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just a place to discuss paradoxes (an immpossible statement) like The statement to the left is false the statement to the left is true

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You have an infinite line, which is a closed-loop, like a circle.
It's all scrunched up in zigzags \/\/\/\/\/\/\/\/\/\/ but really close together, more like |||||||||||||||||. Within the middle of the loop/circle there is an area which can be calculated - the peaks and troughs average out.

Infinite border - finite area?

L

L

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 :-/

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.

No, that's just nonsense. The naïve paradox (with correct punctuation and words) is

The statement to the right is false. The statement to the left is true.

Statements like that are just semantic mumbo jumbo. Far more interesting are the true logical (mathematical) paradoxes, such as Gödel's Indecidability Theorem.

Modern physics (both quantum mechanics and relativity) are replete with apparent paradoxes (such as entanglement) which arise from our attempts to interpret results in a classical way.

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.