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## Farey Sequences and Ford Circles

With thanks to 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: \(\frac{0}{1}, \frac{1}{1} \).

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! This last would be referred to as the Farey Sequence of Order 3.So many patterns! So much to explore...

Plot these

Farey numberson 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 theFord Circles!

AND... If you plot the numerators against the denominators of a, then you get theFarey Sequenceamazing(or justFarey Starburst: Try fareyStar(n)star(n)!)

Exploration 1: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 denominators! \(\frac{a+c}{b+d}\). The resulting fraction is called the

mediantof \(\frac{a}{b}\) and \(\frac{c}{d}\).Be warned, though, that this term may not appear in the next sequence, or even the one after that...

Exploration 2:What do you notice when youcross-multiplynumerators and denominators of pairs of Farey neighbours? (If \(\frac{a}{b}\) and \(\frac{c}{d} \)are consecutive terms of a Farey Sequence, consider \(b*c\) and \(a*d\)...)

Exploration 3:Just for interest... you might consider theof each Farey Sequence.count

How many terms make up the Farey Sequence of Order 1,Use? Order 2,Farey(1)? Order 3,Farey(2)? etc.Farey(3)(or justfareyCount(n)fcount).

Can you see a pattern? (But be careful... first impressions might prove deceiving!)

Exploration 4:And finally, what do you notice about the continued fractions of \(\frac{\sqrt{5}+2n-1}{2}\)? (These are calledNoble Numbers).Some more interesting Noble Numbers may be formed using

tau(\(\tau: \frac{1}{\phi}\)), the reciprocal ofphi(\(\phi\)), the Golden Ratio: For example, \(\frac{\tau}{1+3\tau}\), but what about \(\frac{2+5\tau}{5+12\tau}\)?To better understand the continued fractions of

Noble(and evenNear-Noble Numbers) you may need to delve a little further into Farey Sequences and Ford Circles! In terms of Noble Numbers, you might note that 0 and 1/3 areFarey neighboursinFarey(3), while 2/5 and 5/12 live next door to each other inFarey(12).

Peek behind the scenes...

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About this Utility

Keep playing. Keep exploring. Have fun!

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