Quote: Original post by BlueSalamander
When you switch from a finite number of iterations to an infinite number of iterations, the square with cut corners becomes a perfect circle and the perimeter changes from 4 to pi without warning.

No it doesn't. It only appears to be a perfect circle if your concept of a 'perfect' circle is that of a 2 dimensional structure bent around a point of a 2d plane so you can't actually look close enough to see the fine details. A true perfect circle has no width on the actual 'line', it has space outside the circle, space inside the circle, and values that fall exactly between the two, which get truer and truer as you zoom in.

Cutting the corners will ALWAYS produce a jagged edge, and will always have a perimeter of 4. 1/10^10^10^10 of an inch still counts, and they don't magically go away.

The model presented in the OP that Pi = 4 is flawed because they aren't producing a circle, or even a circle like structure as they approach infinity.

HITLAR didn't forget about Pythagoras! He squared the circle and circled the square and invaded Cube Earth!

Quote: Original post by Way WalkerThat is, why the procedure that yields 3.141... is correct and the one that yields 4 isn't. The only explanation you have is that you knew beforehand that 3.141... is the correct answer, but by what procedure did you come up with that number? How did you decide that that procedure yielded the correct answer?

I don't. I just expect Pi to relate the area, radius and circumference, and two approximations of Pi (which is all we're talking about) can be compared to each other by how well they accomplish that.

If I try to use Pi to get the area from the radius, than figures of 3.14, 3.1 or even 3 all give closer answers than 4.

While you'd be correct pointing out that under my model there are situations where Pi=4 would be "good enough", it is still easy to show that the "Bible says Pi is 3" approach gives a quantifiably superior result. 3 is a closer approximation to Pi than 4, and we know that by trying to use it in any application that demands Pi.

Hmm, somehow I don't doubt that this one would confound Archimedes significantly. It's a disconcerting result.

BlueSalamander's link is the only thing in this thread that really comes close to addressing the paradox. As Eelco has been insisting, the only truly acceptable answer is one born of thorough mathematical rigour: a principled and formal approach which pins down this vague notion of circumference in perfect logical detail, and leaves absolutely no room for conflicting interpretation - lest one be ostracised and cast into the depraved wilderness of pseodumathematics.

JoeCooper: you've touched on some significant notions in parts, but you speak it like a philosophy student, and (seem to) get frustrated when others remain unconvinced of the ideas you attempt to convey (kind-of like a philosophy student >_> ). You've given no good reason why a valid "iterative" approximation of a quantity should necessarily converge (you said "reduce"?) indefinitely to some limiting value, as opposed, to say, starting and remaining at the correct value through all steps; and you haven't explained why the relationship between a circle's circumference and it's area must hold, independent of the results called into question by the paradox.


Anyway, I'll try to formalize this once I've had some sleep, but I expect to say something about metric, Euclidean distance vs the way they do things in Manhattan, and/or something along those lines. For now, I'm not really satisfied that this paradox has been addressed.

Quote: Original post by Hodgman
Is 1/infinity * infinity equal to 1?

God no.

I don't get the argument. It seems fairly obvious to me that as you "remove corners" to inifity it's the area that is converging to be the same area as that of the circle, not the diameter. No matter how small you make the steps, they're still steps.

You can easily draw any sort of shape you want with the same area as that of another shape, but they both can have very different perimeters. A square or any other shape other than a circle doesn't have a "diameter" from which to calculate the area, so trying to calculate its perimeter using a non-existant construct (what's the diameter of a square?) obviously won't work. "Removing corners" doesn't turn a shape with straight edges into a circle with a diameter, no matter how many times you do it.

*Edited to straighten out my own thoughts.

Quote: Original post by Fenrisulvur

Quote: Original post by Hodgman
Is 1/infinity * infinity equal to 1?

God no.

I think someone forgot to use their parentheses. I think Hodgman was asking if (1 / infinity) * infinity was equal to 1, not 1 / (infinity * infinity).

Quote: Original post by Mantear
I think someone forgot to use their parentheses. I think Hodgman was asking if (1 / infinity) * infinity was equal to 1, not 1 / (infinity * infinity).

Eh, I maintain that the inline division operator is an obscure and ambiguous notation.

Anyway, that's indeterminate, and it's not clear just skimming Hodgman's post where he derived the form from.

Quote: Original post by Fenrisulvur

Quote: Original post by Hodgman
Is 1/infinity * infinity equal to 1?

God no.

I think he meant the (1/infinity)*infinity.

crap I was late...

Quote: you speak it like a philosophy student, and (seem to) get frustrated when others remain unconvinced of the ideas you attempt to convey (kind-of like a philosophy student >_> )

Sorry. Incidentally, I'm surrounded by arts & humanities folk and I've never gotten along well with other programmers.

Go figure.

I'm not trying to troll anyone, I'm just having a hard time communicating here.

[Edited by - JoeCooper on December 2, 2010 4:08:34 PM]

Quote: Original post by Fenrisulvur

Quote: Original post by Mantear
I think someone forgot to use their parentheses. I think Hodgman was asking if (1 / infinity) * infinity was equal to 1, not 1 / (infinity * infinity).

Eh, I maintain that the inline division operator is an obscure and ambiguous notation.

Anyway, that's indeterminate, and it's not clear just skimming Hodgman's post where he derived the form from.

I'm assuming that "1/inf" is the smallest value that is still greater than zero.
The "squiggly circle", or "reduced square circle" has an infinite number of bumps on it's surface, each of which adds an infinitely small (but greater than zero) length to the circumference.
length of bump = 1/inf
number of bumps = inf
extra length added by bumps = (1/inf)*inf
perimeter of this "circle" = Pi*D + (1/inf)*inf

If we let D be 1, we know Pi to be 3.1415... and we know that the perimeter of this bumpy circle to be 4, then we can assume that (1/inf)*inf is equal to 4 - 3.1415..., or 0.8584073464102067615373566167205... which is also known as the trollface constant.

I'm glad that's resolved.