Assign a subset of the integers between 0 and 999 to the days of the year, so that the number for a given day corresponds to the approximate portion (per mille) of the year. This should be easy, probably floor(1000*day/365), where day goes from 0 to 364. Not all 1000 numbers get a day.
Other formula possibilities: round instead of floor, noon instead of midnight (0.5 to 364.5).
Bases other than 10: 20^2 = 400, 8^3 = 512, 5^4 = 625, 4^5 = 1024, 3^6 = 729.
Mixed radix 12 and 31 would resemble the ordinary calendar. The following 7 dates (12 * 31 = 372 = 365 + 7) would be omitted: 2/23 4/14 6/5 7/27 9/18 11/9 12/31.
Leap years yield a different set of numbers, which is annoying but reflects the awkwardness of the calendar.
On leap years, the 6 days omitted in (12,31) mixed radix is elegant, one every 2 months: 2/31 4/31 6/31 8/31 10/31 12/31.
Or, assign all 1000 numbers, giving the time of day corresponding to each per mille moment. 0.001 year = 8 hour 45 minute 57 second.
We could do the 400-year Gregorian calendar cycle (146097 days). ((d/146097)*400000) mod 1000. It hits exactly zero at midnight on January 1 once every 400 years.
Compute the (12,31)-mixed radix omitted dates for the Gregorian cycle: 2703 omitted days.
Invent 12 new month names for the (12,31) mixed radix calendar to avoid confusion with the traditional calendar.
31 is an awkward number of days in a month, not having any divisors. Consider (12,60) mixed radix instead, with lots of omitted days. It intriguingly makes the calendar have a structure very similar to a 12-hour clock.
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