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Disclaimer: This might seem like a very dumb question. But I rather ask it and learn than to never ask at all and always wonder. Also, to me it is confusing, you guys are probably much more advance in math than I am.

Let's say you have a "zig-zag step" pattern, you can show that the the total length of the diagonal zig-zag step is 20 for a 10 in by 10 in width and height. As shown in the picture. But as you make the "zig-zag" smaller and smaller the "zig-zag" total should still remain 20, but instead it seems to be $10\sqrt{2}$. I get this it is a triangle now and this is the formula for the diagonal of the triangle, but if the diagonal can really be made of this zig-zag step pattern where the zig-zag are infinitely small, shouldn't it be 20 by that logic?

I get the Pythagorean theorem and everything, but in terms of calculus a the zig zag steps can turn into a triangle by making the length of an individual zig-zag infinitely small, making infinitely many zig-zags. So why is it $10\sqrt{2}$, over $20$.

enter image description here

Dean Bearl
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  • What does this have to do with calculus? There is a calculus tag? – Sterling Night Jan 01 '16 at 01:20
  • @SterlingNight It has to do with limits and making the zig zag steps infinitely small. So, it has to do with calculus, since it is dealing with a limit that is approaching 0 or the number of zig-zag step is approaching infinity. – Dean Bearl Jan 01 '16 at 01:22
  • @MattSamuel Yeah, but shouldn't the total distance of the zig-zag steps be equal to the straight line if the zig-zag steps become infinitely small. Ex:$\lim :_{x\to :0}\left(LengthOfZigZag\left(x\right)\right)=DistanceOfStraight$, where LengthOfZigZag is a function that gives the total length of the LengthOfZigZag based on how many zig-zag exist. – Dean Bearl Jan 01 '16 at 01:28
  • I think your premise is incorrect. You do not get the proper perimeter by successively squaring the edge. You get the area that way, but not the edge. A line/arc integral would be a more appropriate analogy. – johnnyb Jan 01 '16 at 08:24

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