Ford circles are a beautiful geometric representation of rational numbers. Each rational number p over q corresponds to a circle that touches the x-axis. These circles have remarkable tangency properties and connect deeply to number theory.
A Ford circle is defined for each rational number p over q in lowest terms. The circle has its center at coordinates p over q comma one over two q squared, and radius one over two q squared. This means every Ford circle is tangent to the x-axis at the point p over q comma zero.
Ford circles have a remarkable tangency property. Two Ford circles are tangent to each other if and only if the absolute value of p s minus q r equals one, where the circles correspond to fractions p over q and r over s. This condition identifies adjacent fractions in Farey sequences. For example, one third and one half are tangent because the absolute value of one times two minus three times one equals one.
Ford circles provide a beautiful geometric visualization of Farey sequences. A Farey sequence F sub n contains all reduced fractions between zero and one with denominators less than or equal to n. For example, F sub 4 includes zero over one, one over four, one over three, one over two, two over thirds, three over four, and one over one. The Ford circles corresponding to consecutive fractions in a Farey sequence are tangent to each other, creating an elegant geometric pattern.
Ford circles represent a beautiful intersection of geometry and number theory. They elegantly connect rational numbers with geometric tangency properties, revealing the structure of Farey sequences through visual patterns. Each rational number corresponds to a unique circle tangent to the x-axis, and the tangency between circles reveals adjacency relationships in number theory. This creates stunning fractal-like patterns that demonstrate the deep mathematical beauty hidden within simple rational numbers.