## 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 statement to the left is false the statement to the left is true

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active| newest | oldestapparentparadox:You have an

infiniteline, which is a closed-loop, like a circle.It's all scrunched up in zigzags \/\/\/\/\/\/\/\/\/\/ but

reallyclose together, more like |||||||||||||||||. Within the middle of the loop/circle there is an areawhich can be calculated- the peaks and troughs average out.Infinite border - finite area?

L

L

apparentbecause it's not a paradox. Figuring out what's wrong with it is the point, and I think you can see that.L

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 areaa= l×2(pi)r/N / 2. The area of the whole star is thereforeA = 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

isthe answer for your limiting case.wonky mathis precisely in taking that limit :-)