Polyas Orchard Visibility Problem

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Polyas Orchard Visibility Problem and Related
Questions in Geometry and Number Theory
Bruce Cohen Lowell High School,
SFUSD bic_at_cgl.ucsf.edu http//www.cgl.ucsf.edu/hom
e/bic
David Sklar San Francisco State
University dsklar_at_sfsu.edu
Asilomar - December 2008
Ver. 5.00
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Plan
Some new results Allen 1986 Hening
Kelly 2005 Kruskal 2008
A result on seeing out Proof using elementary
geometry
Which trees are needed to insure privacy
Geometry and number theory
A result on not seeing out Proof using
Minkowskis Theorem and elementary geometry
The fraction of trees needed as R gets large
Probability and Analytic number theory
Blichfeldts Lemma and a proof of Minkowskis
Theorem
Bibliography
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Worksheet
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Worksheet
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?
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Polyas 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?
Polyas Formulation of the Problem
(Polya 1918)
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Polyas Lower Bound for r
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Polyas Upper Bound for r
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
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.
The existence of the pair of lattice points
follows from a beautiful theorem of Minkowski.
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Minkowskis Theorem
According to Ross Honsberger (Mathematical
Gems I, p42) this theorem is intuitively
obvious, but not logically evident.

Honsberger presents
an elegant proof of Minkowskis theorem based on
a Lemma due to Blichfeldt (1914).
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Minkowskis Theorem -- Examples
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Minkowskis Theorem -- Example
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Minkowskis Theorem -- Example
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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) (e/2). CDEF is a plane convex
region symmetric about the origin.
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Our proof will be complete if we can show that
the points that exist in the rectangle by
Minkowskis 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.
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A Proof of Minkowskis Theorem
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Recent Results Allen 1986
In a February 1986 American Mathematical Monthly
article Thomas Tracey Allen (of the Department of
Entomology at UC Berkeley) strengthened Polyas
result from
to
In the same article he generalized the problem,
allowing the orchard radius R be a positive
real number, rather than restricting its values
to positive integers, and showed that
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Recent Results Allen 1986
No
No
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Number Theory
Allens 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,
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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.
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Recent Results Kruskal 2008
Gives an alternate proof of Allens results based
on a Stern-Brocot wreath, and generalizes the
results to parallelogram lattices.
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Fewer Trees -- Same Visibility
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 lattice points with coprime coordinates
is easier with square orchards
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Counting Points with Coprime Coordinates
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.
Which by a moderately well known theorem in
analytic number theory (Hardy and Wright, 4th ed,
p268) is
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The Fraction of Points with Coprime Coordinates
This result may be stated more picturesquely in
the language of probability. (Hardy and
Wright, p268)
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Bibliography
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Bibliography