4.4 The Ford circles 127
4.4 The Ford circles ..................
The Ford circle of a rational numberr= pqis the circle of radius 21 q 2 in
the upper half plane tangent to thex-axis at the point(r,0).
The Ford circles of two distinct rational numbers are either disjoint
or tangent to each other externally. In the latter case we say that two
rational numbers areadjacent.
Theorem 4.1.Two rational numbers are adjacent if and only if they are
consecutive terms in a Farey sequence.
Corresponding to each Farey sequence, there is a sequence of tangent
Ford circles. Here are the Ford circles corresponding toF 5.
0 1514 13 25 12 35 23 3445 1Exercise
1.IfpqandPQare adjacent rational numbers, what is the point of tan-
gency of their Ford circles?