Why do fractals and Pi have a special relationship?

[Icebreaker]

This was brought to my attention today, and I haven't had much time to think about it; I think it has something to do with fractals.

If you have half a circle with diameter of 2, the circumference will be \pi.

If you create two circles, each with diameter of 1, the combined length of the circumference is also \pi, and the sum of their diameters will remain at 2.

If you continue in this fashion, the sum of the circumferences will remain at \pi until the semicircles become points, at which point the sum of the circumferences remains at \pi, but the is now the line segment which was the diameter that should actually measure 2 (because the semicircles become points).

Can anyone explain?

 

[Pseudo Statistic]

How are you so sure that this is true?
Diameter of 2, so Pi*d = 2Pi, how's that make Pi 2?

 

[Jameson]

If you have two circles with a diameter of 1, the sum of their circumferences is not \pi , it is 2\pi

C = 2\pi{r}

r is 1/2. so the Circumference of each circle is \pi... add both of these together and you get 2\pi

 

[Crosson]

Your original post contains some trivial errors that make it difficult to read.

This is one of the better limit paradoxes I have seen. It just goes to illustrate that a naive approach to limits is doomed to failure.

Do you know the definition of a limit? If you do, apply it this problem (hint: as the limit of the number of circles goes to infinity, does the sum of the circumferences approach a limit?)

You will find that there is a limit to the sum of the circumferences: pi, not 2.
So in the limit of infinitely many circles, the sum of the diameters is still pi.

The mistake occurs in the phrase "the semi circles become points", this is not true using the definition of a limit. Good one though, I will remember it to show people the perils of having a slippery concept of limits.

 

[scholar]

There is a vaguely similar problem involving limits to infinity and fractals, known as the Koch Snowflake. In this situation the perimeter of a shape goes to infinity, whereas the area converges to some finite value. If you're struggling with limits of this kind then check it out. http://mathworld.wolfram.com/KochSnowflake.html

 

[matt grime]

And another one is the step approximation to te diagonal.

Suppose you want to go from (0,0) to (1,1) in the plane. You can take the diagonal, length sqrt(2). Or you can do it in a series of steps: go 1/n along, then 1/n up, then 1/n along for some integer n and so on. As n tends to infinity the steps approach the diagonal, but the length of the steps is always 2, hence sqrt(2)=2.

 

[DaveC426913]

I'm afraid I don't see the paradox.

A half circle has a diameter of one, and a circumference of pi.
An infinite number of half circles have a summed diameter of one, and a summed circumference of pi.
No paradox so far.

Do you mean that the infinitely small circumferences become a line segment, thus must have a summed length of 1?

No. See Matt's response about the diagonal paradox. It's the same thing, and it's conceptually much simpler.

 

[dextercioby]

I'm afraid I don't see the paradox.

A half circle has a diameter of one, and a circumference of pi.
An infinite number of half circles have a summed diameter of one, and a summed circumference of pi.
No paradox so far.

Do you mean that the infinitely small circumferences become a line segment, thus must have a summed length of 1?

No. See Matt's response about the diagonal paradox. It's the same thing, and it's conceptually much simpler.

THERE IS A PARADOX.Your whole post contains a paradox (more of an oximoron) repeated twice...Be my guest and identify it...

HINT:it's about language...


Daniel.

 

[Icebreaker]

as the diameters of the semicircles become zero, what does the semicircle become if it's not a point?

 

[NateTG]

as the diameters of the semicircles become zero, what does the semicircle become if it's not a point?

They stay (semi-)circles. They're just smaller and smaller.

 

[Icebreaker]

Interesting. Is there a way to prove it?

 

[NateTG]

Interesting. Is there a way to prove it?

Sort of.

If you go to the limit (more on this below) then structure you describe has some odd properties, and it may make as much sense to talk about it as a 'line' as as 'circles' since neither of those is really accurate - it would be more accurate to call the structure as a whole a fractal.

However, no matter how big an n you choose, if you have n half circles lined up along your line segment - they will still be half circles. That is to say, while you're approaching (rather than at) the limit, the structure will always be a line of half circles. In this case, the jump from the approach to the limit is a quantum jump.

The reason mathematicians refer to something as a limit as foo approaches bar is because that this limit is fundementally different than evaluating if foo is equal to bar. Going to the limit (as this example illustrates nicely) can have unexpected results. It's also frequently the case that it's unclear whether the limit exists at all.

Let's say we have a circle, with diameter d. We know that the ratio of the circumference to the diameter is π. Now, if we take your approach to letting the circle go to a point, we get the final result that π should be equal to the circumference of a point (0 or undefined) divided by the diameter of a point (also 0 or undefined) and therefore π is undefined - rather than 3.14...

 

[Felix83]

A half circle has a diameter of one, and a circumference of pi.
An infinite number of half circles have a summed diameter of one, and a summed circumference of pi.
No paradox so far.

Do you mean that the infinitely small circumferences become a line segment, thus must have a summed length of 1?

both of these statements are wrong.

first of all when you say half circles i think you mean circles with half the diameter of the previous mentioned one (1)

an infinite number of circles with a finite nonzero diameter will obviously have a summed circumference of infinity as well as a summed diamter of infinity.

the second statement is unclear. how many circles are you talking about?

say you have a circle with diameter of 1 and circumference of pi. next you have 2 circles but the diamber of each is 1/2. then you have 3 circles with a diameter of 1/3 and so on.

so the sum of the circumferences of n circles is n*(pi*d) = n * (pi * (1/n)) = pi

there for if you follow this trend as n approaches infinity, the sum of the circumferences will tend to pi

you can see from this that following this trend, the diameter of the nth circle will be (1/n)

there fore the sum of the diameters of n circles is n*(d) = n*(1/n) = 1

there as n approaches infinity the sum of the diameters tends to 1

this i what i think you were trying to say

 

[daveyp225]

And another one is the step approximation to te diagonal.

Suppose you want to go from (0,0) to (1,1) in the plane. You can take the diagonal, length sqrt(2). Or you can do it in a series of steps: go 1/n along, then 1/n up, then 1/n along for some integer n and so on. As n tends to infinity the steps approach the diagonal, but the length of the steps is always 2, hence sqrt(2)=2.

How is this a paradox?

By moving 1/n up and 1/n along, you are creating very small right triangles, each with a hypotenuse of sqrt(2/n^2), so the resultant (which is what you are actually looking for) would be N * sqrt(2/N^2). If the number of moves were 10, you'd have a total length of 10( sqrt(2/10^2) ) giving 10sqrt(2)/10 = sqrt(2). The same if it were 1000 moves. 1000srt(2)/1000 = sqrt(2).

Paradox? Where? How does this infer sqrt(2) = 2?

Dave

 

[AKG]

Here's an incredibly crude sketch of what's supposed to be going on. On the first line, you have two bumps, which are in total, the circumference of a circle of diameter 1cm. The next one shows 4 bumps, totaling the circumferences of two circles, each of diameter 1/2 cm. For the first line, the curve made up of two bumps will be pi cm long, and the horizontal line is obviously 2cm long. In the second line, the curve of 4 bumps is again pi cm long, and the horizontal is again 2cm long. As we make more and more circles (or bumps), their radii get smaller (to fit on the 2cm segment), and we can see that the bumpy curve appears to be approaching the straight line segment. If the length of the curve follows the sequence {pi, pi, pi, ...} and the line segment follows the sequence {2, 2, 2, ...} then the limit of the sequence of length of curves is pi, and the limit of the length of segments is 2. But since the bumpy curve appears to approach the line segment in the limit, their lengths "should" approach one another, so pi must be 2. Of course, this is not really a good argument, so it doesn't give us good reason to believe pi = 2, it is based on vague notions of curves appearing to approach other curves.

 

[Archosaur]

This was brought to my attention today, and I haven't had much time to think about it; I think it has something to do with fractals.

If you have half a circle with diameter of 2, the circumference will be \pi.

If you create two circles, each with diameter of 1, the combined length of the circumference is also \pi, and the sum of their diameters will remain at 2.

If you continue in this fashion, the sum of the circumferences will remain at \pi until the semicircles become points, at which point the sum of the circumferences remains at \pi, but the is now the line segment which was the diameter that should actually measure 2 (because the semicircles become points).

Can anyone explain?

Sure, I can explain. If by "continue in this fashion" you mean "keep dividing the diameter by 2 and doubling the amount of circles", the circles would never "become points" because a point has no diameter and no matter how many times you divide, you will never reach zero.

That's enough to derail this thought experiment, but let's go on.

Assume the circles did "become points". A point has no dimensions. No dimensions means no radius, which means no circumference. "Circumference" is a term we use when we're talking about circles. A point is just as much a circle as it is a square as it is a line as it is a Labrador retriever.

So, there's another nail in the coffin, but let's assume that was okay too.

Assume that from this logic, we could come to the conclusion that pi = 2.
But wait, you started the whole thing off by saying that "If you have half a circle with diameter of 2, the circumference will be \pi." You used C=PiD to find the circumference of a circle with diameter 2. You said the circumference came out Pi. Which means that, for the first component of your thought experiment, you assumed Pi = Pi, not 2. You used Pi = Pi as a logical step toward concluding that Pi = 2. That's a big no no.

So, I hope that clears it all up...

 

[Archosaur]

If you have two circles with a diameter of 1, the sum of their circumferences is not \pi , it is 2\pi

C = 2\pi{r}

r is 1/2. so the Circumference of each circle is \pi... add both of these together and you get 2\pi

True, the math is wrong, but the principle isn't affected by that. See? Back up. He also said that ONE circle with diameter 2 would have circumference Pi. That's also wrong. It would have circumference 2Pi, just like the added circumferences of the two R=1/2 circles.

His point is that, if you keep halving the radius while doubling the number of circles, the total circumference of all the circles will never change.

The only thing this mathematical correction does is make it come out as
2Pi = 2 instead of Pi = 2.

So, you're right, the math is wrong, and it does change the outcome, but it doesn't make it anymore true. The flaw is in his logic, not his calculations.

 

[Archosaur]

And another one is the step approximation to te diagonal.

Suppose you want to go from (0,0) to (1,1) in the plane. You can take the diagonal, length sqrt(2). Or you can do it in a series of steps: go 1/n along, then 1/n up, then 1/n along for some integer n and so on. As n tends to infinity the steps approach the diagonal, but the length of the steps is always 2, hence sqrt(2)=2.

Nope. Keep in mind that tending to infinity is not the same thing as being at infinity (because the latter isn't even a "thing"). Therefore the steps "approaching the diagonal" isn't the same thing as the steps being the diagonal. The steps are still steps, albeit very small and very many. So, yes, your total distance traveled is 2, but you haven't traveled on the diagonal at all. Radical 2 is still just radical 2.

 

[Archosaur]

THERE IS A PARADOX.Your whole post contains a paradox (more of an oximoron) repeated twice...Be my guest and identify it...

HINT:it's about language...


Daniel.

Sure, he said "an infinite number" and "infinitely small".

Haha, those aren't just oxymorons. They're downright grammatically incorrect!

Most people will agree that infinity is not a number, but not many people truly grasp what that means. It means there is no such thing as "an infinite number", "infinitely many", "at infinity", or anything of the sort.

I don't mean to dig on you, Dave. It actually is a really difficult thing to grasp.

For example, how many points are there on a line segment that is 1 unit long?

How can you answer this, except by saying "infinitely many"?

The fact is, there isn't an "infinite number" of points, nor is there a finite number of points.
There isn't a number of points.

It can't be expressed as a number of any kind. The points don't exist in "quantity".
They can only be described as a range.

In fact, it isn't even correct to ask the question "how many points are there on a line segment that is 1 unit long?"
Because asking "how many" suggests that I want you to give me a number as your answer.

I might as well ask you "how many degrees Celsius is purple?"

It doesn't make sense.

 

[Office_Shredder]

Assume that from this logic, we could come to the conclusion that pi = 2.
But wait, you started the whole thing off by saying that "If you have half a circle with diameter of 2, the circumference will be \pi." You used C=PiD to find the circumference of a circle with diameter 2. You said the circumference came out Pi. Which means that, for the first component of your thought experiment, you assumed Pi = Pi, not 2. You used Pi = Pi as a logical step toward concluding that Pi = 2. That's a big no no.

So, I hope that clears it all up...

So you're not a fan of proof by contradiction?

Besides, he never stated what pi is... it would be like assuming every circle has a diameter to circumference ratio x, which he doesn't know, and then proving that x=2. There's no logical flaw in this part

 

[Archosaur]

So you're not a fan of proof by contradiction?

Besides, he never stated what pi is... it would be like assuming every circle has a diameter to circumference ratio x, which he doesn't know, and then proving that x=2. There's no logical flaw in this part

But he does assume there is a ratio x. He starts with a circle of radius 1, then the first thing he does is calculate the circumference using the ratio x. Not only that, he assumes that x= Pi = 3.14159...

Also, I am a huge fan of proof by contradiction. However, this is not an example of proof by contradiction.

Proof by contradiction is used when you can't directly prove something to be true. Instead, you prove that it can't be false. In order to prove something true in this fashion, you first assume that the opposite is true and then prove, by arriving at a contradiction, that the opposite cannot be true.

I see what you're saying though. He wishes to prove that Pi = 2, so he assumes that it isn't and tries to arrive at a contradiction. But the contradiction he arrives at is that Pi = 2. The contradiction he uses to disprove the opposite of his theorem is his theorem!

The flaw is not in our understanding of the number Pi.

There is not a paradox.

This is a fallacy! The flaw is in his logic!

Circles don't become points. Ever! No matter how many times you divide, you get smaller circles. There is no such thing as dividing an "infinite number of times". To say that isn't even grammatically correct! This isn't a proof at all. It's just a lie! It's just wrong!

 

[pallidin]

Circles don't become points. Ever! No matter how many times you divide, you get smaller circles. There is no such thing as dividing an "infinite number of times". To say that isn't even grammatically correct! This isn't a proof at all. It's just a lie! It's just wrong!

Why?

A 2-d circle has a certain internal area. As one reduces the circle the area inside also reduces. The same is true for a sphere with respect to its volume.

When either is reduced enough, there is no longer any internal area or volume.

 

[Hurkyl]

Most people will agree that infinity is not a number, but not many people truly grasp what that means. It means there is no such thing as "an infinite number", "infinitely many", "at infinity", or anything of the sort.

Simply put, Archosaur, you've got the wrong idea. It is true that there is not an infinite real number, nor is there a real number named "infinity". But it is simply wrong to take the next step and assert that there are no infinite numbers, or that there is no place called "at infinity".

If you're willing to learn, you should look for information on mathematical structures like:
(1) Extended real numbers -- this number system contains the \pm \infty symbols you see in calculus
(2) Cardinal numbers -- this number system contains the numbers we need to express the size of infinite sets
(3) Projective geometry -- the projective plane is a geometric object that extends the Euclidean plane. In this context, "at infinity" is used to refer to those additional points of the projective plane that do not lie in the Euclidean plane.

 

[Archosaur]

Why?

A 2-d circle has a certain internal area. As one reduces the circle the area inside also reduces. The same is true for a sphere with respect to its volume.

When either is reduced enough, there is no longer any internal area or volume.

If you reduce by subtraction, yes, but no matter how many times you divide a finite number by another finite number, the quotient will never equal zero.

 

[Archosaur]

Simply put, Archosaur, you've got the wrong idea. It is true that there is not an infinite real number, nor is there a real number named "infinity". But it is simply wrong to take the next step and assert that there are no infinite numbers, or that there is no place called "at infinity".

If you're willing to learn, you should look for information on mathematical structures like:
(1) Extended real numbers -- this number system contains the \pm \infty symbols you see in calculus
(2) Cardinal numbers -- this number system contains the numbers we need to express the size of infinite sets
(3) Projective geometry -- the projective plane is a geometric object that extends the Euclidean plane. In this context, "at infinity" is used to refer to those additional points of the projective plane that do not lie in the Euclidean plane.

I got in a little trouble on another thread by insisting that 1 / \infty could not be defined as zero. After having my head chewed off, probably rightfully so, I was eventually informed that, to say "1 / \infty = 0" Is kinda like shorthand for "the limit as n -> \infty of 1 / n = 0"

Which I'm totally okay with.

Is what you're telling me something different?

I am definitely willing to learn. Newtonian physics is my bag. Math... not so much. I like numbers that are less than a billion. This area is not my forte, and I am certainly not loyal to any model of the number system. I will look up the examples you gave me.

...But I still insist that Pi is not 2.

 

[csprof2000]

Just a few interesting points...

1. Not a huge fan of proof by contradiction. If another method of proof is possible, I prefer it.

2. Love the discussion about infinity. I don't follow a lot of the popular mathematics based on these very liberal notions of infinity either, as they are either (a) not present in my field or (b) not necessary where they are present in my field.

You know, it wasn't always such a clear-cut matter whether or not infinity was a valid object of mathematical discourse. In fact, it seems almost a historical accident that it happened at all. There's a fascinating history surrounding Cantor and his detractors, and who are we to say that we know? Great mathematicians have denied the existence of infinity.

 

[Hurkyl]

I got in a little trouble on another thread by insisting that 1 / \infty could not be defined as zero.

In the extended real number system, as well as projective numbers (over any field), division is defined so that 1 / \infty is, in fact, well-defined and equal to 0.

But you do have to clearly state what numbers you're working with. If you were just working with real numbers, then 1 / \infty is meaningless. On the other hand, if you were working with the hyperreal numbers, and H was an infinite number, then 1/H is well-defined, but not zero.

I don't know the context of the other thread, but I imagine (hope) there is some semantic point along those lines that you missed.

 

[Office_Shredder]

But he does assume there is a ratio x. He starts with a circle of radius 1, then the first thing he does is calculate the circumference using the ratio x. Not only that, he assumes that x= Pi = 3.14159...

I don't see where he states the value of pi in his first post

I see what you're saying though. He wishes to prove that Pi = 2, so he assumes that it isn't and tries to arrive at a contradiction. But the contradiction he arrives at is that Pi = 2. The contradiction he uses to disprove the opposite of his theorem is his theorem!

If someone could show me: if statement P is false, then I can logically conclude statement P is true, I would be forced to either scrap the whole logical statement I'm working in, or accept that statement P is true. Are you suggesting there's an alternative?

This is a fallacy! The flaw is in his logic!

Circles don't become points. Ever! No matter how many times you divide, you get smaller circles. There is no such thing as dividing an "infinite number of times". To say that isn't even grammatically correct! This isn't a proof at all. It's just a lie! It's just wrong!

I agree, but just because there's a flaw in his logic doesn't mean all of his logic is flawed

 

[Hurkyl]

You know, it wasn't always such a clear-cut matter whether or not infinity was a valid object of mathematical discourse. In fact, it seems almost a historical accident that it happened at all. There's a fascinating history surrounding Cantor and his detractors, and who are we to say that we know? Great mathematicians have denied the existence of infinity.

It never used to be clear cut just what a derivative was either, or even the notion of 'real number'. One of the great advances of the past couple centuries was that mathematicians started formalizing stuff with sufficient precision so that these things do become clear-cut.

e.g. there is absolutely no question about what "at infinity" actually means in the projective plane, nor is there any question of its logical consistency¹. (And it's practical use was known before it was even discovered -- and continues to be demonstrated)

Cantor's discovery that there infinite cardinals could be distinct was interesting because it was in an entirely different direction: other uses of the infinite tend to have more geometric flavors.


1. At least, beyond the usual questions: projective geometry is consistent if and only if Euclidean geometry is, if and only if the theory of real numbers is.

 

[Archosaur]

I don't see where he states the value of pi in his first post

He doesn't state it, but he uses it to find the circumference of the circle. Then again, his math was wrong, so technically the value of \pi he first uses is \pi/2

So he starts with \pi = \pi/2
and ends up with \pi = 2

If someone could show me: if statement P is false, then I can logically conclude statement P is true, I would be forced to either scrap the whole logical statement I'm working in, or accept that statement P is true. Are you suggesting there's an alternative?

I don't really understand what you're asking...

I agree, but just because there's a flaw in his logic doesn't mean all of his logic is flawed

I agree, but just because not all of his logic is flawed doesn't mean \pi = 2

 

[Office_Shredder]

I agree, but just because not all of his logic is flawed doesn't mean \pi = 2

My point wasn't that he's correct, my point was that your criticism was incorrect; i.e. the act of assuming the existence of pi, and then having an argument that concludes pi=2, does not in and of itself invalidate the argument; rather you have to find a flaw in the argument itself to show the conclusion is incorrect.

 

[Archosaur]

My point wasn't that he's correct, my point was that your criticism was incorrect; i.e. the act of assuming the existence of pi, and then having an argument that concludes pi=2, does not in and of itself invalidate the argument; rather you have to find a flaw in the argument itself to show the conclusion is incorrect.

I think it does invalidate the argument.

If someone were to say, "The sky is blue, so... (some questionable leaps of logic) ...therefore the sky is green.", their argument would be invalid. You couldn't have proven that the sky is purple by first stating that it is green.

He didn't just "assume the existence" of \pi,
he defined \pi as \pi/2.
He concluded that \pi = 2.

That's not a valid argument.

 

[pallidin]

Pi is a consistent mathematical ratio that is simply non-controversial in that respect.

 

[Take_it_Easy]

well, there is uniform convergence of the succession of functions of circles to the function

constant = 0 in the interval (say) [0,1] now the elements of the succession have a graph that has constant length \pi, but the limit has a graph of length 1.

Well that's not a paradox, also more complicate things can happen when you consider uniform convergence.

think of the functions defined in [0,1]

f_n(x) = \sin(1/x)/n when x \in (0,1] and

f_n(0) = 0 they all have infinite length yet they converge uniformly to the costant 0 in [0,1]
that has finite (unit) lenght.

That is no paradox: just something not very intuitive, that happen!