In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 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

Square lattice

In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice. It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group is known symbolically as p4m.Two...

. 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

Coprime

In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

, i.e., where the fraction ^m⁄_n is in reduced form

Irreducible fraction

An irreducible fraction is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent vulgar fraction...

. The name Euclid's orchard is derived from the Euclidean algorithm

Euclidean algorithm

In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...

.

If the orchard is projected

3D projection

3D projection is any method of mapping three-dimensional points to a two-dimensional plane. As most current methods for displaying graphical data are based on planar two-dimensional media, the use of this type of projection is widespread, especially in computer graphics, engineering and drafting.-...

 relative to the origin onto the plane x+y=1 (or, equivalently, drawn in perspective

Perspective (graphical)

Perspective in the graphic arts, such as drawing, is an approximate representation, on a flat surface , of an image as it is seen by the eye...

 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

External links

• Euclid's Orchard Grades 9-11 activities and problem sheet, Texas Instruments Inc.