Editing Farey sequence (section)

===Ford circles===
[[File:Comparison_Ford_circles_Farey_diagram.svg|thumb|250px|link={{filepath:Comparison Ford circles Farey diagram.svg}}|Comparison of Ford circles and a Farey diagram with circular arcs for ''n'' from 1 to 9. Each arc intersects its corresponding circles at right angles. In [[Media:Comparison Ford circles Farey diagram.svg|the SVG image]], hover over a circle or curve to highlight it and its terms.]]

There is a connection between Farey sequence and [[Ford circle]]s.

For every fraction {{math|{{sfrac|''p''|''q''}}}} (in its lowest terms) there is a Ford circle {{math|''C''[''p''/''q'']}}, which is the circle with radius <math>\tfrac{1}{2q^2}</math> and centre at <math>\bigl(\tfrac{p}{q}, \tfrac{1}{2q^2}\bigr).</math> Two Ford circles for different fractions are either [[Disjoint sets|disjoint]] or they are [[tangent]] to one another—two Ford circles never intersect. If {{math|0 < {{sfrac|''p''|''q''}} < 1}} then the Ford circles that are tangent to {{math|''C''[''p''/''q'']}} are precisely the Ford circles for fractions that are neighbours of {{math|{{sfrac|''p''|''q''}}}} in some Farey sequence.

Thus {{math|''C''[2/5]}} is tangent to {{math|''C''[1/2]}}, {{math|''C''[1/3]}}, {{math|''C''[3/7]}}, {{math|''C''[3/8]}}, etc.

Ford circles appear also in the [[Apollonian gasket]] {{math|(0,0,1,1)}}. The picture below illustrates this together with Farey resonance lines.<ref name=Tomas2020>{{cite arXiv |last=Tomas |first=Rogelio |eprint=2006.10661|title=Imperfections and corrections |class= physics.acc-ph|year=2020 <!-- |access-date=4 July 2020 --> }}</ref>

[[File:Apolloinan gasket Farey.png|650px|thumb|center|[[Apollonian gasket]] {{math|(0,0,1,1)}} and the Farey resonance diagram.]]