Season 7 Episode 8

Matt Parker

Matt Parker’s Leap Year video is here; Leap Years: we can do better.

And Matt’s new book Love Triangle can be pre-ordered here.

Matt also mentioned a video about UK tax law I Discovered a Maths Loophole in UK Tax and showed us a jointed triangle-cube-$\sqrt{3}$ thing made by Dr Sabetta Matsumoto, who you might have seen on Numberphile in Butterflies and Gyroids or The Girl with the Hyperbolic Helicoid Tattoo.

Continued Fractions

The Theorem of the Week blog has an article about how these work; continued fractions.

The fractions that you generate along the way (they’re called the convergents) are really good approximations to the true value of the number. Dirichlet’s approximation theorem says that for an irrational number $\alpha$ there are infinitely many pairs $(p,q)$ such that the fraction $\frac{p}{q}$ is very close to $\alpha$, in the sense that $\left|\alpha-\frac{p}{q}\right|<\frac{1}{q^2}$.

The continued fraction convergents help you to actually find such fractions. For details, see, for example, these lecture notes from Rutgers University.

You could extend Dirichlet’s approximation theorem by asking if you could replace $1/q^2$ with something smaller, to get better approximations. The Duffin-Schaeffer theorem on rational approximations says precisely when you can do this. It was proved by James Maynard and Dimitris Kaukoulopoulos July 2019.

The idea of using continued fractions to find an approximation to the length of a year appears to be due to Adam P Goucher, who proposes a 128-year calendar here. I also recommend the rest of Adam’s blog Complex Projective 4-Space.

I first read about the 500-year system in this paper by Yury Grabovsky at Temple University.

If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.