Ford Circle
المؤلف:
Conway, J. H. and Guy, R. K
المصدر:
"Farey Fractions and Ford Circles." The Book of Numbers. New York: Springer-Verlag
الجزء والصفحة:
...
23-10-2019
2348
Ford Circle
Pick any two relatively prime integers 
and 
, then the circle 
of radius 
centered at 
is known as a Ford circle. No matter what and how many 
s and 
s 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 
and 
,
 |
(1)
|
Let 
be the sum of the radii
 |
(2)
|
then
 |
(3)
|
But 
, so 
and the distance between circle centers is 
the sum of the circle radii, with equality (and therefore tangency) iff 
. Ford circles are related to the Farey sequence (Conway and Guy 1996).
If 
, 
, and 
are three consecutive terms in a Farey sequence, then the circles 
and 
are tangent at
 |
(4)
|
and the circles 
and 
intersect in
 |
(5)
|
Moreover, 
lies on the circumference of the semicircle with diameter 
and 
lies on the circumference of the semicircle with diameter 
(Apostol 1997, p. 101).
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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