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## Ford Circles and Farey Sequences

By Dan McKinnon: Please see the blog post here.

Suppose you wanted to make a list of all the rational numbers between 0 and 1. How would you start?

One approach might be to begin with the denominators of the fractions: first, list those with denominator 1 - zero and 1.

Next, add those with denominator 2: \(\frac{0}{1}, \frac{1}{2}, \frac{1}{1} \).

Add those with denominator 3 ( \(\frac{0}{1}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{1}{1} \)) and you are on your way to the

Farey Sequence!So many patterns! So much to explore...

For example: suppose you wanted to know the next term of the sequence between two values, \(\frac{a}{b}\) and \(\frac{c}{d}\)?

Easy! Just add the numerators and the deniminators! \(\frac{a+c}{b+d}\)

Now plot these Farey Numbers on the number line: for each \(\frac{a}{b}\) draw a circle of radius \(\frac{1}{2b^2}\) and centre \(\left(\frac{a}{b},\frac{1}{2b^2}\right)\). These are the

Ford Circlesthat you see above!Now try plotting the numerators of a Farey Sequence against the denominators. The result (when reflected in the axes and the diagonals) is a Farey Sunburst!

Keep playing. Keep exploring. Have fun!

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