
By Royal Decree: An Exact Calendar! (+ Estimating reals by 1/N's  Egyptian style)
Posted:
Jan 8, 2014 8:44 PM


The Gregorian Calendar, if I recall correctly has not just 3 provisions but a 4th, that go something like this: (1) Add an extra day every years. (2) Remove one of these days every 100 years. (3) Put back the extra day removed every 400 years. (4) Take away an extra day every 3000 years or so. That comes out to 365 + 1/4  1/100 + 1/400  1/3000 (or so) days per year.
Ignoring provision #4, that put you at 365 days + 6 hours  864 seconds + 216 seconds, or 365 days, 5 hours, 49 minutes and 12 seconds.
The mean terrestrial year  as of 1900  is 365 days, 5 hours, 48 minutes and 46 seconds. I believe it's going down about 1/2 second every century so that this seconds should be around 45.5 now.
HASN'T ANYONE NOTICED THE OBVIOUS REGULARITY?!
Taking advantage of this, I hereby issue the following decree: (1) Every 4th year shall continue to be a leap year as before (2) Every 32nd leap year, the extra day shall be removed. This yields a total of 365 + 1/4  1/128 days, which is 365 days, 5 hours, 48 minutes and 45 seconds.
This decree is to take effect starting with the year 2100, in which provision (2) shall take effect. The first difference from the Gregorian Calendar shall occur in 2200 which (under the new calendar) will be a leap year (with 2228 being the 365day year instead), while under the Gregorian Calendar, 2200 will be an ordinary 365day year.
This provision will not affect any of you  unless you expect to live another 186 years to March of 2200.
This provision may be arrived at by the following: every number between 0 and 1 can be estimated by a sum and difference of fractions of the form 1/N (for positive integers N), where the accuracy of the estimate (in number of digits) goes up exponentially in the number of fractions used ... just like Newton's method.
0 fractions (and 1 whole number) yield an estimate accurate to 1/2 1 fraction yields an estimate accurate to 1/12 = 1/2 (1/2) (1/3) or less. 2 fractions to 1/352 = 1/2 (1/12) (1/13) ... N fractions to 2/(5^{2^N}  1).
The tabulation of a calendar for a year is arrived at by optimizing the number of fractions to yield 365 + 1/4  1/128 (+ 1/86400 as of 1900).

