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  to , where i and j are positive integers.

The trees visible from the origin are those at lattice points , where m and n 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

See also
Opaque forest problem

References

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

Category:Greek_mathematics
Category:Lattice points