In mathematics, informally speaking, Euclid's orchard is an array of one-dimensional "trees" of unit height planted at the lattice points in one quadrant of a square lattice. More formally, Euclid's orchard is the set of line segments from (x, y, 0) to (x, y, 1), where x and y are positive integers.
The trees visible from the origin are those at lattice points (x, y, 0), where x and y are coprime, i.e., where the fraction x/y is in reduced form. The name Euclid's orchard is derived from the Euclidean algorithm. If the orchard is projected relative to the origin onto the plane x + y = 1 (or, equivalently, drawn in perspective from a viewpoint at the origin) the tops of the trees form a graph of Thomae's function. The point (x, y, 1) projects to
( x x + y , y x + y , 1 x + y ) . {\displaystyle \left({\frac {x}{x+y}},{\frac {y}{x+y}},{\frac {1}{x+y}}\right).}
The solution to the Basel problem can be used to show that the proportion of points in the n × n {\displaystyle n\times n} grid that have trees on them is approximately 6 π 2 {\displaystyle {\tfrac {6}{\pi ^{2}}}} and that the error of this approximation goes to zero in the limit as n goes to infinity.
See also Opaque forest problem
References
External links Euclid's Orchard, Grade 9-11 activities and problem sheet, Texas Instruments Inc. Project Euler related problem