Ford Circles

Ford Circles
This is an example of a fractal image called Ford Circles which is a special case of the Apollonian gasket

Basic Description

Ford Circles are a special case of the Apollonian gasket using two lines and a circle instead of three circles to generate the fractal. We start with the two lines y=0 and y=1, while the circle is centered at the point (0,1/2).

The line y=0 can be thought of as a number line such that there is a Ford circle at each rational point along the line. Rational numbers are very diverse and could be anything from 5 to .67543322 to 184756927/32674637. This means that there would be a Ford circle at both 1 and 2; but also a Ford circle at both .1 and .2; even more there is one at both .0000001 and .000002.

Interestingly though, there are no Ford circles that intersect! In other words, Ford circles are all either tangent at one point or completely disjoint. They are named after Lester R. Ford, Sr., an American mathematician who first described them in 1938.

A More Mathematical Explanation

In general, every Ford circle is tangent to a rational number, denoted p/q, along the x-axis where p [...]

In general, every Ford circle is tangent to a rational number, denoted p/q, along the x-axis where p and q are relatively prime integers.

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The circle that is associated with $p/q$ has center at ($p/q, 1/(2q^2))$ and a radius $1/(2q^2)$.

Interestingly, Ford circles are more than just circles tangent to each other and the line y=0; they are actually a geometrical representation of the Farey sequence where Ford circles are tangent to fractions in the Farey sequence on the number line.

The Farey Sequence

The Farey sequence is named after the British geologist, John Farey, Sr. Although, the French mathematician Augustin Louis Cauchy was the first to write the proof for Farey sequences based off of Farey's work.

Fourteen years earlier, however, the mathematician C. Haros introduced and proved Farey sequences in 1802. Haros was able to show the Farey sequence up to F99.

The Farey sequence a sequence of reduced fractions arranged in increasing order between 1 and 0 of which all of the fractions' denominators are less than some number we pick, which is called n. The Farey sequence then associated with n will be called Fn. We always write 0 as 0/1 and 1 as 1/1.

So, for example if we pick n equal to 3, F3= { 0/1, 1/3, 1/2, 2/3, 1/1}. Picking n=7 gives us the sequence F7={ 0/1, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1/1}. Notice that the fraction 1/2 appears exactly in the middle of both F3 and F7. This property will always be true for every Farey sequence except of course F1.

Terms that appear right beside each other in the Farey sequence also have interesting properties about them. If we take the term p/q and also its neighbor, r/s such that p/q is less than r/s, we can find that $\frac{r}{s} - \frac{p}{q} = \frac{1}{sq}$. This is the same as saying that $rq - ps = 1$ because simply subtracting the fractions give us $\frac{r}{s} - \frac{p}{q} = \frac{rq - ps}{sq}$.

Again looking at consecutive terms p/q, p'/q' and p''/q'' where p/q < p'/ q'< p''/q'' in the Farey sequence, we can find another useful property such that $\frac{p +p''}{q + q''} = \frac{p'}{q'}$. This is called the mediant property. The mediant property can be used as a method for computing a Farey sequence. Just insert each of the mediants such that $q + q'' < n$.

Ford Circles and The Farey Sequence

The Farey sequence relates to Ford circles in unique ways. Think of three terms that are consecutive in a Farey sequence: p/q, p'/q', and p''/q''. The Ford circles associated with each of these Farey sequence terms are denoted C(p,q), C(p',q'), C(p'',q'') and behave such that C(p,q) is tangent to C(p',q') and C(p',q') is tangent to C(p'',q'').

Properties include:

• $C(p,q)$ is tangent to $C(p',q')$ at $(\frac{p'}{q'} - \frac{q}{q'(q'^2+q^2)}$ , $\frac{1}{q'^2 + q^2})$

• $C(p',q')$ is tangent to $C(p'',q'')$ at $(\frac{p'}{q'} + \frac{q''}{(q'^2+q''^2)}$,$\frac{1}{q'^2 + q''^2})$