Pick any two relatively prime integers h and k, then the circle C(h, k) of radius 1/(2k^2) centered at (h/k, ± 1/(2k^2)) is known as a Ford circle. No matter what and how many hs and ks are picked, none of the Ford circles intersect (and all are tangent to the x-axis). This can be seen by examining the squared distance between the centers of the circles with (h, k) and (h', k'), d^2 = (h'/k' - h/k)^2 + (1/(2k^(, 2)) - 1/(2k^2))^2.

adjacent fraction | Apollonian gasket | Farey sequence | Stern-Brocot tree