Euclid's orchard

In mathematics 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 (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

\left ( \frac {m}{m%2Bn}, \frac {n}{m%2Bn}, \frac {1}{m%2Bn} \right ).

References

  1. ^ Weisstein, Eric W., "Euclid's Orchard" from MathWorld.

External links