英和和英

語源

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

名詞

Ford circle (複数形 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 かつ q are coprime integers).

   Every Ford circle is tangent to the horizontal axis ${\displaystyle y=0}$, 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 ${\displaystyle y=0}$ can be considered a Ford circle with infinite radius, corresponding to the case ${\displaystyle p=1,\ q=0}$.

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

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

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

   • 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 ${\displaystyle \mathbb {H} }$ 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 ${\displaystyle C_{a/b}}$ and ${\displaystyle C_{c/d}}$ are tangent if and only if ${\displaystyle \vert ad-bc\vert =1}$.

Further reading

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