Monday, August 1, 2011

Efficient Rules & A Proposed Calendar Reform

Too much dissent is likely to be harmful to a nation and can disrupt the government's need to make decisions. But we also need to consider the quality of the decisions made. Let's assume that a given rule has a given lifetime. Change within that lifetime is unnecessary and arbitrary change may be considered a nuisance. So we might approach the problem from the perspective of least action.

Let us look at calandar reform as an example. The Julian Calendar adopted 365 + 1/4 days for the year. It was found that this led to a deviation from the natural year or the mean time between equinoxes. The Gregorian Calendar corrected this and adopted a mean year of 365 + 1/4 - 1/100 + 1/400 days. But it turns out that this system does not result in the most efficient set of rules.

The value that The Explanatory Supplement to the Astronomical Almanac gives for the mean number of days per year is 365.2421897. There are a number of ways of representing a fraction one of which is continued fractions. The number of terms in the continued fraction can be terminated after a given number of terms and can be converted to a mixed fraction. This simplifies the design of gear trains which can reproduce a given ratio. A simplier system is that of a series of alternating unit fractions. Each unit fraction tells us how frequently we have to break the set of rules defined by the subset of unit fractions. The sequence of alternating unit fractions that corresponds to the number of days in a year is 365 + 1/4 - 1/128 + 1/454545...

Evaluating the expressions for the various number of days in a year shows that even two terms of the alternating unit fraction provides a better approximation for the actual number of days in a year than that does the Gregorian Calendar. So we may ask if all the hassle of Y2K was really necessary. One might conclude that the Catholic Church, the scientific community and government are all "fallible" to some extent.

It might be wise to consider alternatives to this set of rules but if the world wanted to this calendar reform could be adopted by the United Nations and approved by individual nations. A nominal starting year would be the year 2001* since the new proposal and the Gregorian Calendar agree on leap years until the year 2100.

*Edit: You would need to count the number of years from the year 2000 and base the decision for the leap day on whether the number of years is zero modulo 4, 128 (, etc.)

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