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Polygonal Spiral


PolygonalSpiral

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/R=(cot(pi/n))/(csc(pi/n))=cos(pi/n).
(1)

The total length of the spiral for an n-gon with side length s is therefore

L=1/2ssum_(k=0)^(infty)cos^k(pi/n)
(2)
=s/(2[1-cos(pi/n)]).
(3)
PolygonalSpiralSolid

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=1/4s^2cot(pi/n).
(4)

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


See also

Continuous Line Illusion, Nested Polygon, Rep-Tile, Theodorus Spiral

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References

Sandefur, J. T. "Using Self-Similarity to Find Length, Area, and Dimension." Amer. Math. Monthly 103, 107-120, 1996.

Referenced on Wolfram|Alpha

Polygonal Spiral

Cite this as:

Weisstein, Eric W. "Polygonal Spiral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolygonalSpiral.html

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