Ford Circle
Pick any two Integers 
and 
, then the Circle of Radius 
centered at 
is known as a Ford circle. No matter what and how many s and s 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 thecenters of the circles with 
and 
,
 | (1) |
be the sum of the radii | (2) |
 | (3) |
, so 
and the distance between circle centers is 
the sum of the CircleRadii, with equality (and therefore tangency) Iff 
. Ford circles are related to theFarey Sequence (Conway and Guy 1996).See also Adjacent Fraction, Farey Sequence, Stern-Brocot Tree
References
Conway, J. H. and Guy, R. K. ``Farey Fractions and Ford Circles.'' The Book of Numbers. New York: Springer-Verlag, pp. 152-154, 1996. Ford, L. R. ``Fractions.'' Amer. Math. Monthly 45, 586-601, 1938. Pickover, C. A. ``Fractal Milkshakes and Infinite Archery.'' Ch. 14 in Keys to Infinity. New York: W. H. Freeman, pp. 117-125, 1995. Rademacher, H. Higher Mathematics from an Elementary Point of View. Boston, MA: Birkhäuser, 1983.
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