In
mathematics, a Ford circle is a
circle in the
Euclidean plane, in a family of circles that are all tangent to the -axis at
rational points. For each rational number , expressed in lowest terms, there is a Ford circle whose
center is at the point and whose radius is . It is tangent to the -axis at its bottom point, . The two Ford circles for rational numbers and (both in lowest terms) are
tangent circles when and otherwise these two circles are disjoint.[1]
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.[2] In the 17th century
René Descartes discovered
Descartes' theorem, a relationship between the reciprocals of the radii of mutually tangent circles.[2]
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:[3]
Ford circles are named after the American mathematician
Lester R. Ford, Sr., who wrote about them in 1938.[1]
Properties
The Ford circle associated with the fraction is denoted by or There is a Ford circle associated with every
rational number. In addition, the line is counted as a Ford circle – it can be thought of as the Ford circle associated with
infinity, which is the case
Two different Ford circles are either
disjoint or
tangent to one another. No two interiors of Ford circles intersect, even though there is a Ford circle tangent to the
x-axis at each point on it with
rational coordinates. If is between 0 and 1, the Ford circles that are tangent to can be described variously as
the circles associated with the fractions that are the neighbors of in some
Farey sequence,[1] or
the circles where is the next larger or the next smaller ancestor to in the
Stern–Brocot tree or where is the next larger or next smaller ancestor to .[1]
If and are two tangent Ford circles, then the circle through and (the x-coordinates of the centers of the Ford circles) and that is perpendicular to the -axis (whose center is on the x-axis) also passes through the point where the two circles are tangent to one another.
The centers of the Ford circles constitute a discrete (and hence countable) subset of the plane, whose closure is the real axis - an uncountable set.
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.[1]
Ford circles are a sub-set of the circles in the
Apollonian gasket generated by the lines and and the circle [4]
Note that as a matter of convention, the previous calculations excluded the circle of radius corresponding to the fraction . It includes the complete circle for , half of which lies outside the unit interval, hence the sum is still the fraction of the unit square covered by Ford circles.
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 , the diameter of this sphere should be where represents the
complex conjugate of . The resulting spheres are
tangent for pairs of Gaussian rationals and with , and otherwise they do not intersect each other.[6][7]
See also
Apollonian gasket – a fractal with infinite mutually tangential circles in a circle instead of on a line
^Fukagawa, Hidetosi; Pedoe, Dan (1989), Japanese temple geometry problems, Winnipeg, MB: Charles Babbage Research Centre,
ISBN0-919611-21-4,
MR1044556.
^Marszalek, Wieslaw (2012), "Circuits with oscillatory hierarchical Farey sequences and fractal properties", Circuits, Systems and Signal Processing, 31 (4): 1279–1296,
doi:
10.1007/s00034-012-9392-3,
S2CID5447881.