Pi=4 Paradox (Explain This!)

Replies

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  silverscorpion

  Re: Calling all CE's - Explain this!!
  
  Hmm, yeah. Interesting paradox!!
  
  I know of a way to explain this.. I'll wait for other explanations before giving mine!! 😀😀
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  Kaustubh Katdare

  I've a problem with the 'Repeat To Infinity' part. It's still zig-zag and can never be a circle.
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  gohm

  Biggie is correct
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  silverscorpion

  Yes. This arises due to the difference between countably infinite and uncountably infinite points..
  
  In case of removing corners from a square, even after infinite iterations, only a finite number of points of the square will be on the circle.
  But a circle has infinite number of points.
  
  So, even at infinity, the square with corners removed will approach a circle, but never rally become a circle.
  
  So, we can say that after infinite iterations, maybe, the area of the two curves are equal. ie., the square converges to a curve with the same area as the circle, but their perimeters are still different.
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  makru921

  Biggie, repeating to infinity means considering that a circle is made of zig zag lines with infinitesimally small length. We can counter the paradox by stating that a circle has infinite points and after infinite iterations the area might become equal to that of the square, like silverscorpion did. However it is quite possible to imagine that a circle is formed by such very small lines. In that case, the paradox wins! I tried to convince myself that this is impossible but the thought of that possibility remains.. It is really scary!
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  Kaustubh Katdare

  Consider this -
  
  1) /\/\/\/\/\/\/\/\/\/\/..... (infinity)
  2) ------------------.....(infinity) [Consider this as a continuous line 😁 ]
  
  The first zig-zag line is almost twice (just an approximation) that of straight line #2 for any section cut at equal distance from the start point.
  
  If I make the first line small enough, still maintaining the zig-zag nature; the length will still be twice that of line 2. That's exactly I meant. Even if you go about infinity - it will NEVER be a 'straight' line; thus these two lines can never be of the same length.