Polya’s Orchard Visibility Problem and Related Questions in Geometry and Number Theory Asilomar - December 2008 Bruce Cohen Lowell High School, SFUSD David Sklar San Francisco State University Ver. 5.00

A result on seeing out Proof using elementary geometry Plan Blichfeldt’s Lemma and a proof of Minkowski’s Theorem Some new results Allen 1986 Hening & Kelly 2005 Kruskal 2008 Bibliography Worksheet Statement of the problem and an exploration of the case A result on not seeing out Proof using Minkowski’s Theorem and elementary geometry Which trees are needed to insure privacy Geometry and number theory The fraction of trees needed as R gets large Probability and Analytic number theory

Worksheet

ANSWER: Their centers are closest to the line from the origin to ANSWER: Those whose center coordinates are not relatively prime. ANSWER:

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Polya’s Orchard Visibility Problem “How thick must the trunks of the trees in a regularly spaced circular forest grow if they are to block completely the view from the center?” Polya’s Formulation of the Problem Let R be a given positive integer. Let every lattice point (p, q) that satisfies the inequality be the center of a circle of radius r. If r is sufficiently small, there exist rays running from the origin to infinity that don’t intersect any of the circles (the forest is transparent); such rays no longer exist if r is sufficiently large. Let be the value of r which divides the two cases (the limit of transparency). Then we have (Polya 1918)

Polya’s Orchard Visibility Problem:

1 (R,1) r Polya’s Lower Bound for  1R 1

Polya’s Upper Bound for  We want to show that if the tree radius r exceeds 1/R then every ray from the origin is blocked by a tree inside the orchard. Our strategy is to chose an arbitrary diameter AB of the orchard r r R The existence of the pair of lattice points follows from a beautiful theorem of Minkowski. and show that if the tree radius r exceeds 1/R then there exists a pair of lattice points in the orchard such that trees (circles) centered at them intersect AB.

Minkowski’s Theorem According to Ross Honsberger (Mathematical Gems I, p42) this theorem is “intuitively obvious, but not logically evident”. Honsberger presents an elegant proof of Minkowski’s theorem based on a Lemma due to Blichfeldt (1914). Any plane region with area greater than n units, n a positive integer, can be translated so that it covers at least n+1 lattice points. Lemma A set is convex if the line segment joining any two points of the set lies entirely inside the set. Recall: Any plane convex region symmetric about the origin, with area greater than 4, contains lattice points other than the origin.

Minkowski’s Theorem -- Examples

Minkowski’s Theorem -- Example

Let AB be an arbitrary diameter of the circular orchard of radius R and Consider the rectangle CDEF with CF tangent to the orchard boundary at A, DE tangent at B, with AC, DB, BE, and FA all of length (1/R) + (  /2). CDEF is a plane convex region symmetric about the origin. r r R

r r R Our proof will be complete if we can show that the points that exist in the rectangle by Minkowski’s theorem actually lie in the orchard and not in the small piece of the rectangle that lies outside of the orchard. The fact that R is an integer plays a key role here. ()

A Proof of Minkowski’s Theorem

to In a February 1986 American Mathematical Monthly article Thomas Tracey Allen (of the Department of Entomology at UC Berkeley) strengthened Polya’s result from In the same article he generalized the problem, allowing the orchard radius R be a positive real number, rather than restricting it’s values to positive integers, and showed that Recent Results – Allen 1986 (Note that in the integer case. Also note that is the square of the distance to the first point with coprime coordinates lying outside the orchard.) where is the first integer greater than that can be written as the sum of squares of coprime integers.

Recent Results – Allen 1986 Example: So 13 is the first integer greater than the can be written as the sum of squares of coprime integers. No

Number Theory Allen’s generalization leads naturally to the question: Which positive integers can be written as the sum of squares of coprime integers? Which, following a dictum of Polya, leads to the easier question: Which positive integers can be written as the sum of squares of at most two integers? Which leads to three, four, five, …? All of which lead to beautiful theorems of classical number theory. Theorem A number can be written as a sum of coprime squares if and only if it is not divisible by any prime congruent to 3 mod 4 and is not divisible by any power of 2 greater than 2 itself. (Fermat circa 1640) 1, 2, 5, 10, 13, 17, 26, 29, 34, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, …

More Recent Results – Hening & Kelly 2005 They include a proof the theorem that a positive integer can be written as a sum of coprime squares if and only if it is not divisible by any prime congruent to 3 mod 4 and is not divisible by any power of 2 greater than 2 itself.

Recent Results – Kruskal 2008 Gives an alternate proof of Allen’s results based on a Stern-Brocot wreath, and generalizes the results to parallelogram lattices.

Fewer Trees -- Same Visibility Counting lattice points with coprime coordinates is easier with square orchards At what fraction of the lattice points do we need trees to block the view out of the orchard from the origin? We know that we only need trees at lattice points with coprime coordinates. What fraction of the lattice points in the orchard have coprime coordinates? As R gets large?

Counting Points with Coprime Coordinates This number is usually denoted and called the Euler phi function of n The number of points in the nth column with coprime coordinates is also the number of positive integers less than or equal to n and relatively prime to n. The number of points in quadrant I with coprime coordinates is about Which by a moderately well known theorem in analytic number theory (Hardy and Wright, 4 th ed, p268) is

The Fraction of Points with Coprime Coordinates “This result may be stated more picturesquely in the language of probability.” (Hardy and Wright, p268) The probability that two randomly chosen integers are relatively prime is

Bibliography