"Adam P. Goucher" <apgoucher@gmx.com> wrote:

http://cp4space.wordpress.com/2012/09/12/lunisolar-calendars/

I briefly mentioned the use of the Rabbit sequence to optimally approximate a leap year of length 364 + phi.

Interesting. Thanks. And very appropriate for Easter. :-) You may have noticed that all my rules involve divisibility. That's because that's the most obvious property integers have that approaches a limit, i.e. some proportion of the first N integers have this property (e.g. are odd), and that proportion approaches a limit other than 0 or 1 as N grows large. It's difficult to come up with other examples of such a property. (There's the divisibility of the number of 1-bits in the binary representation of the number, but that has nothing to recommend it over the divisibility of the number itself.) You've come up with such a property that I hadn't previously heard of Thanks. Can anyone think of any others? Responding to your discussion (at the above URL) of our calendar: Pope Gregory's intention when he established the current calendar wasn't to track the average length of the tropical year as you implicitly assume, but to track the average time between vernal equinoxes, since his goal was to make sure Easter is always celebrated on the right day. (To us moderns, the idea that there's a "right day," that the placement of holidays isn't completely arbitrary, seems odd, but not to scholars in those days, with the exception of Kepler.) Yes, these are the same thing over the very long term, but not during any given millennium, thanks to the eccentricity of Earth's orbit and the precession of the equinoxes. The average time between consecutive vernal equinixes over the past 2000 years is significantly different from the average time between consecutive autumnal equinixes over the same period. And the difference will be about the same over the next 2000 years too. Also, there's no point in making a calendar more accurate than one day in a thousand years, since the number of days in a year is dropping quickly and unpredictably enough due to tidal drag that we'll need a new calendar rule every two or three millennia anyway. J.R.R. Tolkein made this mistake. In appendix D of _The Lord of the Rings_, he said: The Calendar of the Shire differed in several features from ours. The year no doubt was of the same length,^1 for long ago as those times are now reckoned in years and lives of men, they are not very remote according to the memory of the Earth. ... ^1 365 days, 5 hours, 48 minutes, 46 seconds This was wrong. Well, strictly speaking, the length of the *year* may have been the same within a few seconds, but the length of the *day* certainly wasn't. So the number of days per year was different. I've even seen a few silly speculations as to whether the number of days in a year is transcendental. The "correct" answer is that it usually is, but there are infinitely many times during every nanosecond when it isn't. :-)