Subtitle: Math Dorkiness.
So a couple of days ago I was staring at tile and noticed something funny. If you trace out a square with sides of length L, then the distance traveled from the top left corner to the bottom right corner is 2L (step 1). Now, if you again start in the top left, go 1/2 L to the right, 1/2 L down, 1/2 L to the right, 1/2 L down, then again the total distance traveled is 2L (step 2). Continue this process as many times as you want and the distance traveled is always 2L. In the limit of an infinite number of iterations each vertex gets within an epsilon ball of (e.g., can become arbitrarily close to…) a straight line connecting the top left corner with the bottom right. But we all know that this straight line has a length of sqrt(2)*L. So how is it that every point can be arbitrarily close to a straight line but the total length be larger than the straight line?
I originally thought that through calculus or some well-known series limit, I could prove that in the limit of infinite iterations of this process, the pythagorean theorem would be recovered. I failed and felt bad that I couldn’t do calculus anymore. However, a few days later I was watching a Nova and episode and realized that I had invented a fractal (doubtfully the first one to discover this particular fractal, but google couldn’t find a name for it).
Yay math!









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