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Polygonal Spiral

\begin{figure}\begin{center}\BoxedEPSF{PolygonalSpiral.epsf scaled 1000}\end{center}\end{figure}

The length of the polygonal spiral is found by noting that the ratio of Inradius to Circumradius of a regular Polygon of $n$ sides is

{r\over R}={\cot\left({\pi\over n}\right)\over\csc\left({\pi\over n}\right)} = \cos\left({\pi\over n}\right).
\end{displaymath} (1)

The total length of the spiral for an $n$-gon with side length $s$ is therefore
L={\textstyle{1\over 2}}s\sum_{k=0}^\infty \cos^k\left({\pi\...
...ght)={s\over 2\left[{1-\cos\left({\pi\over n}\right)}\right]}.
\end{displaymath} (2)


Consider the solid region obtained by filling in subsequent triangles which the spiral encloses. The Area of this region, illustrated above for $n$-gons of side length $s$, is

A={\textstyle{1\over 4}}s^2\cot\left({\pi\over n}\right).
\end{displaymath} (3)


Sandefur, J. T. ``Using Self-Similarity to Find Length, Area, and Dimension.'' Amer. Math. Monthly 103, 107-120, 1996.

© 1996-9 Eric W. Weisstein