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In Mathematics , a Ford circle is a Circle with Centre at (''p/q'', 1/(2''q''2)) and Radius 1/(2''q''2), 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 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 is 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 {Link without Title} 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. Any two Ford circles are Congruent in Hyperbolic Geometry . If C and C[''r''/''s'' are tangent Ford circles, then the half-circle joining (''p''/''q'', 0) and (''r''/''s'', 0) that is 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 {Link without Title} . SEE ALSO EXTERNAL LINKS
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