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Pi = 4. Discuss.

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I once discovered the diagonal paradox when I was studying reflective mirrors that are limits of right angle zigzag turns. In real life the type of mirror I was constructing is impossible because light will always have a wavelength larger than the 'resolution' of the bumpy mirror. As a math problem I showed it to my dad and then to an uncle is are really big into math (My uncle currently doesn't believe in real numbers). Needless to say it really bothered him; he then showed it to a bunch of his coworkers and mathematics graduate students. They thought it was really cool and couldn't figure it out, but knew something was weird. Eventually a professor in functional analysis was asked about it, who was excited that other people were thinking about this kind of thing. He then showed a similar "proof" that could show that any number equals any other number.

The problem lies in the fact that this is what is called a 'monster curve'. A particular type of curve defined as the limit of a series of curves. The limit curve doesn't need to be the same curve as another curve (or some other construction) that has the limit of the total difference distance of zero. This was worked on by mathematicians in the 18th, 19th and 20th centuries. Eventually we got Rigorous Functional Analysis and Fractals.

Fun stuff

A bit off topic but, now after a few years as a programmer in the video game business, a math tutor, or recently unemployed, I'm getting psyched for finally getting myself into graduate school. I'm planning on studying up in pure mathematics. My bachelors was in applied / numerical math.

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The fractal surrounding the sphere will always be on the outside. The area between the circle and fractal will have infinitely many very small rectangles that are on the outside.

The circumference obtained using this method is strict upper bound but does not define lower bound. So whatever value is obtained using this method, it's guaranteed to be strictly larger than circumference of circle.


PI, as per such definition, is computed using circumference of circumscribed shape (can be rectangle) and inscribed shape (which is missing in this definition).

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Original post by Antheus
The circumference obtained using this method is strict upper bound but does not define lower bound. So whatever value is obtained using this method, it's guaranteed to be strictly larger than circumference of circle.

Yes it is strictly larger, but it is still very useful. The number is accurate when performed properly, and it is a precise solution at the limit.

This was half of the classical methods of estimating pi to a certain precision.

They would subdivide slices or regular polyhedra on the outside and again on the inside. This gave an upper bound and lower bound. The mathematician could iterate until the solution converged enough for their necessary precision.

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Original post by Antheus
The fractal surrounding the sphere will always be on the outside. The area between the circle and fractal will have infinitely many very small rectangles that are on the outside.

The circumference obtained using this method is strict upper bound but does not define lower bound. So whatever value is obtained using this method, it's guaranteed to be strictly larger than circumference of circle.


PI, as per such definition, is computed using circumference of circumscribed shape (can be rectangle) and inscribed shape (which is missing in this definition).


Can maybe you explain this in a different way? If the problem is that there is smaller areas between the circle and fractal, why doesn't the approximation get better as we approach infinity?

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Imagine if you drew tiny squiggles and spirals along the path, but they were so tiny that if you zoomed out so they're too small to see, it looked like a curved line and made a big circle.

This is that.

But it's not a circle, so if you trace the perimeter, you'll get an inflated number. You might write "This is not a circle" under it and put it up as post-modern art.

That's the flaw in it; it's specious reasoning. It's dressed up to look like it'd work, but it is intentionally designed to give an artificially inflated figure, and as long as Pi is expected to give us the area, it is quantifiably wrong; you can see that the area of your shape does approach 3.14*0.5^2, veering off sharply from 4*0.5^2. So the figure in question can disprove its own result.

Increasing the sides of a polygon also doesn't give us a true circle, it only gives us Pi to the nearest however many digits. So there isn't a right way. But there is a quantifiably useful way.

So I guess what I'm trying to say is that it doesn't matter what the flaw in it is; the important thing is that it only looks like there isn't a flaw because through our limited perception it looks circleish.

[Edited by - JoeCooper on December 1, 2010 2:00:54 PM]

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But, everybody knows that pi = 3 + 1/7. All modern mathematical sophistries aside, this has been known for nearly 5000 years. :-)

You wouldn't want to doubt the guys who built the Great Pyramid, who were admittedly the most fucking awesome mathematicans and architects ever living, considering that their only technical means were wax tablets, clay, ropes, wooden poles, and an awful lot of slave hands.

But, jokes aside, the diagonal paradox is a funny one, it's something I never really grasped either (but eventually one just accepts that something can be wrong even if it looks right) :-)

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The limit curve is not a circle.

Is it even differentiable at any point?

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Quote:
Original post by frob
Quote:
Original post by Antheus
The circumference obtained using this method is strict upper bound but does not define lower bound. So whatever value is obtained using this method, it's guaranteed to be strictly larger than circumference of circle.


They would subdivide slices or regular polyhedra on the outside and again on the inside. This gave an upper bound and lower bound. The mathematician could iterate until the solution converged enough for their necessary precision.


But would you get the correct result if you tried to create a lower bound with a similar construction with an inscribed square? With what little I know about fractals, I wouldn't be surprised if you told me that "lower bound" turns out to be infinite.

Quote:
Original post by nilkn
The limit curve is not a circle.

Is it even differentiable at any point?


Quote:
Original post by JoeCooper
So I guess what I'm trying to say is that it doesn't matter what the flaw in it is; the important thing is that it only looks like there isn't a flaw because through our limited perception it looks circleish.


But these criticisms also apply to circumscribing regular polygons (e.g. is an infinity-gon differentiable at any point?), which does yield pi. This is why the explanation does matter. Without understanding why it doesn't work, you don't really know when it will or won't work. It's why people use "...and I" as the object of a sentence after being told not to use "...and me" as the subject.

Also, because "I said so" isn't a satisfying mathematical explanation.

EDIT: This is actually wrong. The infinity-gon is differentiable. However, the construction in the OP still seems to fulfill the usual definition of a circle: all the points in a plane that are a given distance away from a given point. That definition is apparently incomplete.


[Edited by - Way Walker on December 1, 2010 7:05:05 PM]

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Anti-proof by demonstration:

construct physical circle, diameter = x.
construct physical square over circle, length of side = x.

Wrap string around physical square exactly once. Cut to length.
Wrap string around physical circle exactly once. Cut to length.

Compare strings.

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Original post by JoeCooper
So I guess what I'm trying to say is that it doesn't matter what the flaw in it is; the important thing is that it only looks like there isn't a flaw because through our limited perception it looks circleish.


That makes more sense to me if I never took calculus. I spent 3 years learning that little rectangles = curved lines.

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Original post by samoth
You wouldn't want to doubt the guys who built the Great Pyramid... [snip] ...and an awful lot of slave hands.

Apparently the pyramids were not built by slaves at all. Have a look at the Who Built the Pyramids? section here.

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Original post by ChurchSkiz
Quote:
Original post by JoeCooper
So I guess what I'm trying to say is that it doesn't matter what the flaw in it is; the important thing is that it only looks like there isn't a flaw because through our limited perception it looks circleish.


That makes more sense to me if I never took calculus. I spent 3 years learning that little rectangles = curved lines.


no you didn't. You learned that little rectangles approximate areas involving curved lines. At least I hope that's what you learned.

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Original post by szecs

Remove more corners, perimeter is still 4.



That's the trick. The relation between the side and diagonal of a square isn't rational.
The act of removing the corners implies reducing the perimeter of the resulting shape.

Could anyone draw the formula that represents "removing the corners" from the square so we can appreciate the resulting error?

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Original post by Way Walker
But these criticisms also apply to circumscribing regular polygons


That was the exact previous thing I went into. There isn't a correct method. Just approximations of varying usefulness. Pi = 4 isn't normally going to be a useful figure.

As for the shape, we can plain see that its area does approach that of a circle (implying the familiar Pi value) even if the perimeter is artificially inflated.

With Pi=4, you can't use its diameter to find its area. As long as we agree that Pi is supposed to relate the diameter and area, that's a problem.

But if you work it the other direction, you actually can find Pi from its area.

Quote:
Also, because "I said so" isn't a satisfying mathematical explanation.


Excuse me, but that's not what I said.

[Edited by - JoeCooper on December 2, 2010 1:35:29 AM]

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Original post by JoeCooper
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Original post by Way Walker
But these criticisms also apply to circumscribing regular polygons


That was the exact previous thing I went into.

Quote:
Also, because "I said so" isn't a satisfying mathematical explanation.


Excuse me?

I said that if you try to calculate it through the area, you get a radically different figure, and while using the regular polygon method also isn't perfect, the difference is dramatically smaller to the point of being useful.


I didn't mean to single out your comment since it was a comment on the whole discussion (including the linked thread) and why I included another post there as well. A lot of the explanations of why it doesn't work are no more insightful than "I said so". For example, in the linked discussion the explanations are, "you cannot interchange limits and lengths," and, "Suppose that X(n) is a sequence of objects that have a meaningful limit X. If all of hte X(n) have a property P, then [...] most people will accept that the limit must have P without thinking about." The first isn't entirely true because you can if you have the right limiting sequence (e.g. regular polygons in this case) and the second gives no reason as to why it doesn't work in this case while there are still "numerous theorems in maths that follow this pattern." Basically, it doesn't work because they said so.

Students (and others) new to a particular area of math sometimes get the right answer by doing something wrong or even completely irrelevant. Their answer is quantifiably correct (there being literally 0 difference between their answer and the correct answer), but there's not necessarily any reason it should be or that it will be anywhere near correct in other cases. Why do regular polygons produce a better result? Or is it just chance, like a coder who makes a working program by randomly copy-and-pasting code from the internet? Or is it just that it was on the internet so it must be true?

I think it's related to the fact that the limiting set really is a circle so it will be pi and the derivative of the limiting set exists everywhere, but the limit of the derivative is undefined everywhere. Maybe the full explanation requires a deeper knowledge of fractals than one can give in a single post to a technical but still general audience?

And you're excused. [smile]

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The "because I said so" bit has me thinking.

There are a lot of assumptions going into this.

The whole exercise in question (in the OP's pic) is founded on the assumptio that the circle has less perimeter than the square, but we can do this reducing trick so that its shape approaches that of a circle.

But we can see immediately from the first step that the perimeter is artificially maintained - and I'd posit that this is no different than drawing horns onto the side of the square - therefore it doesn't meet it's own goal; its perimeter is not reduced.

We can also see that we're actually adding a lot of angles.

So as far as the parimeter is concerned, the reduction trick is not actually being done on the property in question.

Does that work?

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Having fun?

Actually it's an interesting problem, because as pointed out:
Quote:
However, the construction in the OP still seems to fulfill the usual definition of a circle: all the points in a plane that are a given distance away from a given point. That definition is apparently incomplete.


It has to do something with fractal geometry, I believe you can make a object that looks like a line segment (infinitely thin), yet it can have any arbitrary length, even infinite.

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Original post by szecs
Having fun?


Yessir.

Quote:
the construction in the OP still seems to fulfill the usual definition of a circle: all the points in a plane that are a given distance away from a given point


Maybe that's not the case. Again, if you just zoom in, it's stair-steps, it's not a circle. The shape on your screen isn't really a circle either, given that it's also painted onto such a grid.

The regular polygon isn't a circle either, and I think we can all agree on that.

Thing is, the regular polygon is designed to reduce its perimeter toward that of a circle while this, isn't.

The whole point of reducing is to approximate so that the figure's properties approach a circle, and this one fails because it doesn't attempt to do this in whole. It attempts to reduce one property, its area, while inflating the other, as a joke. If you look at the other property, you do get Pi, which is consistent with its circleish appearance.

[Edited by - JoeCooper on December 2, 2010 3:29:47 AM]

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So far, I havnt really seen any 'explanations' that go beyond 'but the polygon method confirms to our expectations but this one does not!'

How are they different? Both reduce the distance between the curve and the circle arbitrarily. Both are equally piecewise linear.

I think the fundamental difference is that the one arbitrarily closely approximates the tangent of the circle while the other does not. A sufficiently fractal curve could have any length while satisfying the above properties, while one with constraints on its tangent direction could not.

Still not a proof, but I think its a correct intuition.

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Original post by Eelco
How are they different? Both reduce the distance between the curve and the circle arbitrarily. Both are equally piecewise linear.


They're different because it doesn't reduce the perimeter, and the perimeter is what we're measuring. Therefore nothing measured is being reduced. Nothing at all. We might as well not do any reduction, just leave it as a square, and refute any complaints with "I don't see any 'explainations' other than a square isn't a circle."

It's the area that's reduced, so you're only using the reduction method if you measure that.

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Original post by JoeCooper
Quote:
Original post by Eelco
How are they different? Both reduce the distance between the curve and the circle arbitrarily. Both are equally piecewise linear.


They're different because it doesn't reduce the perimeter, and the perimeter is what we're measuring. No reducing is done. The figure's perimeter isn't changed at all. You might as well not change the figure.

The area is, and if you measure that, you'll find it most certainly does give us Pi.


Is that a valid distinction though? We can conceive of perimeters that do change, yet do not converge to the correct value; say, a grid outline on one side and a polyhedron on the other, or a more subtle instance of such a curve. Just because it changes, doesnt make it right, nor does the fact that it doesnt change tell you anything, without a preconceived notion of the quantity you are trying to figure out.

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We do have preconceived notions, assumptions and specifications, which is perfectly OK and in fact mandatory. If we can't agree on what a circle is and what Pi is supposed to do, than we might as well skip the diagrams, make up numbers and call it a day.

This approach is only being questioned because it looks like the reduction approach. Since the perimeter is not reduced, it is not the reduction approach, and is therefore one of an infinite number of totally random activities that also don't do anything relevant or useful, like watching Stargate. Your arguments are equally valid in support of me watching Stargate, counting the minutes, adding a break to refill my coffee, dividing it by 9 and calling it Pi; I can't tell that this is the wrong approach unless I have some idea of what I'm looking for.

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We know the circle has Pi*diameter circumference and Pi*(0.5*diameter)^2 area.

The reduced square will always have an area greater than the circle's, plus it will always have a circumference of 4*diameter (also greater than the circle's).

If we take the points of the reduce square which are touching the circle's edge, and join those points into a polygon, then that polygon will always have an area and circumference less than the circle's.

One gives us maximum values, the other minimum values. Pi is somewhere between the two.
[edit]Another way to realise that this reduce square thingy is obviously going to have a large circumference is to take the idea to the extreme... Imagine a circle, then pick a point on it and travel around clockwise. You travel along in infinitely small steps, at each step, you move the edge inwards and outwards, creating a bump of an infinitely small distance. We just added an infinite number of bumps to the edge, which have a greater than zero length... What's the new circumference? Is 1/infinity * infinity equal to 1? If so, the new circumference is... ~Pi*Diameter+1??

[Edited by - Hodgman on December 2, 2010 8:37:03 AM]

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