Paradoxes Answered

Are you taking the limit of zero angle for each zig-zag? If so, I think you've got an array of pseudo-delta-functions with finite peak height, and therefore infinitesimal (read zero) area, but covering the plane. How close to the center do the peaks go? The enclosed area should just be the space strictly interior to the peaks. If I'm wrong, could you post the derivation? Feel free to use LaTeX and a GIF or PDF, since it's much easier and clearer than ASCII math :-/

I used the word apparent because it's not a paradox. Figuring out what's wrong with it is the point, and I think you can see that.

L

Yes, indeed! I wasn't arguing :-) I don't think you're wrong -- a fractal, after all, is a boundary of infinite length which encloses or covers a finite area. I am interested in the computational details, though -- how do you represent the zig-zags, and how are you doing the integration?

I don't - if it's infinite the angles must be zero, so obviating the thickness of any lines drawn or represented the enclosed area must be zero. Or without the zig-zagging infinite. It looks better on paper, but I didn't seek it out and post an image because these things work better for confusion without something to look at. It took me a little while to work it out (but I like thinking). L

But you know as well as I do that limits don't always come out intuitively. I'm interested in how you represent the zig-zag in the finite angle limit (i.e., an N-pointed star). Let's use the inner vertices of the star as the baseline. Give that a radius r, so a circumference of 2(pi)r.

For N points, each point has a base of 2(pi)r/N. In the small-angle limit (appropriate for large N), we can treat the sides as having a fixed, constant length l, equal to the height, and hence an area a = l×2(pi)r/N / 2. The area of the whole star is therefore

A = N×a + 2(pi)r
= N×2(pi)rl/N / 2 + 2(pi)r
= 2(pi)rl/2 + 2(pi)r

So, A = (2+l)(pi)r. Notice that the N cancelled out before we ever took the actual limit, so this result is the answer for your limiting case.

You'd have N as infinity, rather than l - so you get a result. Hmmm, there must be some wonky-maths in here somewhere... L

That's correct. By construction I kept the height of the zig-zigs finite (think of one of those kitschy '60's era wall clocks), but made them narrower and narrower (limit as N -> infinity). The wonky math is precisely in taking that limit :-)

Ah thanks. L

. ROFL

well then how about this: nothing is imposible

Again with the semantic mumb jumbo (to borrow Kelsey's phrase). Or shall we start up again with oxymorons? Okay, my turn. Thoughtful guy! Polite evolution debater! Um...intelligent knexer! Oo, that last one might have been too much...>ducks barrage of non-lethal plastic toys<

Creation science! Oo, that last one might have been too much...>ducks barrage of non-lethal bible verses<

It's not the verses you have to worry about. It's the sharp corners on all of the badly mistranslated books.

And the stones :D

I think Mick and Keith are old enough that you can out-run them if you really have to.

. That answer is not very satisfactory. _{link is for you young punks that didn't get the Mick & Keith reference}

Ummm..you proposed paradoxes, self-contradictory statements or groups of statements, or situations which are logically or physically inconsistent. "Nothing is impossible" is simply a falsehood, at least in this universe.

"With God, all things are possible" I have that on a bookmark somewhere..

"With Ignorance, all things seem possible"

Just a related quote; of absolutely no hold on your particular comment's content.

Due to your explanation i will not flame. What you say is also perfectly true, alot of people through ignorance think things are possible. Me and my faith, is not an example.

Flamebait.

Nope : )

Yeah, bookmarks are always a great source for Truth. But how, exactly, do you write the citation?

But what if I have a pair of entangled bookmarks? Every time I click on the one, the other changes!

Unfortunately, they're only entangled as long as you don't click on them. Once you do, you project them out into orthogonal (or parallel, depending on the specifics of the entanglement) eigenstates, and after that they evolve independently according to the Schrödinger equation.

_{^{You do know you were asking for it, don't you?}}

if nothing is impossible, that means everything is possible, meaning that it is possible to make something that is impossible, but, that would be impossible because everything is possible, except making something impossible even though it's possible because everything is not not possible.

Thank you!

Those stupid "paradoxes" have always annoyed me!

_{>Waits for the inevitable God-rock one<}