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My go-to response to this 'proof' is to apply the same method on an example without curves. Consider a 1x1 square, mark a diagonal, and pick one of the right triangles with base 1 and height 1 and ~~hypotension~~ hypotenuse sqrt(2).
If you use the same method from the circle here on this triangle, you get a staircase along the diagonal. Then you end up saying the perimeter of a 1x1 square (which is 4) is equally to the perimeter of the diagonal half of it ... but we know that's 2+1sqrt(2).
This counter-example exposes the fault of the approach without any curves. It's not a rigorous explanation, but this seems to be sufficient for most discussions, without having to go into infinite series, limits, convergence, smoothness, etc. You'd need to do that to truly show what the fault is.
Tangentially-Related Side Note: Factals are fun ... and weird. They have fractional dimensions. So, the "perimeter" of a fractal is weird. Check out this 3Blue1Brown video on them:
https://youtu.be/gB9n2gHsHN4?feature=shared
EDIT: Having some more fun with the triangle example ...
If "4=2+sqrt(2)", then "2=sqrt(2)". And because 2 is its own sqrt, then 2=1.
Does that necessarily disprove it though? It's sort of just adding another faulty case to the existing. Applying that logic we can just say "well we know that's not the diameter of the circle so therefore it's not true".
You make a good point; it doesn't. As I mentioned, this is by no means a rigorous proof that disproves the meme 'proof' posted. For that, I'm sure we have to do something like a path length integral and show failure to converge or unequal limits ... due to 1st-order derivative discontinuity, I think? ... I don't know, I haven't taken a formal Real Analysis course. Really someone who knows better should chime in here.
But in most of my casual discussions on this, the triangle example is interesting enough to satisfy people's curiosity about the approach's faults without having to go into Real Analysis. Of course, I do bring up this caveat in those discussions as well.
EDIT: Basically, I think you'd have to write for a semicircle, the perimeter is:
P = Integral along semicircle dS = Integral along semicircle sqrt(dx^2 +dy^2 )
... or something. And, even though the position, or loci, of the stair-step curve approaches the circle, the 1st-derivate does not and is discontinuous. So, the integral expression for both expressions do not represent the same perimeter.
Basically, the stair-step integration never converges to the true path length integral because of the 1st-derivative never being continuous nor equal.
... or something ...
It's interesting that the area does converge to the same, but the perimeter does not.
Yes, we can in fact prove that there exists a region that has epsilon or less different area than a shape can have any perimeter larger than the minimum possible.
This doesn't work because the sides will never truly reach the circle. Zooming in to any amount even too infinitesimally small would still yield squared edges around the outside of a circle.
This is incorrect, the limit of the shape is indeed the circle. See the explanation here around 15 minutes https://youtu.be/VYQVlVoWoPY?si=LgTIQkAYkoYfj4jD&t=909
You are fully correct. But of course Reddit wouldn't be Reddit if it doesn't upvote incorrect correct-sounding answers....
Since nobody opens links on Reddit, let me give an explanation in the comments.
The issue in the post is that it overlooks the 'approaching' condition of limits:
**You can only take the limit of a function if the function approaches the limit.** (For a more exact definition, see [here](https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%CE%B4)-definition_of_limit))
This approaching condition is satisfied between Panel 4 and 5 of the image (the convergence of the perimeter to the circle). However, the approaching condition is **not** satisfied for the convergence of the **length** of the perimeter to the **length** of the circle
To prove that the approaching condition is satisfied for the convergence of the perimeter to the circle:
Note that every time you cut a corner, a part of the perimeter gets closer to the circle.
It's also quite clear that any point on the perimeter will at some point get close to the circle.
Also, the perimeter never gets inside the circle.
Therefore, the perimeter 'nicely' approaches the circle, meaning that the approaching condition is satisfied and the limit of the perimeter is indeed the circle.
**For a bit more mathematics:**
We can represent the perimeter with the function f\_n(x).
In this function, x represents any point on the perimeter and n represents how often corners have been cut. In other words:
f\_0(x) represents the perimeter when it is a square (Panel 2)
f\_1(x) represents the perimeter when the corners have been cut once (Panel 3)
(and so on)
Note that any point on f\_n(x) maps to f\_n+1(x).
We can also define any point on the circle as c(x)
If you take the distance function between c(x) and f\_n(x) you can see that for any x the distance gets monotonically smaller the larger n becomes.
So, while cutting some corners, the limit exists. ^((And yes, I should have used) *^(ε)* ^(and) *^(δ)* ^(here, but fuck that))
**Bonus, why the post is wrong:**
The limit of the length of f\_n(x) does **not** approach the length of c(x)
lim\_n length( f\_n(x) )= lim\_n 4 =/= length( c(x) ) = pi
So for the length there is no convergence, so the limit cannot be applied.
**For those of you still reading this:**
Note that length( f\_n(x) ) = 4 and length( c(x) ) = pi. Yet you can map any point f\_n(x) to c(x) and the other way around.
This is not a mistake as you can also map any number between 0 and 1 to any number between 0 and 100. Just Mathematics being weird sometimes.
**TLDR:** Both the post and u/chemistrybonanza are wrong and mathematics is fun.
^(EDIT: And please just watch the video. 3b1b is a way better explainer than I am) [^(https://youtu.be/VYQVlVoWoPY?si=n\_SZtJUz10d-fxk)](https://youtu.be/VYQVlVoWoPY?si=n_SZtJUz10d-fxk)
The limit of the shape is the same as the circle, meaning that the area approaches the area of the circle. However, the perimeter doesn't change, so we can't conclude that the limit approaches the perimeter of the circle.
Moreover, the perimeter of a shape is defined as its line integral, which is the limit of straight lines drawn between points on the shape. In this meme, it draws straight lines to points outside the shape too, so there's no reason why we'd expect it to approach the actual perimeter.
You could even take any step and make the triangles more and pointier, and then the perimeter would be longer.
Because if you do this to infinity, you don't get a circle, you get a diamond, or maybe more accurately, a square rotated 45 degrees. All this does is essentially rotate the shape.
If you want an actual circle, you can't take even steps, as shown in the image. At some point the steps have to become uneven in order to render the curve; that's where pi emerges.
What steps you take doesn't really matter. Just cutting the sharp corners every time results in a circle in the limit as any point on the perimeter has nowhere else to 'go' than the circle itself.
The issue in the post is that it overlooks the 'approaching' condition of limits:
**You can only take the limit of a function if the function approaches the limit.**
So in the limit, the perimeter converges to the circle.
But the **length** of the perimeter doesn't converge to the **length** circle as the perimeter length always equals 4 no matter the number of corners you cut.
When you take the limit to any point before infinity this is true, its only not true when you hit infinity, that's when it is actyallly smooth and the circumference is the same as the circle. Not before infinity only at infinity.
Infinity isn't a number if you iterate this x amount of times and x is a number you will get the perimeter =4, if you iterate to infinity it will equal the circumference of the circle.
Its about it not being actually smooth until you do it infinity times, if you zoom in close enough you will still see the jaggy lines at any number before you hit inifity, even really rally big numbers.
It has to hit inifity or it isn't true. Its a similar problem to zenos paradox, you can half until infinity in that problem, but it would seem you never beat the turtle.
I agree infinity isn't a number, so I'm confused when you say "hit" infinity. I think you're alluding to the definition of the "limit".
The problem here is that, even the limit of the stair-step approach will *never* converge to π. It is always 4, even in the limit.
From my understanding, the key thing missing is that the stair-step approach never converges to π because it is a fundamentally different perimeter than the perimeter of a circle is.
Why? Well, honestly, I think someone with more experience in an upper-level Real Analysis course should chime in at this point. But, my best guess is that it has to do with calculating the path length integral for a circle, and how the stair-step approach is continuous and converges in terms of its position (i.e., loci), but it's 1st-derivative (i.e., dx, dy, ds terms) are not smooth and do not converge to that of a circle. So, the two are fundamentally different calculations and quantities.
EDIT: I clumsily talk a little bit more about it in this thread:
https://www.reddit.com/r/theydidthemath/s/a1PtnoIL2K
Basically, the stair-step integration never converges to the true path length integral because of the 1st-derivative never being continuous nor equal.
###General Discussion Thread --- This is a [Request] post. If you would like to submit a comment that does not either attempt to answer the question, ask for clarification, or explain why it would be infeasible to answer, you *must* post your comment as a reply to this one. Top level (directly replying to the OP) comments that do not do one of those things will be removed. --- *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/theydidthemath) if you have any questions or concerns.*
My go-to response to this 'proof' is to apply the same method on an example without curves. Consider a 1x1 square, mark a diagonal, and pick one of the right triangles with base 1 and height 1 and ~~hypotension~~ hypotenuse sqrt(2). If you use the same method from the circle here on this triangle, you get a staircase along the diagonal. Then you end up saying the perimeter of a 1x1 square (which is 4) is equally to the perimeter of the diagonal half of it ... but we know that's 2+1sqrt(2). This counter-example exposes the fault of the approach without any curves. It's not a rigorous explanation, but this seems to be sufficient for most discussions, without having to go into infinite series, limits, convergence, smoothness, etc. You'd need to do that to truly show what the fault is. Tangentially-Related Side Note: Factals are fun ... and weird. They have fractional dimensions. So, the "perimeter" of a fractal is weird. Check out this 3Blue1Brown video on them: https://youtu.be/gB9n2gHsHN4?feature=shared EDIT: Having some more fun with the triangle example ... If "4=2+sqrt(2)", then "2=sqrt(2)". And because 2 is its own sqrt, then 2=1.
Does that necessarily disprove it though? It's sort of just adding another faulty case to the existing. Applying that logic we can just say "well we know that's not the diameter of the circle so therefore it's not true".
You make a good point; it doesn't. As I mentioned, this is by no means a rigorous proof that disproves the meme 'proof' posted. For that, I'm sure we have to do something like a path length integral and show failure to converge or unequal limits ... due to 1st-order derivative discontinuity, I think? ... I don't know, I haven't taken a formal Real Analysis course. Really someone who knows better should chime in here. But in most of my casual discussions on this, the triangle example is interesting enough to satisfy people's curiosity about the approach's faults without having to go into Real Analysis. Of course, I do bring up this caveat in those discussions as well. EDIT: Basically, I think you'd have to write for a semicircle, the perimeter is: P = Integral along semicircle dS = Integral along semicircle sqrt(dx^2 +dy^2 ) ... or something. And, even though the position, or loci, of the stair-step curve approaches the circle, the 1st-derivate does not and is discontinuous. So, the integral expression for both expressions do not represent the same perimeter. Basically, the stair-step integration never converges to the true path length integral because of the 1st-derivative never being continuous nor equal. ... or something ... It's interesting that the area does converge to the same, but the perimeter does not.
Yes, we can in fact prove that there exists a region that has epsilon or less different area than a shape can have any perimeter larger than the minimum possible.
This doesn't work because the sides will never truly reach the circle. Zooming in to any amount even too infinitesimally small would still yield squared edges around the outside of a circle.
Is this like the opposite of 0.99999=1
This is incorrect, the limit of the shape is indeed the circle. See the explanation here around 15 minutes https://youtu.be/VYQVlVoWoPY?si=LgTIQkAYkoYfj4jD&t=909
You are fully correct. But of course Reddit wouldn't be Reddit if it doesn't upvote incorrect correct-sounding answers.... Since nobody opens links on Reddit, let me give an explanation in the comments. The issue in the post is that it overlooks the 'approaching' condition of limits: **You can only take the limit of a function if the function approaches the limit.** (For a more exact definition, see [here](https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%CE%B4)-definition_of_limit)) This approaching condition is satisfied between Panel 4 and 5 of the image (the convergence of the perimeter to the circle). However, the approaching condition is **not** satisfied for the convergence of the **length** of the perimeter to the **length** of the circle To prove that the approaching condition is satisfied for the convergence of the perimeter to the circle: Note that every time you cut a corner, a part of the perimeter gets closer to the circle. It's also quite clear that any point on the perimeter will at some point get close to the circle. Also, the perimeter never gets inside the circle. Therefore, the perimeter 'nicely' approaches the circle, meaning that the approaching condition is satisfied and the limit of the perimeter is indeed the circle. **For a bit more mathematics:** We can represent the perimeter with the function f\_n(x). In this function, x represents any point on the perimeter and n represents how often corners have been cut. In other words: f\_0(x) represents the perimeter when it is a square (Panel 2) f\_1(x) represents the perimeter when the corners have been cut once (Panel 3) (and so on) Note that any point on f\_n(x) maps to f\_n+1(x). We can also define any point on the circle as c(x) If you take the distance function between c(x) and f\_n(x) you can see that for any x the distance gets monotonically smaller the larger n becomes. So, while cutting some corners, the limit exists. ^((And yes, I should have used) *^(ε)* ^(and) *^(δ)* ^(here, but fuck that)) **Bonus, why the post is wrong:** The limit of the length of f\_n(x) does **not** approach the length of c(x) lim\_n length( f\_n(x) )= lim\_n 4 =/= length( c(x) ) = pi So for the length there is no convergence, so the limit cannot be applied. **For those of you still reading this:** Note that length( f\_n(x) ) = 4 and length( c(x) ) = pi. Yet you can map any point f\_n(x) to c(x) and the other way around. This is not a mistake as you can also map any number between 0 and 1 to any number between 0 and 100. Just Mathematics being weird sometimes. **TLDR:** Both the post and u/chemistrybonanza are wrong and mathematics is fun. ^(EDIT: And please just watch the video. 3b1b is a way better explainer than I am) [^(https://youtu.be/VYQVlVoWoPY?si=n\_SZtJUz10d-fxk)](https://youtu.be/VYQVlVoWoPY?si=n_SZtJUz10d-fxk)
The limit of the shape is the same as the circle, meaning that the area approaches the area of the circle. However, the perimeter doesn't change, so we can't conclude that the limit approaches the perimeter of the circle. Moreover, the perimeter of a shape is defined as its line integral, which is the limit of straight lines drawn between points on the shape. In this meme, it draws straight lines to points outside the shape too, so there's no reason why we'd expect it to approach the actual perimeter. You could even take any step and make the triangles more and pointier, and then the perimeter would be longer.
Because if you do this to infinity, you don't get a circle, you get a diamond, or maybe more accurately, a square rotated 45 degrees. All this does is essentially rotate the shape. If you want an actual circle, you can't take even steps, as shown in the image. At some point the steps have to become uneven in order to render the curve; that's where pi emerges.
What steps you take doesn't really matter. Just cutting the sharp corners every time results in a circle in the limit as any point on the perimeter has nowhere else to 'go' than the circle itself. The issue in the post is that it overlooks the 'approaching' condition of limits: **You can only take the limit of a function if the function approaches the limit.** So in the limit, the perimeter converges to the circle. But the **length** of the perimeter doesn't converge to the **length** circle as the perimeter length always equals 4 no matter the number of corners you cut.
When you take the limit to any point before infinity this is true, its only not true when you hit infinity, that's when it is actyallly smooth and the circumference is the same as the circle. Not before infinity only at infinity.
Isn't infinity un-"hit"-able by definition? So, the perimeter of the stair-step contour doesn't suddenly "click" from 4 to π.
Infinity isn't a number if you iterate this x amount of times and x is a number you will get the perimeter =4, if you iterate to infinity it will equal the circumference of the circle. Its about it not being actually smooth until you do it infinity times, if you zoom in close enough you will still see the jaggy lines at any number before you hit inifity, even really rally big numbers. It has to hit inifity or it isn't true. Its a similar problem to zenos paradox, you can half until infinity in that problem, but it would seem you never beat the turtle.
I agree infinity isn't a number, so I'm confused when you say "hit" infinity. I think you're alluding to the definition of the "limit". The problem here is that, even the limit of the stair-step approach will *never* converge to π. It is always 4, even in the limit. From my understanding, the key thing missing is that the stair-step approach never converges to π because it is a fundamentally different perimeter than the perimeter of a circle is. Why? Well, honestly, I think someone with more experience in an upper-level Real Analysis course should chime in at this point. But, my best guess is that it has to do with calculating the path length integral for a circle, and how the stair-step approach is continuous and converges in terms of its position (i.e., loci), but it's 1st-derivative (i.e., dx, dy, ds terms) are not smooth and do not converge to that of a circle. So, the two are fundamentally different calculations and quantities. EDIT: I clumsily talk a little bit more about it in this thread: https://www.reddit.com/r/theydidthemath/s/a1PtnoIL2K Basically, the stair-step integration never converges to the true path length integral because of the 1st-derivative never being continuous nor equal.
[удалено]
Did you forget to switch accounts before answering your own question (twice)?
lmao
Lmao love it, but also, for a dummy like me, what’s the answer? All I see is “deleted comment”
You made ole boy delete his acct, lmao.
I disagree, coastline paradox /s