Euclid's orchard

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 Template:Math.

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 Template:Math to Template:Math, where Template:Mvar and Template:Mvar are positive integers.

One corner of Euclid's orchard Template:Nowrap 2-dimensional, the proportion of trees being visible from the origin (blue) tends towards Template:Nowrap

Perspective view of Euclid's orchard from the origin. Red trees denote rows two off the main diagonal.

The trees visible from the origin are those at lattice points Template:Math, where Template:Mvar and Template:Mvar are coprime, i.e., where the fraction Template:Math 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 Template:Math (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 Template:Math projects to

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

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

See also

• Opaque forest problem

External links

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

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