B What could prove this wrong? I'm having a dispute with friends

[phinds]

... it would be nice if the English language didn’t use the same word “suit” to describe the thing I wear to my friend’s wedding, the thing lawyers file in courts, one of the four categories of playing cards…. But that’s just not how natural languages evolve.

It also means "fits well" as in "That color will suit you well", and "is OK with me (or isn't OK with me)" as in "What you are suggesting does not suit me".


There are likely other uses that I'm not thinking of at the moment.

In others words, as we already agree:

Context is everything, as they say ...

SO ... @ducknumerouno, get with the program

 

[fresh_42]

I find it somehow telling that there isn't a mathematical definition for infinity. Even a definition of the form "A is infinite if ..." does not exist.

The most common examples are probably $\displaystyle{\lim_{n \to \infty}a_n}$ and $\displaystyle{\sum_{n=1}^\infty f_n(x)}.$ The first one does not use anything infinite in its definition, only arbitrarily high indices, and the second one is defined by the first, and so again, without using the term infinite.

The other natural association is the number of natural numbers. Infinities are noted by $\aleph_\alpha$ and are defined recursively. So $\aleph_0$ is the only occasion where the word infinite can, but does not need to be used. One can either define $\aleph_0=|\mathbb{N}|$ without using the term, or one can say that $\aleph_0$ is the least infinite ordinal number. However, even the infinity of natural numbers is defined by the simple statement that every natural number has a successor, again avoiding infinity.

In this sense, infinity isn't a mathematical concept. It is an abbreviation to make notation easier.

That's typical in mathematics. Logical problems like Russell's paradox are bypassed by more specific definitions instead of running into a wall. I suspect that infinities in physics are even more problematic and that physicists refer to mathematics instead of trying to define it. But there is no mathematical book that tells us what infinity is, only one that requires the property $s(n)=n+1.$ It remains a problem for philosophy and linguistics.

The mathematical answer to the original question is simply a statement about convergence
$$
4=\lim_{n \to \infty}\sum_{k=1}^n r_k \neq \lim_{n \to \infty}\sum_{k=1}^n c_k=L(\text{circle})=\pi.
$$
Crossing the limit is prohibited since the limit of the rectified curve is different from the limit of the parts of the curve.

 

[Klystron]

rue, infinity definitely depends on the context, but that just shows how confusing it can be to some people . If it can change meaning based on the situation, maybe it’s more of an issue than we know?

Having read this thread and reread the OP's posts, the original poster often confuses symbol with referent, that which the symbol points or refers to. Suggest reading introductory texts on semiotics. The clarity gained from this knowledge will help understand puzzles such as 'squaring the circle' while providing a firm basis for communication.

 

[mathwonk]

That (post #1) is a beautifully simple example! For a similar, (but harder to visualize), example where area approximation of surfaces also fails, see Spivak, Calculus on manifolds, p. 129.

 

[DaveC426913]

try it.

Sure.

Quick question: What's epsilon?

Follow up question: what is 'a beautifully simple example'?

 

[mathwonk]

@fresh_42:
In reference to post #52, I don't know if this is satisfactory as a definition, but, as you no doubt know, a common characterization of infinite sets is of course:
A set is infinite if and only if there is a bijection from the set to a proper subset of itself.

 

[fresh_42]

@fresh_42:
In reference to post #52, I don't know if this is satisfactory as a definition, but, as you no doubt know, a common characterization of infinite sets is of course:
A set is infinite if and only if there is a bijection from the set to a proper subset of itself.

Yes, that's an infinite set. The object provides a context. And it doesn't even distinguish between countable and uncountable infinity, so even though there is a context, it is still insufficient. What I meant is what I said in post #35. There is no definition for infinity itself, stripped of any objects, other than greater or equal $\aleph_0$ for sets. https://ncatlab.org/nlab/show/HomePage lists 371 pages that have "infinity" in their titles. If we search for "infinity," then we get this list of 371 possible choices. The concept of infinity is void without further information. I meant that an infinite sequence is something different than the infinity of the continuum, or infinities as integration limits. We cannot use infinity without further specification.

Mathematics is of no help when it comes to philosophically explaining infinity. The original question about the convergence of area versus perimeter only shows how carefully the term infinite has to be used. For example, I think of infinite towers of modules when someone says infinite, probably something people had not in mind.

 

[mathwonk]

Ascending or descending towers? I guess one already has descending towers in the integers: Z, 2Z, 4Z, 8Z,..... In fact these are nice examples of bijectively equivalent subsets.

 

[fresh_42]

Ascending or descending towers? I guess one already has descending towers in the integers: Z, 2Z, 4Z, 8Z,..... In fact these are nice examples of bijectively equivalent subsets.

Lol, depends on whether I have an Artin day or a Noether day, I admire both. But I think more often about a tower of Lie algebras that are not included in each other.
https://www.physicsforums.com/insig...-representation-of-mathcalBmathfraksl2mathbbR

 

[Lnewqban]

TL;DR Summary: Pi = 4?
I know it theoretically never touches the circle, but does the circle ever really become a circle?

Is impossible to create squares that follow the shape of the circle.
If you do, the line formed by the corners of the smaller squares never becomes a circle.

 

[Filip Larsen]

Is impossible to create squares that follow the shape of the circle.

The interesting thing is that it is possible for a square curve of length 4 folded in the given way (thus conserving length) to be brought infinitely close to a circular curve of length $\pi$. This does of course not mean that $\pi = 4$ as this "puzzle" teases, but it is instructive for readers to understand why not, perhaps especially if you already know and use that in R it is true that $0.9... = 1$ (another brain-teasing puzzle related to the real numbers) to wrongly conclude that "infinitely close to a circle" must imply it is a circle.

 

[PeroK]

The interesting thing is that it is possible for a square curve of length 4 folded in the given way (thus conserving length) to be brought infinitely close to a circular curve of length $\pi$. This does of course not mean that $\pi = 4$ as this "puzzle" teases, but it is instructive for readers to understand why not, perhaps especially if you already know and use that in R it is true that $0.9... = 1$ (another brain-teasing puzzle related to the real numbers) to wrongly conclude that "infinitely close to a circle" must imply it is a circle.

Be careful not to confuse the elements of a sequence with its limit. In both cases, numerical and geometric, we have an infinite sequence of things, none of which is equal to the limit. The limit is not part of the sequence, but a number or shape that has a well-defined relationship to the sequence.

 

[Filip Larsen]

Be careful not to confuse the elements of a sequence with its limit.

Yes indeed. I was trying to describe the (to me) most obvious pitfall this puzzle is nudging (lay-) people to fall into, that is, to make them wrongly conclude that a curve brought arbitrarily close to another curve implies the two curves must have same length. And I got prompted to say this because I imagine anyone in this situation wouldn't feel it much of a full explanation to be told that a square shape cannot match a circular shape.

 

[gmax137]

In cities with perpendicular streets and avenues, it doesn't matter where you make the turns right and left, you still have to walk the "right angle distance.

 

[fresh_42]

In cities with perpendicular streets and avenues, it doesn't matter where you make the turns right and left, you still have to walk the "right angle distance.

Manhattan metric in German, taxicab geometry in English.

 

[DaveC426913]

Yes indeed. I was trying to describe the (to me) most obvious pitfall this puzzle is nudging (lay-) people to fall into, that is, to make them wrongly conclude that a curve brought arbitrarily close to another curve implies the two curves must have same length.

A piece of string might be another way for @ducknumerouno to make the point to his friends in a very physical and intuitive way.

He could put a piece of string along the perimeter of the square, then ask his friends to tease the string into an ever better approximation of the circle's perimeter.

They will eventually have to concede that, no matter how closely it matches the circle, the string never gets shorter; it just bunches up (i.e. zig zags).

 

[DaveC426913]

In cities with perpendicular streets and avenues, it doesn't matter where you make the turns right and left, you still have to walk the "right angle distance.

Well, yeah, but i'm not sure this helps the OP's friends understand the problem/solution.

Surely, when they think through the city streets scenario they will arrive at right-left turns that are so infinitesimally small that they'll see it as just walking diagonally.


Above, I suggested using a piece of string. The advantage with the string scenario is that intuitively they know that the string never actually gets shorter, no matter how many 90 degree bends they put in it.

 

[fresh_42]

Well, yeah, but i'm not sure this helps the OP's friends understand the problem/solution.

My impression is that he has long left the discussions.

Surely, when they think through the city streets scenario they will arrive at right-left turns that are so infinitesimally small that they'll see it as just walking diagonally.

Changing the metric changes the entire problem statement, beginning with the definition of $\pi.$

Above, I suggested using a piece of string. The advantage with the string scenario is that intuitively they know that the string never actually gets shorter, no matter how many 90 degree bends they put in it.

One could halve the peaks of the squares with every step, give up orthogonality, and gain convergence.

 

[DaveC426913]

My impression is that he has long left the discussions.

Maybe. Or maybe he just visits his friend every few weeks.

Changing the metric changes the entire problem statement, beginning with the definition of $\pi.$

Sure, but that is not an intuitive property that his friends are going to get. I'm sure we'd lose them before the end of "Changing the metric..."

One could halve the peaks of the squares with every step, give up orthogonality, and gain convergence.

Again, I'm considering this from a layperson's POV. We shouldn't assume the OP's friends know any more than elementary math (else this thread wouldn't exist).

 

[fresh_42]

Again, I'm considering this from a layperson's POV. We shouldn't assume the OP's friends know any more than elementary math (else this thread wouldn't exist).

True, but the entire discussion is about lengths of polygons that do not approach the circumference. If we change the polygon into one with half the distance between the vertices and the circle at every step, we will receive a situation with convergence of lengths.

I think it would be much clearer to look at the situation of a diagonal in a square of length one. The same "trick" resulted in
$$
\sqrt{2}=\sqrt{1^2+1^2}\stackrel{?}{=}1+1=\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}=\ldots=2
$$

 

[Gavran]

No matter how big the number n is, the angle between any two adjacent sides of the polygon in the original post will always be 90 degrees or 270 degrees. The angle does not approach 180 degrees as n approaches infinity, so the polygon does not tend to the circle.
The number n is the number of sides of the polygon.

 

[PeroK]

No matter how big the number n is, the angle between any two adjacent sides of the polygon in the original post will always be 90 degrees or 270 degrees. The angle does not approach 180 degrees as n approaches infinity, so the polygon does not tend to the circle.

By what definition of convergence?

 

[Gavran]

By what definition of convergence?

The limit of any constant is the same constant. There are two constants and there are two limits in this case: 90 degrees and 270 degrees.
I speak about properties of the polygon which must approach particular values as n approaches infinity. One of them is the angle between adjacent sides.

 

[PeroK]

The limit of any constant is the same constant. There are two constants and there are two limits in this case: 90 degrees and 270 degrees.
I speak about properties of the polygon which must approach particular values as n approaches infinity. One of them is the angle between adjacent sides.

That's not a definition of convergence.

 

[jackjack2025]

From the initial image which contains 6 little squares and some insects for no clear reason
1. Fine
2. Fine
3. Fine
4. No... actually you are adding corners, so logically wrong here.
5. What do you mean by that?
6. Massive leap without any logic behind it.

As others have said, the perimeter is 4, 4, 4, 4, ... ad infinitum, it converges to 4. Does not converge to Pi.

 

[Filip Larsen]

As others have said, the perimeter is 4, 4, 4, 4, ... ad infinitum, it converges to 4. Does not converge to Pi.

You have just restated the puzzle without really explaining anything. The apparent contradiction the puzzle establish (for the layperson) is that a curve which is obviously fixed at length 4 at the same time is also is made to follow a circumference of length $pi$, obviously infinitely close, thus teasing the layperson about how both can be true.

To add to the confusion, or perhaps rather deepen the trap, one could even throw in the observation that the area inside the two curves "in the end" is exactly the area of a circle with circumference $pi$, so now you have a circular area with the same area as a geometric circle but with a much larger boundary curve length? And if you took this circular area with "square" boundary and rolled along a flat line (without slipping) it would still roll with same rate as the geometric circle. What gives??

So, a nice explanation would ideally explain this variant of the puzzle as well. I personally prefer the fractal dimension explanation, since this directly relates to the "practical" observation made in the coastline paradox which does not require much math for the layperson to understand.

 

[jackjack2025]

It does not follow a circumference of length pi infinitesimally (not infinitely!) close

 

[Filip Larsen]

It does not follow a circumference of length pi infinitesimally (not infinitely!) close

In the process of "flipping corners" of the square curve you will notice that the maximum distance between any point on one of the curves and the closest point on the other curve goes toward zero, thus in that sense the two curves are infinitely close to each other. This is a relevant distance measure since the puzzle shows a picture of the two curves as a whole, i.e if you were to draw the two curves on a piece of paper then no matter how fine a pen you choose the two drawings would be identical.

 

[jackjack2025]

In the process of "flipping corners" of the square curve you will notice that the maximum distance between any point on one of the curves and the closest point on the other curve goes toward zero, thus in that sense the two curves are infinitely close to each other. This is a relevant distance measure since the puzzle shows a picture of the two curves as a whole, i.e if you were to draw the two curves on a piece of paper then no matter how fine a pen you choose the two drawings would be identical.

No, that isn't convergence in any sensible way.

 

[DaveC426913]

So, we all know what the solution is because we all know the maths involved. All the proposed solutions require some degree of abstraction of the problem, talking about convergence and infinitesimals, bound to lose listeners who are already struggling.

The novelty of this particular thread is that it is a challenge for us to provide a solution to non-math-adepts**. Take it as an exercise in making math more accessible. Are you comfortable enough with the subject that you can explain it to a grade school student with no advanced maths?

**(Not to mention being technically on-topic, OP's absence aside)


I invite posters to take this as a challenge to find non-math, analogue solutions that evoke an intuitive understanding.

I throw my own solution back into the ring - a piece of string along the perimeter. No matter how many zigs and zags you add to a piece of string, it will never get shorter. A grade school child can intuit this.

Alternatives?

 

[fresh_42]

Alternatives?

You cannot beat a string. Every other alternative ultimately comes down to this example. I like my version with the diagonal in the unit square "refuting" Pythagoras and the simple equation
$$
1+1=\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}=\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}=\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}=\ldots=2
$$
whose value never decreases. But, of course, that's again a string.

 

[jackjack2025]

So, we all know what the solution is because we all know the maths involved. All the proposed solutions require some degree of abstraction of the problem, talking about convergence and infinitesimals, bound to lose listeners who are already struggling.

The novelty of this particular thread is that it is a challenge for us to provide a solution to non-math-adepts**. Take it as an exercise in making math more accessible. Are you comfortable enough with the subject that you can explain it to a grade school student with no advanced maths?

**(Not to mention being technically on-topic, OP's absence aside)


I invite posters to take this as a challenge to find non-math, analogue solutions that evoke an intuitive understanding.

I throw my own solution back into the ring - a piece of string along the perimeter. No matter how many zigs and zags you add to a piece of string, it will never get shorter. A grade school child can intuit this.

Alternatives?

Give a student a square pizza box. Say: "You can only cut out squares from this box, and your cuts need to be at right angles, i.e. can you can cut out a corner. Now, how much you wanna bet you can create a perfect circle?" I think they will not take the bet and won't need any fancy explanations. But the string is a nice one :)

 

[Filip Larsen]

No, that isn't convergence in any sensible way.

Well, since you don't provide any arguments I will just reply like manner and say sure it is! And I will even claim it models fairly well how a typical layperson would think about the two curves.

Note again that like Dave I am trying to point towards explanations that would make sense for the layperson, so just saying "there is no convergence" is not going to cut it.

 

[BWV]

So does this fix the OP?

 

[DaveC426913]

So does this fix the OP?

View attachment 361093

Sorry, I don't follow it.

 

[Filip Larsen]

So does this fix the OP?

Repeating the puzzle with a different shape, here with curve length $\sqrt{8}$, is perhaps a nice way to illustrate that the circle has not much to do with the cause of the confusion and that something fishy is going on no matter what shape you "wiggle" the square curve to match up with. In this case the relative geometry between the diagonal and the square curve remains the same no matter what part of the curve you zoom in at so this should simplify things when trying to establish an explanation for the average layperson.

 

[Gavran]

The novelty of this particular thread is that it is a challenge for us to provide a solution to non-math-adepts**. Take it as an exercise in making math more accessible. Are you comfortable enough with the subject that you can explain it to a grade school student with no advanced maths?

What about using a microscope?

 

[BWV]

Sorry, I don't follow it.

In L1 (manhattan or taxicab metric) a circle is a diamond so ‘pi’, the ratio of circumference to diameter, is 4

 

[DaveC426913]

In L1

200 pixels among 200,000....

 

[BWV]

200 pixels among 200,000....

View attachment 361116

Is that what you saw of my post?

 

[PeroK]

What about using a microscope?

That raises the question, what does $\pi$ look like under a microscope?

 

[DaveC426913]

Is that what you saw of my post?

No. That's the 0.1% of the pixels in your post 84 I would have had to notice, in order to get the joke.

'Subtle' is an understatement. ;)

 

[Gavran]

That raises the question, what does π look like under a microscope?

It looks like a circle with a diameter of 1. The same can not be said for the figure in the original post.

 

[A.T.]

TL;DR Summary: Pi = 4?
I know it theoretically never touches the circle, but does the circle ever really become a circle?

This iterative approximation reduces the area-error in each step, but it doesn't reduce the perimeter-error at all. So, it can used to derive the area of the circle but not its circumference (and pi from it).

See also this video:

 

[wrobel]

Old thread but it dives up as a hot thread every time . I just want to say that there are several definitions of arc length. Math begins from definitions.