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Square Division by Lines

The average number of regions N(n) into which n lines divide a square is

N^_(n)=1/(16)n(n-1)pi+n+1

(Santaló 1976; Finch 2003, p. 481).

SquareDivisionbyLines

The maximum number of sequences is presumably the same as for circle division by lines, namely

N_(max)(n)=1/2(n^2+n+2).

For n=1, 2, ..., this gives 2, 4, 7, 11, 16, 22, ... (OEIS A000124), which is the same number into which the plane, a circle, etc. can be divided.

See also

Circle Division by Lines

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References

Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Goudsmit, S. "Random Distribution of Lines in a Plane." Rev. Mod. Phys. 17, 231-322, 1945.Santaló, L. A. "Sobre la distrabucion de planos en el espacio." Rev. Unión Mat. Argentina 13, 120-124, 1948.Santaló, L. A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976.Sloane, N. J. A. Sequence A000124/M1041 in "The On-Line Encyclopedia of Integer Sequences."Trott, M. "Coloring Closed Curves." §1.6 in The Mathematica GuideBook for Graphics. New York: Springer-Verlag, pp. 393-430, 2004. http://www.mathematicaguidebooks.org/.

Referenced on Wolfram|Alpha

Square Division by Lines

Cite this as:

Weisstein, Eric W. "Square Division by Lines." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SquareDivisionbyLines.html

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