## Fairy Tales

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##### By Robin Kirkham
and Sinje Bulter

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**W**ho was **J**ohn **F**arey?

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.**

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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

F_{5}=
[, , , , , , , , , , ]

Is
this just another **F**arey **T**ale?

**Farey Tales Rolling By ….**

#### Let us look at some of these
**F**arey **T**ales.

#### This is a proof that Fairies do exist.

####

#### 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

‘round
and ‘round.

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*.*