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 (*i*, *j*, 0) to (*i*, *j*, 1), where *i* and *j* are positive integers.

The trees visible from the origin are those at lattice points (*m*, *n*, 0), where *m* and *n* are coprime, i.e., where the fraction *m*/*n* 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 (*m*, *n*, 1) projects to

(
m
m
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,
n
m
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,
1
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n
)
.