Dear Irv, Helios & Calendar People
For those wanting to know about my cutting algorithm I show below my original note I sent about it to the list. The actual definition of the cutting algorithm is in
I realise that the cutting algorithm can be applied to any cycle where the leap years are spaced a smoothly as possible and eventually yield the first year of a symmetrical (Helios) or quasi-symmetrical (quasi-Helios) cycle. To do so, the cycle must be treated as a cycle with no end or beginning. One can represent the cycle as a linear sequence of common years and leap years provided the first year is treated as the year after the last year.
I show it for the Hebrew leap year cycle
First iteration cut as follows:
The cycle is then cut into five parts of either ‘cLc’ or ‘cLcLc’, noting that the final ‘cLcL’ is followed by the ‘c’ at the start and so they form another ‘cLcLc’. There is no implicit cut at the end of the cycle representation. If we do cut at the end of a cycle representation that cut must be shown (as in a later example).
We now have two symmetrical types of part ‘cLc’ and ‘cLcLc’ and they are spread as smoothly as possible around the cycle.
The ‘cLcLc’ is the minority type of the two types of parts and so is treated as a leap year in the second iteration. The 2nd and 3rd ‘clc’ or the only two consecutive ‘cLc’s and so one second iteration cut is made there.
Now we have only one part (starting after the ||)
‘cLc|cLcLc|cLc|cLcLc|cLc’ and it is symmetrical. It begins with the 13th year of the original 19-year cycle.
This will also work if the number of years in the cycle is even, then we end up with two parts, which in either order form a quasi-symmetrical cycle. I show it with a Tabular Islamic cycle.
cLccLcLccLccLcLccLccLccLcLccLc (leap years 2,5,7,10,13,15,18,21,24,26,29)
We have the same two types of part as the Hebrew leap year cycle and the same minority type ‘cLcLc’.
We then have the two parts as previously described. The fact that we have a final cut at the end indicates that the cycle is already quasi-symmetrical. The other quasi-symmetrical cycle can be made by interchanging the two parts
This is the variant in which year 16 is a leap year instead of year 15.
Curiously I before I let the list know about the cutting algorithm, I let Irv know about it and in his reply he mentioned the 353-year rectified Hebrew calendar cycle with K=269.
No I show how the cutting algorithm would apply to this.
This 353-year cycle consists of nine 19-year cycles ‘LccLccLcLccLccLcLcc’ followed by the same truncated to 11 years followed by nine more of these 19-year cycles.
The first two steps cut this 19-year cycle to
And the truncated 19-year cycle to
Then the parts separated by the ‘||’ after the first ‘||’ are nine 19-year parts followed by an 11-year part followed by nine more 19-year parts. Then the third and final iteration cuts between year 2 & 3, so indicating that the cycle would be symmetrical if the start were postponed by two years. So if we reduced the year numbers by 2, the resulting 353-year cycle (starting from year 1) would be symmetrical. I then check the resulting K as 269+2*130-353=176=(C-1)/2.
Note that for cycles with an odd number of years, the cutting cycle has one additional iteration, because there is no initial implicit cut between the last and first year.
Dear Calendar People
Firstly, just to remind you of the Helios cycle and quasi-Helios cycle:
On this list I have defined a Helios Cycle as a cycle of two types of year (which we can refer to as common and leap), such that the two types of years a spread as evenly as possible and are symmetrically placed so the last year is the same type as the first year, the penultimate years is the same kind as the second year etc.
If the cycle has C years of which L are leap years then year Y is a leap year if and only if
((L*Y + (C-1)/2 ) mod C ) < L
I’ve named such a cycle a Helios cycle, because Helios has produced a large number of examples of such. Most recent being the 25-year and 33-year cycles of his CODE-29 calendar.
The advantage of a Helios cycle is that the first year start coincides with the mean year start. This is useful to know when choosing an epoch.
The ARC period of the recently mentioned Archetypes calendars is a Helios cycle for long years and also for leap years.
A Helios cycle that is not a repetition of a shorter cycle must have an odd-number of years. Irv discovered a similar cycle with an even number of years, which is symmetrical except for the middle two years one of which is a common year and the other is a leap year. I call such a cycle a quasi-Helios cycle. The first year of such a cycle begins as close to the mean year as possible.
I have found a way of cutting a Helios cycle into parts such that each part is a Helios cycle and there are just two types of parts and the sequence of parts also forms a Helios cycle.
This method can also be applied to a quasi-Helios cycle for which the parts are Helios cycles, but their sequence is a quasi-Helios cycle.
The divisions of the 1803-year and 3150-year leap week cycles into seven and eight parts respectively that I recently showed (and show again below) are examples of divisions obtained by this method. So is the division of both into 11-year and 17-year Helios cycles shown by Irv.
I’ve already explained this method to Irv and he has been working on a computer implementation of it. However, I can use the algorithm in my head.
The cutting algorithm goes as follows:
Get the sequence of leap years and common years L and c for a Helios or quasi-Helios cycle, wherever there is even-number of consecutive common years between two leap years cut the sequence in the middle of the consecutive common years.
E.g. ccLcccLccccLcccLcccLccccLcccLcc is cut to ccLcccLcc|ccLcccLcccLcc|ccLcccLcc
There should be just two types of period in the cut. In this example, ‘ccLcccLcc’ and ‘ccLcccLcccLcc’ both are symmetrical and one is more common than the other.
This is the first division by the cutting algorithm.
For leap week calendars, it is the division into 11s and 17s.
Now treat the more common type of period as a common year and the less common period as a leap year and apply the cutting algorithm again to get the second division. This leads to 231+293+231+293+231+293+231 for the 1803-year cycle and 417+355+417+355+417+417+355+417 or its reverse for the 3150-year cycle.
The idea can be applied to leap year cycles other than leap week cycles. If applied to the 19-year leap month cycle cLccLcLccLccLcLccLc and we get cLc|cLcLc|cLc|cLcLc|cLc for the first division.
The more common period is ‘cLc’ and there are never more than one consecutive occurrence of it between the others (‘cLcLc’), so the second division is the complete 19-year cycle.
The second division of the 353-year cycle would be into 19-year cycles and one 11-year cycle in the middle ( whose first division is cLc|cLcLc|cLc ).
In the Archetypes calendar, the 1803-year ARC period of 664 leap-month years has a third division is (353,353,391,353,353), where 391 is the same as 353 but with a 19 added to the beginning and another 19 added to the end.
Helios’s 33-year and 25-year cycles from the CODE-29 calendar behave quite differently under the cutting algorithm. The 33-year cycle of 8 leap cycles won’t cut at all, because all the leap years within the cycle are four years apart so have an odd-number of common years in between. Indeed, every part in a first division has either one leap year or several leap years placed an equal even number of years apart, which is equal to twice the ordinal position of the first leap year. The 25-year cycle divides into cLc|cLc|cLc|cLcccLc|cLc|cLc|cLc as its first division. Its second division is the complete 25-year cycle, because it has only one part counting as a leap part.
The 1803-year ARC period of 350 leap years has a first division into periods of five ‘ccLcc’ and 11 ‘ccLcccccLcc’ years. This second division has 21 parts of alternating between 103 years and 67 years, starting and ending with 103 years. The 103-year part is (5,5,11,5,5,5,5,5,11,5,5,5,5,5,11,5,5) and the 67-year part is (5,5,11,5,5,5,5,5,11,5,5) in the first division.
I wonder whether Helios can use this to construct Magic squares for long Helios cycles out of magic squares for short Helios cycles (the parts and the sequence of the two types of parts).
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