1. FORD CIRCLES
Katelyn Jessie
State University of New York at Potsdam
jessiek196@potsdam.edu
Hudson River Undergraduate Mathematics Conference
April 11, 2015
2. HOROCIRCLE
A 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
(x − p)2
+ (y − r)2
= r2
p
(p,r)
r C
5. FORD CIRCLES
The Ford Circle 𝐶 𝑥 of a rational number x =
𝑎
𝑏
is the
horocircle with unique radius
1
2𝑏2 and at base point x
Claim: Two ford circles
Cx and Cy, where x =
a
b
and y =
c
d
, are tangent
iff ad − bc = 1
6. Proof of claim
x =
a
b
y =
c
d
r =
1
2b2 s =
1
2d2
x − y 2 = 4rs
a
b
−
c
d
2
= 4(
1
2b2
)(
1
2d2
)
ad
bd
−
bc
bd
2
= 4(
1
4b2d2
)
(ad − bc)2
b2d2
=
1
b2d2
(ad − bc)2 = 1
7. CONTINUED FRACTIONS
A continued fraction is an expression of the form
b0 +
1
b1+
1
b2+
1
b3+ …
b0 is an integer, bi are positive integers
The sequence b0, b1, b2, …. is finite or infinite
Let
𝐴 𝑛
𝐵 𝑛
= 𝑏0+
1
𝑏1+⋯
8. Then we have
A0
B0
= b0
A1
B1
= b0+
1
b1
=
b0b1 +1
b1
We define integers A0, A1, A2, …and positive integers B0,
B1, B2, … by the matrix recurrence relation
An An−1
Bn Bn−1
=
An−1 An−2
Bn−1 Bn−2
bn 1
1 0
for n 2
9. We are given
A0
B0
= b0
A1
B1
=
b0b1 +1
b1
We see that
An An−1
Bn Bn−1
=
A1 A0
B1 B0
=
b0b1 +1 b0
b1 1
Then for any n 2
An An−1
Bn Bn−1
=
An−1 An−2
Bn−1 Bn−2
bn 1
1 0
10. So
An An−1
Bn Bn−1
=
An−1 An−2
Bn−1 Bn−2
*
bn 1
1 0
for any n 2
An An−1
Bn Bn−1
=
An−1 An−2
Bn−1 Bn−2
But =
A1 A0
B1 B0
=
b0b1 +1 b0
b1 1
= 1
11. So for any n,
An An−1
Bn Bn−1
= 1
Therefore when we take the determinants in these
equations we see that AnBn−1 − An−1Bn = 1
12. CONTINUED FRACTIONS
The value of a finite continued fraction is the final term
𝐴 𝑁
𝐵 𝑁
and the value of an infinite continued fraction is the limit
of the sequence
𝐴 𝑛
𝐵 𝑛
15. CONVERGENT OF
The rationals
𝐴 𝑛
𝐵 𝑛
are known as the convergents of
A0
B0
<
A2
B2
<
A4
B4
< … < < … <
A5
B5
<
A3
B3
<
A1
B1
The convergents of can alternate from one side of
to the other
16. INFINITE CONTINUED FRACTION CONT.
By using the quadratic equation we find that
x = 1+
1
x
x =
−b b2−4ac
2a
x2 = x +1 x =
1+ 5
2
x2
-x -1 = 0
Note: Since we are dealing
with positive integers
the golden ratio will
converge to positive
integers
18. BEST APPROXIMATION OF
A rational
𝑎
𝑏
is a best approximation of provided that, for
each rational
𝑐
𝑑
such that d b, we have
bα − a dα − c with equality iff
𝑐
𝑑
=
𝑎
𝑏
19. THEOREM 1.1
A 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 𝐶 𝐴0
𝐵0
, 𝐶 𝐴1
𝐵1
, 𝐶 𝐴2
𝐵2
, … where
𝐴 𝑛
𝐵 𝑛
are the convergents of
20. Since AnBn−1 − An−1Bn = 1, we see that any two
consecutive Ford circles in the continued fraction
chain of are tangent
21. Given a rational x =
𝑎
𝑏
and a real , we define
𝑅 𝑥() =
1
2
bα − a 2 =
b2
2
α − x 2
as the radius of the unique horocircle with base point
that is tangent to Cx
Note: Rx(x) = 0
x − y 2 = 4rs
− 𝑥 2 = 4(
1
2b2
)s
− 𝑥 2 = (
2
b2
)s
b2
2
− 𝑥 2 = s = Rx()
22. 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. Cx 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 [Cz] rad [Cx]
then
Rx() Rz(), with equality iff z = x
23. Lemma 2.1: Given a rational x =
a
b
,
1. if α − x < − x then Rx() < Rx();
2. if z is a rational distinct from x then rad[Cz] Rx(z),
with equality, iff Cz and Cx are tangent
Lemma 2.2: Let Cx and Cx be tangential Ford Circles. If a rational z lies
strictly between x and y then Cx has smaller radius than both Cx and Cx
Lemma 2.3: Let Cx and Cx be tangential Ford Circles such that rad[Cz] >
rad[Cz], 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 Rx() <
Rx()
Corollary 2.4: Let Cx and Cx be tangential Ford Circle such that rad[Cx] >
rad[Cy], and suppose that a real number lies strictly between x and y. If z is a
rational such that rad[Cz] rad[Cx], then Rx() Rz(), with equality,
iff z = x
24. 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 Rx() =
1
2
bα − a 2 =
b2
2
α − x 2, and the fact
that the radius of C 𝑎
b
is
1
2b2
Statements 1 and 4 are equivalent using Lemmas
and Corollary 2.4
25. 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.
Editor's Notes
Ford Circles and how they are applied to best rational approximations of irrationals.
Define a Horocircle
The blue line labeled d is the distance between the two radii of circle x and circle y
In order for the two circles to be tangent, the distance must equal the sum of the two radii: r+s
If d > r+s then the circles will be disjoint
If d < r+s then the circles will overlap
Here I have the formula for the distance, d
Now plug in r+s for d
After computing some algebra, I have found that (x-y)^2 = r+s
When computing in reverse, I substituted 4rs for (x-y)^2 into the original distance equation
After computing some algebra, I was able to prove that d = r+s
Therefore, we can conclude that the two circles are tangent //
Define Ford Circles
Make claim
First, state my variables
Next, use the equation that states that two circles are tangent: 4rs = (x-y)^2
Substitute variables into the equation and I find that (ad-bc)^2 = 1
Therefore we can conclude that the two ford circles are tangent.//
Define Continued Fractions
So, when we take the determinant of the original recurrence relation, we see that the determinant (bn,1,1,0) = 1
Therefore, the determinant of the first matrix is equivalent to the determinant of the second matrix by the product of the determinant=the determinant of the product definition
But, the determinant of the first = the determinant of the second = 1
Therefore, we can conclude that the matrix (An, An-1, Bn, Bn-1) =1 and that the determinant of the matrix is =1.//
Here is an example of an infinite continued fraction, and we are going to call it X
I found that x = 1+1/x
I then found all An/Bn for n=1,2,3,…,10
Define Converget of alpha
Using the quadratic equation, I found that x = 1+sqrt(5)/2
Since we are dealing with positive integers, x must be positive and the golden ratio must be 1+sqrt(5)/2
Now, using the equation of alternating convergents, we must let alpha = 1+sqrt(5)/2
We then see that the convergence of the golden ratio alternates
Define Best Approximation of Alpha
State Theorem
This theorem is a classic and very important theorem that is used by many mathematicians. It is applied in many different ways either algebraically or by using plane lattices. Many people like Ian Short sue it geometrically based on the theory of Ford Circles.
Theorem jumps right into the first definition of continued fraction chains
Use the equation from a few slides back that stated |AnBn-1 – An-1Bn| = 1
In my second proof, I showed that two ford circles are tangent if and only iff |ad-bc| = 1
This is the second, and last, definition of Theorem 1.1
By using the equation from my first proof |x-y|^2 = 4rs that states that two horocircles are tangents
We can conclude that a ford circle is a horocircle.//
Our main theorem is the following:
Statement 2 is a geometric reformation of 1
4 is a geometric reformation of 3
The equivalence of 1 and 3 yields thrm 1.1
To prove this theorem, Ian Short uses some lemmas and a corollary
2.1 gives basic properties of the function Rx
2.2 and 2.3 lemmas about basic properties of Ford Circles
2.4 proves that ford circle Cx and the rational in 2.3 satisfy statement 4
If z is a rational such that rad [ 𝐶 𝑧 ] rad [ 𝐶 𝑥 ] then R x () R z (), with equality iff z = x
Unfortunately, I do not have enough time to go into detail about this proof
I have provided the names of the articles if you wish to read more about Ford Circles, and to look further into the proof of Theorem 1.2
If you
CURRENT EMAIL:
katenick@roadrunner.com
jessik@rpi.edu
![HOROCIRCLE
A 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
(x − p)2
+ (y − r)2
= r2
p
(p,r)
r C](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)