Everything about Ford Circle totally explained
In
mathematics, a
Ford circle is a
circle with
centre at (
p/q, 1/(2
q2)) and
radius 1/(2
q2), where
p/q is an
irreducible fraction - a fraction in its lowest terms, where
p and
q are
coprime integers).
History
Ford circles are named after American mathematician
Lester R. Ford, Sr., who described them in an article in
American Mathematical Monthly in
1938, volume 45, number 9, pages 586-601.
Properties
The Ford circle associated with the fraction
p/
q is denoted by C[
p/
q] or C[
p,
q]. There is a Ford circle associated with every
rational number. In addition, the line
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,
q = 0.
Two different Ford circles are either
disjoint or
tangent to one another. No two interiors of Ford circles intersect - even though there's a Ford circle tangent to the
x-axis at each point on it with
rational co-ordinates. If
p/
q is between 0 and 1, the Ford circles that are tangent to C[
p/
q] are precisely those associated with the fractions that are the neighbours of
p/
q in some
Farey sequence.
Ford circles can also be thought of as curves in the
complex plane. The
modular group of transformations of the complex plane maps Ford circles to other Ford circles.
By interpreting the upper half of the complex plane as a model of the
hyperbolic plane (the Poincaré half-plane model) Ford circles can also be interpreted as a
tiling of the hyperbolic plane by
horocycles. Any two Ford circles are
congruent in
hyperbolic geometry. If C[
p/
q] and C[
r/
s] are tangent Ford circles, then the half-circle joining (
p/
q, 0) and (
r/
s, 0) that's perpendicular to the
x-axis is a hyperbolic line that also passes through the point where the two circles are tangent to one another.
Ford circles are a sub-set of the circles in the
Apollonian gasket generated by the lines
y = 0 and
y = 1 and the circle C[0/1].
Total area of Ford circles
There is a link between the area of Ford circles,
Euler's totient function and the
Riemann zeta function.
As no two Ford circles intersect, it follows immediately that the total area of the Ford circles
This sum was discussed on
es.ciencia.matematicas..
Further Information
Get more info on 'Ford Circle'.
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