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The length of the polygonal spiral is found by noting that the ratio of inradius to circumradius of a regular polygon of sides is

$ \frac{r}{R}= \frac{\cot(\frac{180}{n})}{\csc(\frac{180}{n})}= \cos(\frac{180}{n}) $

The total length of the spiral for an n-gon with side length is therefore $ J= \frac{s}{2[1-\cos(\frac{180}{n})]} $

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 , is $ A= \frac{s^2}{4}\cot(\frac{180}{n}) $

The shaded triangular polygonal spiral is a rep-4-tile.

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