Please show me the flaw in this.

The problem itself; clicky, and some discussion about it; clicky.

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.

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).

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.

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.

Quote: 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?

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]

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) :-)

The limit curve is not a circle.

Is it even differentiable at any point?

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]