You are currently browsing the tag archive for the ‘ford circles’ tag.

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.

Screen Shot 2018-05-26 at 7.42.08 PM

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

Screen Shot 2018-05-26 at 7.51.54 PM

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

Circle 1:

Screen Shot 2018-05-26 at 7.51.57 PM

has centre:

Screen Shot 2018-05-26 at 7.50.54 PM

and radius:

Screen Shot 2018-05-26 at 7.50.58 PM

Circle 2:

Screen Shot 2018-05-26 at 7.53.22 PM

has centre:

Screen Shot 2018-05-26 at 7.53.28 PM

and radius:

Screen Shot 2018-05-26 at 7.53.31 PM

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:

Screen Shot 2018-05-26 at 4.29.09 PM

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 Fsequence.  So do other consecutive terms in the Farey sequence also generate circles touching at a point?

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

Screen Shot 2018-05-26 at 8.02.27 PM

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

Screen Shot 2018-05-26 at 8.10.26 PM

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

Screen Shot 2018-05-26 at 8.15.35 PM

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

Website Stats

  • 5,594,997 views

Recent Posts

Follow IB Maths Resources from British International School Phuket on WordPress.com