Ford Circle

Ford circles up to q = 20

Farey series Ford circles of the fifth order

The Ford circles are circles in the real plane, one for each rational number and one for the point infinite. The circles are named after the American mathematician Lester R. Ford , who discovered them in 1938.

definition

The Ford circle to break with coprime , whole numbers and is usually referred to with or . He has for the radius and its center is in the point . In addition, the Ford circle is defined as the straight line ( projectively , this is a circle with its center at infinity). ${\displaystyle \textstyle {\frac {p}{q}}}$ ${\displaystyle p,q}$ ${\displaystyle q\geq 0}$ ${\displaystyle C[p/q]}$ ${\displaystyle C[p,q]}$ ${\displaystyle q\not =0}$ ${\displaystyle \textstyle {\frac {1}{2q^{2}}}}$ ${\displaystyle \textstyle \left({\frac {p}{q}},{\frac {1}{2q^{2}}}\right)}$ ${\displaystyle C[1,0]}$ ${\displaystyle y=1}$

Properties of the Ford circles

The interior of two different Ford circles is disjoint , i.e. H. the circles do not overlap. However, they can touch. In addition, every rational point on the x-axis is touched by a Ford circle.

If the fraction is in the open interval , then the Ford circles touching correspond to the neighbors of in a Farey series . ${\displaystyle \textstyle {\frac {p}{q}}}$ ${\displaystyle (0;1)}$ ${\displaystyle C[p/q]}$ ${\displaystyle \textstyle {\frac {p}{q}}}$

Ford Bullets (3D)

A generalization results with Gaussian numbers p = p '+ ip' 'and q = q' + iq ''. The division of two complex numbers by integer coefficients results in rational coefficients: With integers | q | ² = q '* q' + q '' * q '', n '= p' * q '+ p' '* q' 'and n' '= p''q'-p'q' 'the quotient can be written as p / q = (n' + in '') / | q | ². For all integers p ',' p '', q ', q' 'with coprime p, q spheres with radius r = at the point ((p / q)', (p / q) '', r) caused Ford balls . ${\displaystyle \textstyle {\frac {1}{2|q|^{2}}}}$

Two balls and are tangent exactly when . ${\displaystyle P/Q}$ ${\displaystyle p/q}$ ${\displaystyle |Pq-pQ|=1}$

Ford spheres over the complex plane

See also

• Star Brocot Tree

literature

• John H. Conway , Richard K. Guy : Number magic - of natural, imaginary and other numbers . Birkhäuser Verlag 1997. (Original: The Book of Numbers , New York 1996, ISBN 0-387-97993-X )

Individual evidence

1. ^ LR Ford: Fractions . In: The American Mathematical Monthly . tape 45 , no. November 9 , 1938, ISSN  0002-9890 , pp. 586-601 , doi : 10.1080 / 00029890.1938.11990863 ( tandfonline.com [accessed May 19, 2020]).