Assuming paint molecules are infinitely small, how much paint do you need to fill the volume of Gabriel's Horn? What happens when you pour the paint out? Have you coated the entire inside surface area?
(Do not reference the Wikipedia explanation, which I found after asking this question)
We were reviewing stuff in my AP Calculus BC class at my high school (we're preparing for the AP Test).
@naraku: Theoretical paint molecules are infinitely small in this case.
You're right that you can coat the inside. With infinitely small paint molecules, an infinite number of molecules would fill the finite volume, and with infinite paint molecules you have enough molecules to coat the infinite inside surface area and even turn it inside out.
Wikipedia's solution is that the paint would have to get thinner infinitely, and it also says it can't all be with the same thickness. This is, of course, not with theoretical infinitely small paint molecules.
Oh, and after you filled it, you'd have to make sure it didn't create a black hole, being infinitely dense and all.
You're right, that's my mistake. Just because they're infinitely small doesn't mean the spacing between them is too - I guess that complicates my explanation quite a bit...in fact I think my whole explanation is wrong because of that. I'll do some thinking.
I just looked up Gabriel's horn. It looks like a 3d version of the Dirac delta function. The Dirac delta function is a basic principle in calculus which says that the integral (area) under 1 dimensional line is 1. We use that in communication theory quite a bit when representing a simple sinusoid in the frequency domain.
> The Dirac delta function is a basic principle in calculus
¿? it's a really weird function, f(x) = 0, \forall x not_eq 0 and the integral in all the domain is 1
> when representing a simple sinusoid in the frequency domain
I suppose that you use the discrete version
Hmm... Well, OP, it would appear that you answered your own question. If the paint is infinitely small, then the question does not apply. If the paint is not infinitely small, then it simply cannot fill the entirety of the horn.
So, since the question appears to have been answered, I feel it to be appropriate to pose a new question- what is the sum where n goes from 1 to infinity of 1/n^{2}? Preferably without trying to look up the solution on, say, Wikipedia. That takes all of the mathematics out of the mathematics- knowing the solution beforehand defeats the point.
ne555, it doesn't have to be discrete. The theoretical or analogue stuff can be done continuous. It's only when you go digital that you need to take the discrete FTs.
Ah, heh, yeah. So, it was when n started at 1 - I was remembering it backwards. Still pretty good seeing as I didn't even do anything in my head, I just happened to have done it before and remembered the interesting result because of ho rare it is to see Pi squared.
Well, do you remember how to get to that result? There are several ways not using any calculus or other advanced concepts, but the method I find easiest to understand is the one using Fourier series.
Just go in a loop, when the difference in calculation between this iteration and last iteration is less than your tolerance, stop. NUMERICAL METHODS :)
1 2 3 4 5 6 7 8
double answer = 0, prev = 0;
for (int i = 1; true ; ++i)
{
prev = answer;
answer += 1.0 / ( (double)( i * i ) );
if (abs(answer - prev) < tolerance )
break;
}