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Re: SYMMETRICAL DISTRIBUTION OF MONTHS OVER 4 YEARS:
by Karl Palmen
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Dear Walter and Calendar People
I think I may have dealt with this topic once before.
Amos and I have pointed out that alternating periods of nine (30 31 30 31 30 31 30 31 30) and seven (30 31 30 31 30 31 30) would create months spaced as uniformly
as possible over an Olympiad. This cycle is symmetrical about any month in the middle of a period.
For 33-year cycle we need more periods of seven than of nine months. The number of periods needed is simply the number of 30-day months less the number of 31-day
months, which is 223-173=50. I recall this round number before. We have 396 months in a 33-year cycle. This is 46 more than the 350 that would result if all periods had 7 months, hence we have 23 periods of 9 months and 27 periods of 7 months.
These 50 periods, group into four superperiods three with 13 periods and one with 11 periods.
Long superperiod, 13 periods 7,9,7,9,7,9,7,9,7,9,7,9,7 = 103 months
Short superperiod, 11 periods 7,9,7,9,7,9,7,9,7,9,7 = 87 months
Symmetry occurs about the middle month of the short superperiod and the middle month of the middle long superperiod.
When would a 366-day year occur? If and only if it begins on the 2ndor 4th month of a period, except for the 4th month of
the last period of a superperiod. There are 96 such months in a 33-year cycle.
Therefore a cycle can be symmetrical about the first month of a leap year, if that month is the 4th month of the 7th period of a long
superperiod between two other long superperiods.
So by symmetry, it can be symmetrical about the last month of a leap year, if that month is the 4th month of the 7th period of a long
superperiod between two other long superperiods.
The 366-day years occur every 4 years within a superperiod (like Olympiads). Only the first 366-day year of a superperiod can occur five years after the previous
such year and for only one of the four superperiods.
Happy Easter
Karl
12(10(14
From: Walter Ziobro [mailto:petersonwalter@...]
Sent: 04 April 2012 21:35
To: Palmen, Karl (STFC,RAL,ISIS); CALNDR-L@...
Subject: Re: SYMMETRICAL DISTRIBUTION OF MONTHS OVER 4 YEARS:
Thank you, Karl, for your observations.
A thought occurred to me after reading your response. Another way that the 31 day months can be distributed is over a period of 33 tropical years. There would be 173 31-day months, and
223 30-day months, for a total of 12053 days, but I haven't been able to think how they could be distributed symmetrically.
-Walter Ziobro
From: "karl.palmen@..." <karl.palmen@...>
To: petersonwalter@...; CALNDR-L@...
Sent: Wednesday, April 4, 2012 8:03 AM
Subject: RE: SYMMETRICAL DISTRIBUTION OF MONTHS OVER 4 YEARS:
Dear Walter and Calendar People
I show Walter’s suggestion but highlighted in red the even-numbered months of 30 days and the odd-numbered months of 31 days .
Year/
Olympiad: 1 2 3 4
January 30 31 31 31
February 31 30 30 30
March 30 30 31 31
April 31 31 30 30
May 30 30 30 31
June 31 31 31 30
July 30 30 30 30
August 30 31 31 31
September31 30 30 30
October 30 30 31 31
November 31 31 30 30
December 30 30 30 31
The black and red periods are all 7 months except one period of 13 months. So the Olympiad is symmetrical about the middle month of the interval of 13 months which is
January of year 1.
So the symmetry like this:
Feb year 1 same length as Dec year 4 with 31 days
Mar year 1 same length as Nov year 4 with 30 days
Apr year 1 same length as Oct year 4 with 31 days
etc.
The 30-day and 31-day months are not spread as uniformly as possible. This would be achieved by changing the pairs of consecutive 30-day months as follows while preserving
the symmetry
Jul/Aug year 1 -> May/Jun year 1 (2 months earlier)
Feb/Mar year 2 -> Dec year 1/Jan year 2 (2 months earlier)
Sep/Oct year 2 -> Sep/Oct year 2 (no change)
Apr/May year 3 -> Apr/May year 3 (no change)
Nov/Dec year 3 -> Jan/Feb year 4 (2 months later)
Jun/Jul year 4 -> Aug/Sep year 4 (2 months later)
The black and red periods are then 7 or 9 months spread as uniformly as possible. Indeed, the back periods have 9 months and the red periods have 7 months.
Year/
Olympiad: 1 2 3 4
January 30 30 31 30
February 31 31 30 30
March 30 30 31
31
April 31 31 30 30
May 30 30 30 31
June 30 31 31
30
July 31 30 30
31
August 30 31 31
30
September 31 30 30 30
October 30
30 31 31
November 31 31 30 30
December 30 30 31 31
However I do notice that year 3 rather than year 4 is the 366-day year of the Olympiad. Such a distribution (uniform as possible) cannot be made symmetrical about January
of the year after a 366-day year, but can be made symmetrical about December of a 366-day year. I leave as a puzzle for calendar people to work this out. A simple manipulation will lead to the answer.
Spreading the months as uniformly as possible across an entire Gregorian 400-year cycle would be much more complicated. 400 years would have 606 black and red periods
(rather than 600).
If one does not care about the months begin distributed as uniformly as possible, one can simplify. Make all even number months have 30 days and all odd-numbered months
have 31 days with three exceptions and distribute the exceptions symmetrically. The exceptions can be Jan year 1, July years 2 and 3. Also year 4 has no exception and so has 366 days.
Year/
Olympiad: 1 2 3 4
January 30 31 31 31
February 30 30 30 30
March 31 31 31 31
April 30 30 30 30
May 31 31 31 31
June 30 30 30 30
July 31 30 30
31
August 30 30 30 30
September 31 31 31 31
October 30 30 30 30
November 31 31 31 31
December 30 30 30 30
There is an even simpler scheme that is symmetrical about December in a leap year. I’ll leave it to other calendar people to work it out.
Karl
12(10(13
From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...]
On Behalf Of Walter Ziobro
Sent: 04 April 2012 01:19
To: CALNDR-L@...
Subject: SYMMETRICAL DISTRIBUTION OF MONTHS OVER 4 YEARS:
SYMMETRICAL DISTRIBUTION OF MONTHS OVER 4 YEARS:
Here's a novel idea I got for distributing the lengths of the months: instead of fixing the lengths of the months in each year, why not distribute the 31 day months symmetrically over a 4 year period, with the leap month at the end? My Olympiad calendar would
look like this:
Year/
Olympiad: 1 2 3 4
January 30 31 31 31
February 31 30 30 30
March 30 30 31 31
April 31 31 30 30
May 30 30 30 31
June 31 31 31 30
July 30 30 30 30
August 30 31 31 31
September31 30 30 30
October 30 30 31 31
November 31 31 30 30
December 30 30 30 31
The basic 7 month pattern is 30-31-30-31-30-31-30, which repeats 6 times, with 30-31-30-31-30-31 at the end.
Walter Ziobro
--
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