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Plane Division by Circles

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PlaneDivisionbyCircles

Plane division by circles is the partition of the plane into regions by an arrangement of n circles, where n is a nonnegative integer. For n>=1, the maximal number of regions is

N(n)=n^2-n+2,

giving values for n=1, 2, ... of 2, 4, 8, 14, 22, 32, 44, 58, ... (OEIS A014206).

The maximum is attained when each pair of circles intersects in two distinct points and no three circles pass through the same point. For n>=2, the nth circle then adds 2(n-1) regions. With no circles, N(0)=1.

See also

Arrangement, Circle, Circle Division by Lines, Plane Division by Ellipses, Plane Division by Lines, Space Division by Spheres

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References

Indiana School Mathematics J. 14, No. 4, p. 4, 1979.Konhauser, J. D. E.; Velleman, D.; and Wagon, S. Which Way Did the Bicycle Go? And Other Intriguing Mathematical Mysteries. Washington, DC: Math. Assoc. Amer., p. 177, 1996.Problem Q736. Parabola 24, 22, 1988.Sloane, N. J. A. Sequence A014206 in "The On-Line Encyclopedia of Integer Sequences."Yaglom, A. M. and Yaglom, I. M. Challenging Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover, pp. 102-106, 1987.

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Plane Division by Circles

Cite this as:

Weisstein, Eric W. "Plane Division by Circles." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PlaneDivisionbyCircles.html

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