Circle Division by Lines


Determining the maximum number of pieces in which it is possible to divide a circle for a given number of cuts is called the circle cutting or pancake cutting problem. The minimum number is always n+1, where n is the number of cuts, and it is always possible to obtain any number of pieces between the minimum and maximum. The first cut creates 2 regions, and the nth cut creates n new regions, so




Evaluating for n=1, 2, ... gives 2, 4, 7, 11, 16, 22, ... (OEIS A000124). This is equivalent to the maximal number of regions into which a plane can be cut by n lines.

See also

Circle Division by Chords, Plane Division by Circles, Plane Division by Ellipses, Plane Division by Lines, Space Division by Planes, Space Division by Spheres, Square Division by Lines

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Santaló, L. A. "Sobre la distrabucion de planos en el espacio." Rev. Unión Mat. Argentina 13, 120-124, 1948.Sloane, N. J. A. Sequence A000124/M1041 in "The On-Line Encyclopedia of Integer Sequences."Sloane, N. J. A. and Plouffe, S. Figure M1041 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.Trott, M. "Coloring Closed Curves." §1.6 in The Mathematica GuideBook for Graphics. New York: Springer-Verlag, pp. 393-430, 2004., A. M. and Yaglom, I. M. Challenging Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover, pp. 102-106, 1987.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 31, 1986.

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Circle Division by Lines

Cite this as:

Weisstein, Eric W. "Circle Division by Lines." From MathWorld--A Wolfram Web Resource.

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