FORD CIRCLESKatelyn JessieState University of New York at Potsdamjessiek196@potsdam.eduHudson River Undergraduate Mathematics ConferenceApril 11, 2015

HOROCIRCLEA circle C in the plane that is tangent to the

real axis at a point p, and that otherwise lies in the upper half-plane

Has base point p and radius r We denote the radius of C by rad[C]

0

+

p

(p,r)r C

HOROCIRCLETwo horocircles with radii r and s, and distinct points x and y, are tangent iff

r

x

d

y

s

Assume the circles are tangent

(r + s) =

( =( + ( = 4rs +

) = 4rs + -2rs +

2rs = - 2rs = +

4rs = d = r + s

FORD CIRCLESThe Ford Circle of a rational number x = is

the horocircle with unique radius and at base point x

Claim: Two ford circles and , where x = and y = , are tangent iff

Proof of claim

x = y =

r = s =

= 1

CONTINUED FRACTIONSA continued fraction is an expression of the

form +

is an integer, are positive integers The sequence , , , …. is finite or infinite

Let =

Then we have = = + =

We define integers , , , …and positive integers , , , … by the matrix recurrence relation

= for n 2

We are given = =

We see that = =

Then for any n 2 =

So = * for any n 2

=

But = = = 1

So for any n, = 1

Therefore when we take the determinants in these equations we see that = 1

CONTINUED FRACTIONSThe value of a finite continued fraction is the

final term and the value of an infinite continued fraction is the limit of the sequence

INFINITE CONTINUED FRACTION1 + x =

x = 1+

= 2 = = 1.6153

= = 1.5 = = 1.6190

= = 1.6666 = = 1.6176

= = 1.6 = = 1.6181

= = 1.625 = = 1.6179

CONVERGENT OF The rationals are known as the convergents of

< < < … < < … < < <

The convergents of can alternate from one side of to the other

INFINITE CONTINUED FRACTION CONT.By using the quadratic equation we find that

x = 1+ x = = x +1 x = -x -1 = 0

Note: Since we are dealing with positive integers the golden ratio will converge to positive integers

CONVERGENTS OF INFINITE CONTINUED FRACTION

< < < … < < … < < <

= x = = 1.61803…

< < < < … < < … < < < <

1.5 < 1.6 < 1.615 < 1.6176 < 1.6179 < … < 1.61803 < … < 1.6181 < 1.619 < 1.625 <1.66 < 2

BEST APPROXIMATION OF A rational is a best approximation of

provided that, for each rational such that d b, we have

with equality iff =

THEOREM 1.1A rational x that is not an integer is a

convergent of a real number iff it is a best approximation of

Theorem 1.1 is a classic and very important result that is used by many. In the article “Ford Circles, Continued Fractions, and Rational Approximation”, Ian Short gives a geometric proof based on the theory of Ford Circles

We define the continued fraction chain of a real number to be the sequence of Ford Circles , , , …

where are the convergents of

Since = 1, we see that any two consecutive Ford circles in the continued fraction chain of are tangent

Given a rational x = and a real , we define () = =

as the radius of the unique horocircle with base point that is tangent to Cx

Note: (x) = 0

= s = ()

THEOREM 1.2 Let be a real number. Given a rational x

that is not an integer, the following are equivalent:

1. x is a convergent of ;2. is a member of the continued fraction

chain of ;3. X is a best approximation of ;4. If z is a rational such that rad [] rad []

then () (), with equality iff z = x

Lemma 2.1: Given a rational x = , 1. if < then () < (); 2. if z is a rational distinct from x then rad[] (z), with equality, iff and are tangent

Lemma 2.2: Let and be tangential Ford Circles. If a rational z lies strictly between x and y then has smaller radius than both and

Lemma 2.3: Let and be tangential Ford Circles such that rad[] > rad[], and suppose that a real number lies strictly between x and y, and a rational z lies strictly outside the interval bounded by x and y. Then () < ()

Corollary 2.4: Let and be tangential Ford Circle such that rad[] > rad[], and suppose that a real number lies strictly between x and y. If z is a rational such that rad[] rad[], then () (), with equality, iff z = x

PROOF OF THEOREM 1.2 Statements 1 and 2 are equivalent by the

definition of a continued fraction chain

Statements 3 and 4 can be seen to be equivalent using () = = , and the fact that the radius of is

Statements 1 and 4 are equivalent using Lemmas and Corollary 2.4

Email: jessiek196@potsdam.edu References:

Irwin, M. C. (1989). Geometry of Continued Fractions. The American Mathematical Monthly, 696-703.

Short, I. (2011). Ford Circles, Continued Fractions, and Rational Approximation. The American

Mathematical Monthly, 130-135.