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Ford Circle

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FordCircles

Pick any two relatively prime integers h and k, then the circle C(h,k) of radius 1/(2k^2) centered at (h/k,+/-1/(2k^2)) is known as a Ford circle. No matter what and how many hs and ks are picked, none of the Ford circles intersect (and all are tangent to the x-axis). This can be seen by examining the squared distance between the centers of the circles with (h,k) and (h^',k^'),

d^2=((h^')/(k^')-h/k)^2+(1/(2k^('2))-1/(2k^2))^2.
(1)

Let s be the sum of the radii

s=r_1+r_2=1/(2k^2)+1/(2k^('2)),
(2)

then

d^2-s^2=((h^'k-hk^')^2-1)/(k^2k^('2)).
(3)

But (h^'k-k^'h)^2>=1, so d^2-s^2>=0 and the distance between circle centers is >= the sum of the circle radii, with equality (and therefore tangency) iff |h^'k-k^'h|=1. Ford circles are related to the Farey sequence (Conway and Guy 1996).

FordCirclesIntersection

If h_1/k_1, h_2/k_2, and h_3/k_3 are three consecutive terms in a Farey sequence, then the circles C(h_1,k_1) and C(h_2,k_2) are tangent at

alpha_1=((h_2)/(k_2)-(k_1)/(k_2(k_2^2+k_1^2)),1/(k_2^2+k_1^2))
(4)

and the circles C(h_2,k_2) and C(h_3,k_3) intersect in

alpha_2=((h_2)/(k_2)+(k_3)/(k_2(k_2^2+k_3^2)),1/(k_2^2+k_3^2)).
(5)

Moreover, alpha_1 lies on the circumference of the semicircle with diameter (h_1/k_1,0)-(h_2/k_2,0) and alpha_2 lies on the circumference of the semicircle with diameter (h_2/k_2,0)-(h_3/k_3,0) (Apostol 1997, p. 101).

See also

Adjacent Fraction, Apollonian Gasket, Farey Sequence, Stern-Brocot Tree

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References

Apostol, T. M. "Ford Circles." §5.5 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 99-102, 1997.Conway, J. H. and Guy, R. K. "Farey Fractions and Ford Circles." The Book of Numbers. New York: Springer-Verlag, pp. 152-154, 1996.Ford, L. R. "Fractions." Amer. Math. Monthly 45, 586-601, 1938.Pickover, C. A. "Fractal Milkshakes and Infinite Archery." Ch. 14 in Keys to Infinity. New York: W. H. Freeman, pp. 117-125, 1995.Rademacher, H. Higher Mathematics from an Elementary Point of View. Boston, MA: Birkhäuser, 1983.

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Ford Circle

Cite this as:

Weisstein, Eric W. "Ford Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FordCircle.html

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