Ford Circles & Farey Series

Lester R Ford, Sr. (1886–1967).

American mathematician Lester Randolph Ford Sr. (1886–1967) was President of the Mathematical Association of America from 1947 to 1948 and editor of the American Mathematical Monthly during World War II. He is remembered today for the system of circles named in his honour.

For any rational number ${p/q}$ in reduced form ( ${p}$ and ${q}$ coprime), a Ford circle is a circle with center at ${(p/q,1/(2q^{2}))}$ and radius ${1/(2q^{2})}$ . There is a Ford circle associated with every rational number. Every Ford circle is tangent to the horizontal axis and each two Ford circles are either tangent or disjoint from each other.

There is a Ford circle tangent to the ${x}$ -axis at every point having rational coordinates. If ${C}$ is the Ford circle associated with ${p/q}$ with ${0 < p/q < 1}$ , the Ford circles that are tangent to ${C}$ are the circles associated with the fractions ${r/s}$ that are the neighbours of ${p/q}$ in some Farey sequence. They are also linked to the Stern-Brocot tree and to the circles in the Apollonian gasket Apollonian gasket, a fractal named after Apollonius of Perga.

Farey Sequences

The Farey Sequence of order ${n}$ is the sequence of (reduced) fractions between 0 and 1 which have denominators less than or equal to ${n}$ , arranged in order of increasing size. Each Farey sequence starts with the value 0 (=0/1) and ends with the value 1 (=1/1). The eighth {Farey Sequence} is

$\displaystyle F_8 = \biggl\{ \frac{0}{1}, \frac{1}{8}, \frac{1}{7}, \frac{1}{6}, \frac{1}{5}, \frac{1}{4}, \frac{2}{7}, \frac{1}{3}, \frac{3}{8}, \frac{2}{5}, \frac{3}{7}, \frac{1}{2}, \frac{4}{7}, \frac{3}{5}, \frac{5}{8}, \frac{2}{3}, \frac{5}{7}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}, \frac{1}{1} \biggr\}$

There are 23 terms in this sequence. John Farey speculated, in 1816, that each new entry is the mediant of its neighbours. That is, for two adjacent fractions ${m_1 / n_1}$ and ${m_2 / n_2}$ , the new number is ${(m_1+m_2) / (n_1+n_2)}$ :

$\displaystyle \mbox{Mediant}\left(\frac{m_1}{n_1},\frac{m_2}{n_2}\right) = \frac{m_1+m_2}{n_1+n_2}$

For example, between 3/5 and 2/3, the next rational to appear is 5/8. Cauchy provided proofs of Farey’s claims and ascribed the credit to Farey. It is a consequence of this relationship that, at any stage, if ${m_1/n_1}$ and ${m_2/n_2}$ are two neighbouring entries with ${m_1/n_1<m_2/n_2}$ , then ${(m_2n_1-m_1n_2) = 1}$ .

Representing Fractions by Circles

Ford’s idea was to represent fractions by circles. Through each rational point ${x=p/q}$ on the real line he constructed a circle of radius ${1/(2q^2)}$ tangent to the ${x}$ -axis and in the upper half-plane. The integers are represented by circles of radius ${1/2}$ and fractions by smaller circles.

By considering the distance between the circles corresponding to two distinct rationals, Ford showed that the circles are either disjoint or tangent to each other. He also showed that for every fraction ${p/q}$ there is an adjacent fraction ${r/s}$ such that their two circles are tangent. Indeed, if ${P_0/Q_0}$ is adjacent to ${p/q}$ then so are all the fractions

$\displaystyle \frac{P_n}{Q_n} = \frac{P_0 + np}{Q_0 + nq} \,, \quad \mbox{\ for all\ }n\in\mathbb{Z} \,.$

Exactly two of these have denominators smaller than ${q}$ .

Fig 2. The circle for the mediant touches the other two circles.

Ford’s paper (Ford, 1938) is still worthwhile reading. He established some results on the approximation of an arbitrary real number by rationals. He also showed the link between his circles and the Farey series. If we consider two tangent circles, associated with ${p/q}$ and ${r/s}$ , then the circle for the mediant, ${(p+r)/(q+s)}$ , is tangent to both of them (See Fig. 2).

Sources

${\bullet}$ Conway, John H. and Richard Guy, 1996: The Book of Numbers. Copernicus. ISBN: 978-0-3879-7993-9.

${\bullet}$ Ford, L. R., 1938. Fractions. Amer. Math. Monthly, 45, (9), 586–601.
https://www.jstor.org/stable/2302799

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