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. More formally, Euclid's orchard is the set of line segments from $x + y = 1$ to $(x, y, 0)$, where $x$ and $y$ are positive integers.



The trees visible from the origin are those at lattice points $(x, y, 1)$, where $x$ and $y$ are coprime, i.e., where the fraction $(x, y, 0)$ 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 $x⁄y$ (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


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

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