Ford circle

English

Some Ford circles

Etymology

Named after American mathematician Lester Randolph Ford, Sr., who wrote about them in 1938.

Noun

Ford circle (plural Ford circles)

1. (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 $y=0$, 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 $y=0$ can be considered a Ford circle with infinite radius, corresponding to the case $p=1,\ q=0$.

   • 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 $\textstyle {p_{n}}/{q_{n}}$, and having radius $\textstyle a_{n+1}^{-1}q_{n}^{-2}$, must contain in its interior some points belonging to $\textstyle {p_{n}}/{q_{n}}$, such as $\textstyle \theta +ia_{n+1}^{-1}q_{n}^{-2}$ whose imaginary part lies between $\textstyle q_{n}^{-2}$ and $\textstyle q_{n+1}^{-2}$.

   • 2008, Jan Manschot, Partition Functions for Supersymmetric Black Holes, Amsterdam University Press, page 78:

     A Farey fraction $\textstyle {\frac {k}{c}}$ defines a Ford circle $\textstyle {\mathcal {C}}(k,c)$ in $\textstyle \mathbb {C}$. Its center is given by $\textstyle kc+i{\frac {1}{2c^{2}}}$ and its radius is $\textstyle {\frac {1}{2c^{2}}}$. Two Ford circles $\textstyle {\mathcal {C}}(a,b)$ and $\textstyle {\mathcal {C}}(c,d)$ are tangent whenever $\textstyle ad-bc=\pm 1$. This is the case for Ford circles related to consecutive Farey fractions in a sequence $\textstyle F_{N}$.

   • 2016, Ian Short, Mairi Walker, “Even-Integer Continued Fractions and the Farey Tree”, in 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 $\mathbb {H}$ 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 $C_{a/b}$ and $C_{c/d}$ are tangent if and only if $\vert ad-bc\vert =1$.

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: Ford-Kreis m

Further reading

• Farey sequence on Wikipedia.Wikipedia
• Ford circle on Wikipedia.Wikipedia
• Ford Circle on Wolfram MathWorld