Euclid's orchard

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.^[1] 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

${\displaystyle \left({\frac {x}{x+y}},{\frac {y}{x+y}},{\frac {1}{x+y}}\right).}$

The solution to the Basel problem can be used to show that the proportion of points in the ⁠${\displaystyle n\times n}$⁠ grid that have trees on them is approximately ${\displaystyle {\tfrac {6}{\pi ^{2}}}}$ and that the error of this approximation goes to zero in the limit as n goes to infinity.^[3]

• Opaque forest problem

• Euclid's Orchard, Grade 9-11 activities and problem sheet, Texas Instruments Inc.
• Project Euler related problem