TITLE: Klein's j-invariant and Ford circles
QUESTION [8 upvotes]: Klein's j-invariant has structure which seems to resemble Ford circles:

The latter show up all over number theory (continued fractions, Rademacher's expansion for p(n), etc.)
Can someone explain the connection?

REPLY [8 votes]: Ford circles are the orbit of a horocycle under the action of the modular group $\Gamma$ on the upper half plane $\frak h$ (see e.g. this entry of SBS).  Modular forms of weight zero (of which the $j$-invariant is an instance) are fully $\mathrm{SL}_2(\Bbb Z)$-invariant. So the periods of $j$ (representing the quotient $H/\Gamma$) tile the hyperbolic plane (of which $\frak h$ is a model) according to the same symmetry as the Ford circle tiling.