Leap years and continued fractions
As a year is not precisely 365 days in length and as it is most favorable for calendars to preserve years as integer multiples of days, the technique of leap years balances the offset. The most primitive form of leap years would be to add a day to every 4th year, resulting in an average year length of 365.25 days, close enough to the ‘real’ year for daily calendar purposes and over the span of anyone’s lifetime. However, more elaborate systems have been introduced to account for a greater accuracy in this regard. In this article, I discuss a system based on the continued fraction expansion of the length of the year, and I prove a small optimality theorem about it.
Tropical years, sidereal years, and the Gregorian calendar
The Gregorian calendar uses a 400-year cycle of leap years whereby a leap year is defined to be a year that is: divisible by 400, or divisible by 4 but not by 100. For instance, the years 2000, and 2004 are leap years, whilst the year 1900 is not a leap year. This results in a cycle of 400 years of which 100 - 3 = 97 are leap years and an average year length of precisely 365.2425 days, which is quite close to the intended length of the tropical year, that is, the time it takes for the sun to appear in the same relative position in the sky once again, which is 365.2422 according to one article by NASA which also calculates that the Gregorian calendar overshoots by one day about once every 3,300 years.
The tropical year differs from the sidereal year, i.e. the length of the orbit of the Earth because of the precession of the Earth’s axis over the period of around 26,000 years. Both are not constant due to perturbations from other planets and stars and whatnot, and it seems that, at least for the tropical year, they can change quite a bit! To explain my ideas below, I will take the number 365.242190402 published by the Paris Observatory as the length of the tropical year. I will consider it constant and exact for simplicity.
Leaping and continued fractions
The concept of picking leap years is inherently connected to continued fractions. Consider the simple Julian rule of the 4th year being a leap year. The intuition behind it is that the length of the year is 365 and 1 out of 4 years have an extra day. Translated to numbers this corresponds exactly to saying that the length of the year is 365 + 1/4, that is, 365 and 1 for every 4.
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In general, for a common year of length \(n\) where every \(m^\text{th}\) year is a leap year, the actual length of the year is naturally described as \(n + \frac 1 m\), the basic pattern which repeats recursively in continued fractions. You can say that the form \(n + \frac 1 m\) is a leapy form.
Recursive leaping
The continued fraction for the length \(l = 365.242190402\) of the tropical year is \([365, 4, 7, 1, 3, 20, 1, 3, 2]\), otherwise written as:
The interpretation of \(365 + \frac 1 4\) being “365 days and one out of 4 years being leap” is now clear; however, the question arises with the interpretation of the remaining terms. The key here is the recursive nature of continued fractions.
To consider the continued fraction up to the third term and then translate naively as above would give “365 days and one out of \(4 + \frac 1 7\) years being leap”. The leapy form \(4 + \frac 1 7\) will correspond to something where we have the sentence “4 \(X\) and one out of 7 \(Y\) being leap”. Where \(4X\) is the length is the length of a common \(Y\), so that \(5X\) is the length of a leap \(Y\). \(X\) here is a year since every 4-years make a cycle, but what then is \(Y\)? We can think of \(Y\) as a “block” of years.
In this analogy, a leap block would mean that it is 5 years long. In other words, for every 6 passings or the 4-year cycle, the 7th passing is a 5-year cycle, so you count “1, 2, 3, 4 is leap” 6 times and “1, 2, 3, 4, 5 is leap” once, and repeat for a cycle of length 29 years.
Here, years replaced days, and blocks replaced years. To interpret the next term 3, we do the same with blocks of blocks. In general, we can define days and years to be blocks too and fit them into a hierarchy \(b_0, b_1, b_2, \cdots\). Here \(b_0\) corresponds to days, so that \(b_1\) corresponds to “blocks of days”, i.e. years, and so on and so forth.
Benefits
The benefits of this system consist of 3 major points.
- Whenever the constants change and/or more accurate measurements are found, the way to adjust is unambiguous, automatic, immediate, and standard.
- The system achieves the shortest possible cycle for a given accuracy. This result is provable because the convergents of a continued fraction are those fractions where the denominator is the least possible for the given accuracy.
- The approximations are gradual and thus respect locality, meaning that on the smaller-scale, the adjustments look like the best approximations for that scale, and as the scale grows this is still respected, so the system is stable on the scale of thousands of years as well as the scale of 4 years, or 29 years, etc…
Issues
A friend of mineCAD made a notable remark about the matter that the system is very complicated that it could not reasonably be a good idea if one is manually tracking leap years. I had thought of using this system completely automatically within clocks in digital devices such as phones and whatnot; however, he noted that the automation of such things would lead to people being oblivious to how time is managed, and that the spread of similar “we will handle it for you” situations to things as basic and date and time would be a dangerous situation with respect to the freedom of the general population.