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. [MathWorld|urlname=EuclidsOrchard|title=Euclid's Orchard] 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

: $left\; (\; frac\; \{m\}\{m+n\},\; frac\; \{n\}\{m+n\},\; frac\; \{1\}\{m+n\}\; ight\; ).$

Euclid's orchard is mentioned in "Brutus", the 10th episode of .

References

External links

* [http://education.ti.com/educationportal/activityexchange/download_file.jsp?cid=US&fileurl=Math%2FMathGeneral%2F7737%2FAct3_OrchardProblem_Brutus_final.pdf Euclid's Orchard Grades 9-11 activities and problem sheet] , Texas Instruments Inc.