Pi=4 Paradox (Explain This!)
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Makes any sense?? How do you explain this? Anyway it is interesting 😀😀
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Makes any sense?? How do you explain this? Anyway it is interesting 😀😀Replies
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Member • Dec 6, 2010
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!! 😀😀Are you sure? This action cannot be undone. -
Administrator • Dec 6, 2010
I've a problem with the 'Repeat To Infinity' part. It's still zig-zag and can never be a circle.Are you sure? This action cannot be undone. -
Member • Dec 6, 2010
Biggie is correctAre you sure? This action cannot be undone. -
Member • Dec 6, 2010
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.Are you sure? This action cannot be undone. -
Member • Dec 6, 2010
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!Are you sure? This action cannot be undone. -
Administrator • Dec 6, 2010
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.Are you sure? This action cannot be undone.