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Problem of the Day #59: Infinite Semicircular Arcs May 17, 2011

Posted by Alex in : potd , trackback

Let image $1$ be the figure formed by drawing four unique and non-overlapping circular arcs, each with a diameter that is a side of a square with side length $1$. Let image $i + 1$ be the figure formed when, for each semicircular arc in image $i$, two new unique semicircular arcs are drawn, each with one point at the end of the original arc and the second point at the midpoint of the arc. See the images below for examples. In each image, the “outside perimeter” is represented by solid lines.

Find the outside perimeter of figure $22$.
Alternatively, find the outside perimeter of figure $i$ as $i$ approaches $\infty$, or prove that the answer does not exist.


Figures:


Image $1$


Image $2$


Image $3$

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