Date: Tue, 25 Jan 2011 18:43:15 -0000
From: "Adam P. Goucher"
To: "Adam P. Goucher"
Subject: [math-fun] Calendars and continued fractions
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Dear all:
In 45 BC, Julius Caesar reformed the Roman calendar to closely
adhere to the tropical year. The Julian calendar has a leap year every
four years, and thus has an average year length of:
1461/4 = 365.25 days.
This is not a particularly accurate approximation to the actual tropical
year, which is 365.2421897 days. Our current calendar, the Gregorian
calendar, has the following rules:
* If 400 | N, then N is a leap year, otherwise;
* If 100 | N, then N is not a leap year, otherwise;
* If 4 | N, then N is a leap year, otherwise;
* N is not a leap year.
It is not too difficult to ascertain that this leads to a mean year of:
146097/400 = 365.2425 days.
This is much better than the Julian calendar, but far from perfect. What
we actually desire is a *best rational approximation* to the length of
the mean tropical year. Calculating the continued fraction expansion of
the mean tropical year yields the following:
365; 4, 7, 1, 3, 27, ...
The large value 27 means that it is wise to truncate immediately before
it, and we obtain the following approximation:
365; 4, 7, 1, 3 = 46751/128 = 365.2421875.
This is impressively close to the mean tropical year, and corresponds
to the following rule system:
* If 128 | N, then N is not a leap year, otherwise;
* If 4 | N, then N is a leap year, otherwise;
* N is not a leap year.
This system is simpler than the Gregorian calendar, has a shorter
period, and is 140 times more accurate. Additionally, a period of
128 years is a power of two, and therefore it is trivial to determine
the number of days in the year based on the year's binary expansion.
The next discrepancy between my calendar and the Gregorian
calendar occurs in the year 2048, which is a leap year in the
Gregorian calendar but not in mine.
We have now identified the fact that best rational approximations
form good calendars. What if a sadistic deity wanted to create a
solar system to make it intentionally difficult for the inhabitants to
design an accurate calendar? Obviously, we would want a number
that has poor rational approximations. The basis is again continued
fractions, but a value of Phi = 1.618033... would be optimal for this
purpose. It has a continued fraction of 1; 1, 1, 1, 1, ..., with the
best rational approximations equal to ratios of consecutive Fibonacci
numbers.
Let's presume we had a mean tropical year of 364+Phi days, so a
year can have 365 or 366 days, in the ratio 1 : Phi. When should the
'leap years' occur? This is rather interesting, and corresponds to the
Golden String:
L R L L R L R L L R L L R L R L L R L R ...
(L = leap year, R = regular year)
Sincerely,
Adam P. Goucher