Ford circle
English
![](Images/wiktionary/Ford_circles_colour.svg.png.webp)
Some Ford circles
Etymology
Named after American mathematician Lester Randolph Ford, Sr., who wrote about them in 1938.
Noun
Ford circle (plural Ford circles)
- (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).
- Every Ford circle is tangent to the horizontal axis , 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 can be considered a Ford circle with infinite radius, corresponding to the case .
- 1949, American Journal of Mathematics, Volume 71, Johns Hopkins University Press, page 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 .
- 2008, Jan Manschot, Partition Functions for Supersymmetric Black Holes, Amsterdam University Press, page 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 .
- 2016, Ian Short, Mairi Walker, Even-Integer Continued Fractions and the Farey Tree, Jozef Širáň, Robert Jajcay (editors), Symmetries in Graphs, Maps, and Polytopes: 5th SIGMAP Workshop, Springer, page 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
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Further reading
Farey sequence on Wikipedia.Wikipedia
- Ford Circle on Wolfram MathWorld