Ford circle

English right|thumb|Some Ford circles Etymology Named after American mathematician Lester R. Ford Lester Randolph Ford, Sr., who wrote about them in 1938. Noun en geometry 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). en Every Ford circle is tangent to the horizontal axis , and any two Ford circles are either disjoint or meet at a tangent. en There is a unique Ford circle associated with every rational number. Additionally, the axis can be considered a Ford circle with infinite radius, corresponding to the case . en 1949 w:American Journal of Mathematics 71 w:Johns Hopkins University Press https://books.google.com.au/books?id=1y3yAAAAMAAJ&q=%22Ford+circle%22%7C%22Ford+circles%22&dq=%22Ford+circle%22%7C%22Ford+circles%22&hl=en&sa=X&ved=2ahUKEwjTqvGn1IbrAhVaT30KHf1lDEkQ6AEwD3oECDIQAg 413 Thus the Ford circle [1], drawn tangent to the real axis at , and having radius , must contain in its interior some points belonging to , such as whose imaginary part lies between and . en 2008 Jan Manschot Partition Functions for Supersymmetric Black Holes w:Amsterdam University Press https://books.google.com.au/books?id=pbnd6IutVwIC&pg=PA78&dq=%22Ford+circle%22%7C%22Ford+circles%22&hl=en&sa=X&ved=2ahUKEwjTqvGn1IbrAhVaT30KHf1lDEkQ6AEwAXoECEwQAg#v=onepage&q=%22Ford%20circle%22%7C%22Ford%20circles%22&f=false 78 A Farey fraction defines a Ford circle in . Its center is given by and its radius is . Two Ford circles and are tangent whenever . This is the case for Ford circles related to consecutive Farey fractions in a sequence . en 2016 Ian Short; Mairi Walker Even-Integer Continued Fractions and the Farey Tree Jozef Širáň; Robert Jajcay Symmetries in Graphs, Maps, and Polytopes: 5th SIGMAP Workshop Springer https://books.google.com.au/books?id=HarWCwAAQBAJ&pg=PA298&dq=%22Ford+circle%22%7C%22Ford+circles%22&hl=en&sa=X&ved=2ahUKEwjTqvGn1IbrAhVaT30KHf1lDEkQ6AEwAHoECAEQAg#v=onepage&q=%22Ford%20circle%22%7C%22Ford%20circles%22&f=false 298 Ford circles are a collection of horocycles in 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 and are tangent if and only if . Translations any of a class of circles, each unique to a rational number and pairwise either tangent to or disjoint from each other circle German: de Ford-Kreis m Further reading Farey sequence Ford Circle on MathWorld Wolfram MathWorld en Circle Fractals Infinity en eponyms