Euclid's orchard should not be confused with Orchard-planting problem.
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 to, where and are positive integers.
The trees visible from the origin are those at lattice points, where and are coprime, i.e., where the fraction 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 (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 projects to
\left(
| x | |
| x+y |
,
| y | |
| x+y |
,
| 1 | |
| x+y |
\right).
The solution to the Basel problem can be used to show that the proportion of points in the grid that have trees on them is approximately
\tfrac{6}{\pi2}