Date: Tue, 25 Jan 2011 18:43:15 -0000 From: "Adam P. Goucher" To: "Adam P. Goucher" Subject: [math-fun] Calendars and continued fractions Message-ID: Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original 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