Ford Circles

This carries on the previous investigation into Farey sequences, and is again based on the current Nrich task Ford Circles.  Below are the Farey sequences for F₂, F₃ and F₄. You can read about Farey sequences in the previous post.

This time I’m going to explore the link between Farey sequences and circles.  First we need the general equation for a circle:

This has centre (p,q) and radius r.  Therefore

Circle 1:

has centre:

and radius:

Circle 2:

has centre:

and radius:

Now we can plot these circles in Geogebra – and look for the values of a,b,c,d which lead to the circles touching at a point.

When a = 1, b = 2, c = 2, d = 3:

Do we notice anything about the numbers a/b and c/d ?  a/b = 1/2 and c/d = 2/3 ?  These are consecutive terms in  the F₃ sequence.  So do other consecutive terms in the Farey sequence also generate circles touching at a point?

a = 1, b = 1, c = 2, d = 3

Again we can see that the fractions 1/1 and 2/3 are consecutive terms in the F₃ sequence. So by drawing some more circle we can graphically represent all the fractions in the F₃ sequence:

So these four circles represent the four non-zero fractions of in the F₃ sequence!

and this is the visual representation of the non-zero fractions of in the F₄ sequence.  Amazing!