In mathematics, a Ford circle is a circle in the Euclidean plane, in a family of circles that are all tangent to the
x
{\displaystyle x}
-axis at rational points. For each rational number
p
/
q
{\displaystyle p/q}
, expressed in lowest terms, there is a Ford circle whose center is at the point
(
p
/
q
,
1
/
(
2
q
2
)
)
{\displaystyle (p/q,1/(2q^{2}))}
and whose radius is
1
/
(
2
q
2
)
{\displaystyle 1/(2q^{2})}
. It is tangent to the
x
{\displaystyle x}
-axis at its bottom point,
(
p
/
q
,
0
)
{\displaystyle (p/q,0)}
. The two Ford circles for rational numbers
p
/
q
{\displaystyle p/q}
and
r
/
s
{\displaystyle r/s}
(both in lowest terms) are tangent circles when
|
p
s
−
q
r
|
=
1
{\displaystyle |ps-qr|=1}
and otherwise these two circles are disjoint.
Contents
History
Ford circles are a special case of mutually tangent circles; the base line can be thought of as a circle with infinite radius. Systems of mutually tangent circles were studied by Apollonius of Perga, after whom the problem of Apollonius and the Apollonian gasket are named. In the 17th century René Descartes discovered Descartes' theorem, a relationship between the reciprocals of the radii of mutually tangent circles.
Ford circles also appear in the Sangaku (geometrical puzzles) of Japanese mathematics. A typical problem, which is presented on an 1824 tablet in the Gunma Prefecture, covers the relationship of three touching circles with a common tangent. Given the size of the two outer large circles, what is the size of the small circle between them? The answer is equivalent to a Ford circle:
1
r
middle
=
1
r
left
+
1
r
right
.
{\displaystyle {\frac {1}{\sqrt {r_{\text{middle}}}}}={\frac {1}{\sqrt {r_{\text{left}}}}}+{\frac {1}{\sqrt {r_{\text{right}}}}}.}
Ford circles are named after the American mathematician Lester R. Ford, Sr., who wrote about them in 1938.
Properties
The Ford circle associated with the fraction
p
/
q
{\displaystyle p/q}
is denoted by
C
[
p
/
q
]
{\displaystyle C[p/q]}
or
C
[
p
,
q
]
.
{\displaystyle C[p,q].}
There is a Ford circle associated with every rational number. In addition, the line
y
=
1
{\displaystyle y=1}
is counted as a Ford circle – it can be thought of as the Ford circle associated with infinity, which is the case
p
=
1
Total area of Ford circles
There is a link between the area of Ford circles, Euler's totient function
φ
,
{\displaystyle \varphi ,}
the Riemann zeta function
ζ
,
{\displaystyle \zeta ,}
and Apéry's constant
ζ
(
3
)
.
{\displaystyle \zeta (3).}
As no two Ford circles intersect, it follows immediately that the total area of the Ford circles
{
C
[
p
,
q
]
:
0
<
p
q
≤
1
}
Ford spheres (3D)
The concept of Ford circles can be generalized from the rational numbers to the Gaussian rationals, giving Ford spheres. In this construction, the complex numbers are embedded as a plane in a three-dimensional Euclidean space, and for each Gaussian rational point in this plane one constructs a sphere tangent to the plane at that point. For a Gaussian rational represented in lowest terms as
p
/
q
{\displaystyle p/q}
, the diameter of this sphere should be
1
/
2
q
q
¯
{\displaystyle 1/2q{\bar {q}}}
where
q
¯
{\displaystyle {\bar {q}}}
represents the complex conjugate of
q
{\displaystyle q}
. The resulting spheres are tangent for pairs of Gaussian rationals
P
/
Q
{\displaystyle P/Q}
and