In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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
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, denoted as . It is one of the five types of two-dimensional lattices as classified by their ...
. More formally, Euclid's orchard is the set of line segments from to , where and are positive
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s.
The trees visible from the origin are those at lattice points , where and are
coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
, i.e., where the fraction is in
reduced form In statistics, and particularly in econometrics, the reduced form of a simultaneous equations model, system of equations is the result of solving the system for the endogenous variables. This gives the latter as functions of the exogenous variables, ...
. The name ''Euclid's orchard'' is derived from the
Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is a ...
.
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
:
The solution to the
Basel problem
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
can be used to show that the proportion of points in the grid that have trees on them is approximately
and that the error of this approximation goes to zero in the
limit as goes to infinity.
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
orchard
An orchard is an intentional plantation of trees or shrubs that is maintained for food production. Orchards comprise fruit tree, fruit- or nut (fruit), nut-producing trees that are generally grown for commercial production. Orchards are also so ...
Lattice points
{{math-hist-stub