Hatcher-Topology-of-Numbers · Problem 1.2.2

Continued Fractions

Mathematics — Algebraic Topology · ford-circles-circular-farey-diagram

Problem

Draw a figure showing how Ford circles are positioned in a circular Farey diagram by the following procedure. Start with a circle

C

of radius 1 which will be the outer boundary of the Farey diagram. Next, draw two tangent circles of radius

1/2

inside

C

and tangent to

C

at two opposite points of

C

. Label these two tangency points

1/0

and

0/1

. Now continue drawing smaller circles inside

C

with the same tangency patterns as the Ford circles in the upper halfplane Farey diagram, and label the tangency points of these circles with

C

according to the mediant rule. After a number of these circles have been drawn, superimpose the semicircles of the Farey diagram itself.