Meaning of Ford circle | Babel Free

Definitions

Any one of a class of circles with centre at (p/q, 1/(2q²)) and radius 1/(2q²), where p/q is an irreducible fraction (i.e., p and q are coprime integers).

Equivalents

Examples

“Every Ford circle is tangent to the horizontal axis y#61;0, and any two Ford circles are either disjoint or meet at a tangent.”

“There is a unique Ford circle associated with every rational number. Additionally, the axis y#61;0 can be considered a Ford circle with infinite radius, corresponding to the case p#61;1,#92;q#61;0.”

“Thus the Ford circle [1], drawn tangent to the real axis at #92;textstyle#123;p#95;n#125;#47;#123;q#95;n#125;, and having radius #92;textstylea#95;#123;n#43;1#125;#123;-1#125;q#95;n#123;-2#125;, must contain in its interior some points belonging to #92;textstyle#123;p#95;n#125;#47;#123;q#95;n#125;, such as #92;textstyle#92;theta#43;ia#95;#123;n#43;1#125;#123;-1#125;q#95;n#123;-2#125; whose imaginary part lies between #92;textstyleq#95;n#123;-2#125; and #92;textstyleq#95;#123;n#43;1#125;#123;-2#125;.”

“A Farey fraction #92;textstyle#92;frackc defines a Ford circle #92;textstyle#92;mathcal#123;C#125;(k,c) in #92;textstyle#92;mathbbC. Its center is given by #92;textstylekc#43;i#92;frac 1#123;2c²#125; and its radius is #92;textstyle#92;frac1#123;2c²#125;. Two Ford circles #92;textstyle#92;mathcalC(a,b) and #92;textstyle#92;mathcalC(c,d) are tangent whenever #92;textstylead-bc#61;#92;pm 1. This is the case for Ford circles related to consecutive Farey fractions in a sequence #92;textstyleF#95;N.”

“Ford circles are a collection of horocycles in #92;mathbbH used by Ford to study continued fractions in papers such as [2, 3].[…]Two Ford circles intersect in at most a single point, and the interiors of the two circles are disjoint. In fact, one can check that the Ford circles C#95;#123;a#47;b#125; and C#95;#123;c#47;d#125; are tangent if and only if #92;vertad-bc#92;vert#61;1.”

CEFR level