## Fairy Tales

##### By Robin Kirkham and Sinje Bulter

Who was John Farey?

John Farey was born 1766 in Woburn, Bedfordshire, England and died in 1826 in London, England.

Although Farey was a geologist, not a mathematician but he was known in mathematics for the “Farey Series” that is discussed in the work.

John Farey’s mathematics was recognized when wrote an article called “On a curious property of vulgar fractions.”   Today vulgar fractions are referred to as Improper Fractions.

The article was merely four paragraphs long yet it changed the way Farey was perceived.  In his first paragraph he talked about how he noted curious properties with fractions and by the end of merely four paragraphs he had introduced these curious properties.

These properties, defined as the Farey series, is when a given number n, is considered then all rational numbers between 0 and 1 when expressed in their lowest terms, have denominators that do not exceed the n.

Each member of the sequence is equal to the rational whose numerator is the sum of the numerators of the fractions on either side, and whose denominator is the sum of the denominators of the fractions on either side.

For example:

### Fractically any Farey Tale

take n = 5

F5= [, , , , , , , , , , ]

Is this just another Farey Tale?

# Farey Tales Rolling By ….

#### Or do they?

Fairy Tales come to life when they are beautifully illustrated.  The following is an illustration of the Farey series invented by Lester R. Ford, Ford’s Touching Circles.

Farey Trails – another path along the way….

Now let us take a look at the progression of the Farey Series in relationship to the actual order.  View the circles as they go

F1 F2 F3 F4 F5 F4 F3 F2 F1

This Farey Tale has proven to be more than just a Tale.  It is time to take it a step further.  It is time to embrace the reality of the Farey Series.  Click here for a journey on the Farey Experience.