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
Therefore,
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 LinesExplore with Wolfram|Alpha
More things to try:
References
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. https://www.mathematicaguidebooks.org/.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 31, 1986.Yaglom, A. M. and Yaglom, I. M. Challenging Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover, pp. 102-106, 1987.Referenced on Wolfram|Alpha
Circle Division by LinesCite this as:
Weisstein, Eric W. "Circle Division by Lines." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CircleDivisionbyLines.html