Ford Circle - Total Area of Ford Circles

Total Area of Ford Circles

There is a link between the area of Ford circles, Euler's totient function φ, the Riemann zeta function ζ, and Apéry's constant ζ(3).

As no two Ford circles intersect, it follows immediately that the total area of the Ford circles

is less than 1. In fact the total area of these Ford circles is given by a convergent sum, which can be evaluated.

From the definition, the area is

A = \sum_{q\ge 1} \sum_{ (p, q)=1 \atop 1 \le p < q } \pi \left( \frac{1}{2 q^2} \right)^2.

Simplifying this expression gives

A = \frac{\pi}{4} \sum_{q\ge 1} \frac{1}{q^4} \sum_{ (p, q)=1 \atop 1 \le p < q } 1 = \frac{\pi}{4} \sum_{q\ge 1} \frac{\varphi(q)}{q^4} = \frac{\pi}{4} \frac{\zeta(3)}{\zeta(4)},

where the last equality reflects the Dirichlet generating function for Euler's totient function φ(q). Since ζ(4) = π 4/90, this finally becomes

Read more about this topic:  Ford Circle

Famous quotes containing the words total, area, ford and/or circles:

    “Unlike Descartes, we own and use our beliefs of the moment, even in the midst of philosophizing, until by what is vaguely called scientific method we change them here and there for the better. Within our own total evolving doctrine, we can judge truth as earnestly and absolutely as can be, subject to correction, but that goes without saying.”
    —Willard Van Orman Quine (b. 1908)

    “Self-esteem is the real magic wand that can form a child’s future. A child’s self-esteem affects every area of her existence, from friends she chooses, to how well she does academically in school, to what kind of job she gets, to even the person she chooses to marry.”
    —Stephanie Martson (20th century)

    “History and experience tell us that moral progress comes not in comfortable and complacent times, but out of trial and confusion.”
    —Gerald R. Ford (b. 1913)

    “And why do you cry, my dear, why do you cry?
    It is all in the whirling circles of time.
    If millions are born millions must die,”
    —Robinson Jeffers (1887–1962)