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
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, WhirlExplore with Wolfram|Alpha
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 PolygonCite this as:
Weisstein, Eric W. "Nested Polygon." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NestedPolygon.html