## Wednesday, February 29, 2012

### Happy Continued Fraction Day

From God Plays Dice:
Why do we have February 29 this year, and not in other years? Of course it’s because the ratio between the Earth’s orbital period and its rotational period is not an integer, but rather is about 365.242199. Let’s call this 365+α. And this is approximated well by the rational number $365{1 \over 4}$, the first convergent of the continued fraction. The convergents of continued fractions give, in a sense, the “best possible” rational approximations to irrational numbers.

Yury Grabovsky observes that the next few convergents to α are 7/29, 8/33, 31/128, 163/673, and indeed the Iranian calendar uses 8 leap years in 33. This is a bit harder to compute in one’s head. 31/128 would be pretty easy to work with — a year is a leap year if it’s divisible by 4, except not if it’s divisible by 128.
(Implicit in the Gregorian calendar rules — no leap years in years divisible by 100, except if they’re divisible by 400 — is the rational approximation 97/400, but that’s not a convergent.)

Therefore I nominate February 29 (in years when it occurs) as a new holiday, to be observed by the consumption and/or use of things that rely on rational approximations to irrational numbers.
1. go look at the moon. The Metonic cycle is a period of 19 years, which is very nearly 235 (synodic) lunar months. So the full moon, for example, falls on (approximately) the same day on the solar calendar in the year N and in the year N + 19. The Hebrew calendar has seven leap years in nineteen, where the leap years have 13 (lunar) months instead of twelve. The Islamic calendar has twelve lunar months in each year, with the result that they fall backwards $235-12 \times 19 = 7$ lunar months in nineteen lunar years. Oh, and on February 29, 1936 the phase of the moon was the same as it is today.
2. play some music. Western music theory is based on the existence of the circle of fifths, which in turn is based on the fact that $(3/2)12 \approx 2^7$ — that is, twelve perfect fifths is very nearly seven octaves. Taking logs this becomes $\log_2 3 \approx 19/12$. The fact that this is not the best approximation ever — it’s off by $0.0016$ — as well as the desire to incorporate other consonances into musical tuning caused lots of trouble....MORE