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.

Plan view of one corner of Euclid's orchard, in which trees are labelled with the x co-ordinate of their projection on the plane x + y = 1 .
One corner of Euclid's orchard, blue trees visible from the origin
Perspective view of Euclid's orchard from the origin. Red trees denote rows two off the main diagonal.