle) rather than straight lines is a red herring; the same paradox can be simulated in "Pythagorean" geometry (see >>11030127).

>> No.11030931

Actually, scratch the part about the area measure being well-defined unlike length, I don't think that's formally correct.

Nonetheless, the point still stands that in the Euclidean plane, you can't hide away "infinitesimal pockets of area" the way you can for path lengths.

>> No.11030938

No, that's not a good explanation for what's going on. You can inscribe a regular n-polygon in the circle and as n increases, the perimeter will approach that of the circle (pi). The shapes will also never be exactly the same distance, so your "thus" doesn't follow at all.
What's actually happening is that in the euclidean space, we know how to calculate the distance of line segments but not of arbitrary shapes. We do have, however, a nice class of curves that we can calculate the length of in a very simple way - called rectifiable curves. We just connect points of the curve by lines and take the supremum of lengths such simpler curves. The circle is a rectifiable curve and the length turns out to be 2 pi r from this definition. The OP is an example of how careful you have to actually be about defining the length of a complex curve (that's not made out of line segments), and how different ways to define it lead to different lengths.
Also, now that I think about it, the taxicab sort of length is probably also well defined for rectifiable curves, which will also be a type of length. But we don't like this definition because it doesn't coincide with our intuition for simple line segments. So in fact, as long as we have a well-defined concept of a length of a curve for simple enough curves which coincides with our intuition for line segments and satisfies a certain number of intuitive properties, the length will always come out as 2 pi r.
OP is a beautiful example of how we need to be careful about our definitions, just like the examples of continuous yet nowhere differentiable functions and all sorts of weird topological spaces.

>> No.11030947

OP is the answer to the following mathematical problem:
Given a curve C(t) 0<=t<=1, is it true that for all sequences simple curves S_n(t) 0<=t<=1 composed of finitely many line segments, if sup(distance between S_n(t) and C) approaches 0, then L(S_n) (length of the curve S_n) approaches a well-defined number dependent only on C?

>> No.11030966
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>> No.11031072


>> No.11031318
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Everything is a social construct and imperfect and future generations will think we were retarded.

>> No.11031322


>> No.11031340

Is Sketchbook /sci/ approved?
Are Fuu and Ryou?
What about Asakura-sensei?
Kurihara is certainly very knowledgeable of insect taxonomy.

>> No.11031388

>You'll agree with me that these two paths are the same length?
>Today we'll be talking about the shortest possible distance.
>And that these two are the same too?

This makes no goddamn sense

>> No.11031392
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you're welcome

>> No.11031411
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The shape at the end of the repetition has an infinite number of corners. A circle doesn't have any corners.

>> No.11031418

Who cares what that newfag thinks

>> No.11031428

There isn't the end of the repetion. The process of adding corners never ends. What do you mean "the end of repetition"? There are infinitely many natural numbers.

>> No.11031468

enjoy yourself with art club antics and infinite cats

>> No.11031511

You made my day. Thank you.

>> No.11031775

I wish mods here would ban you already. You're so stupid...

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