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Paradoxes

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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The Jamalam7 years ago
The next sentence is a lie. I'm lying.
The Jamalam7 years ago
Go back in time and kill yourself? Amy winehouse going back and killing jabba the hutt? Instructables destroyed and me building knex guns now?
lemonie8 years ago
Here's an apparent paradox:
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 :-/
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 :-)
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