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Nested Polygon


NestedPolygon

Beautiful patterns can be created by drawing sets of nested polygons such that the incircle of the nth polygon is the circumcircle of the (n+1)st and successive polygons are rotated one half-turn at each iteration. The results are shown above for nested triangles through heptagons in alternating black and white and in a cyclic rainbow coloring.

Nested octagon animation

The animation above shows successive iterations of a nested octagon.

The black region of a nested square has area

 A=sum_(n=0)^infty((-1)^n)/(2^n)=2/3

if the initial square has unit edge length.


See also

Circular Chessboard, Continuous Line Illusion, Derived Polygon, Midpoint Polygon, Polygon Circumscribing, Polygon Inscribing, Polygonal Spiral, Tangential Polygon, Whirl

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References

Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 165-166, 1984.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, p. 97, 2002.

Referenced on Wolfram|Alpha

Nested Polygon

Cite this as:

Weisstein, Eric W. "Nested Polygon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NestedPolygon.html

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