Square division by lines is the problem of counting the regions formed in the interior of a square
by lines. Let 
be a nonnegative integer,
and write each line as 
, where 
and 
. Choose the 
lines independently by normalizing
the invariant measure 
on the lines that meet the
interior of the square. The
expectation value 
of the number of regions is
(Santaló 1976; Finch 2003, p. 481).
The maximum number of regions is the same as for circle
division by lines, namely
For 
, 2, ..., this gives 2, 4, 7, 11, 16, 22, ... (OEIS
A000124). This is also the maximum number of
regions produced by 
lines in the plane
or a disk.
See also
Circle Division by Lines
Explore with Wolfram|Alpha
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,
321-322, 1945. https://doi.org/10.1103/RevModPhys.17.321.Santaló,
L. A. "Sobre la distribución 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. https://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 Resource. https://mathworld.wolfram.com/SquareDivisionbyLines.html
Subject classifications
