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!
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