Dear Irv, Helios and Calendar People Here I list the Helios Cycles I think have the greatest structural complexity and so have the greatest case for simplifying at the expense of greater jitter, if that jitter can be tolerated. I defined a Helios cycle as a cycle of two types of ‘year’ which can be called ‘common’ or ‘leap’, such that the ‘leap years’ are spread as smoothly as possible in the cycle and the cycle is symmetrical in the sense that the first year
is of the same type as the last year, the second year is of the same type as the penultimate year, etc.. I’ve called such cycles Helios cycles, because Helios produced so many examples of them. A single Helios cycle must have an odd number of years in order to have the symmetry defined here. I next list a series of Helios cycles, each of which is the shortest cycle of its level of structural complexity.
Here ‘c’ indicates a ‘common year’ and ‘L’ a ‘leap year’. Leap years are assumed to be a minority: 1 year 0 leap ‘c’ 3 years 1 leap ‘cLc’ 11 years 4 leap ‘cLccLcLccLc’ 41 years 15 leap ‘cLccLcLccLccLccLcLccLccLcLccLccLccLcLccLc’ Here I show them again but with ‘|’ in the middle of every run of an even number of common years within the cycle: 1 year 0 leap ‘c’ 3 years 1 leap ‘cLc’ 11 years 4 leap ‘cLc|cLcLc|cLc’ 41 years 15 leap ‘cLc|cLcLc|cLc|cLc|cLcLc|cLc|cLcLc|cLc|cLc|cLcLc|cLc’ Now I can see that each is form from the previous by replacing each ‘c’ with a ‘cLc’ and each ‘L’ with a ‘cLcLc’. There the number of years and leap years, which are sufficient to determine the Helios cycle are: 1 year 0 leap 3 years 1 leap (1/3 = 0.333333…) 11 years 4 leap (4/11 = 0.363636…) 41 years 15 leap (15/41 = 0.3658536585…) 153 years 56 leap (56/153 = 0.36601307…) 571 years 209 leap (209/571 = 0.36602452…) 2131 years 780 leap (780/2131 = 0.36602534…) 7953 years 2911 leap (2911/7953 = 0.36602540…) …. Y years L leap 3Y+2L years Y+L leap The ratio of leap years converges from below to a constant about 0.3660254, which is the fractional part of the year in Helios Hell, whose approximating Helios cycles are the most complex possible. This fraction is near the fractional part of the number of lunar months in a year about 12.3683 and the number of days in a 12-month lunar year about 354.367 . This explains why their cycles are structurally complicated, when the leap years
are spaced as smoothly as possible. The same can be done with the quasi-Helios cycles, which have an even number of years and are symmetrical, except for the two middle years then starting with the 2 year cycle with one leap year we get 2 years 1 leap 8 years 3 leap (3/8 = 0.375) Octaeteris 30 years 11 leap (11/30 = 0.366666…) Tabular Islamic 30-year cycle
112 years 41 leap (41/112 = 0.3660714…) 418 years 153 leap (153/418 = 0.3660287…) 1560 years 571 leap (571/1560 = 0.36602564…) 5822 years 2131 leap (2131/5822 = 0.36602542…) This converges to the same number, but from above instead of below also the number of leap years equals the number of years in the series of fully symmetrical Helios cycles. And so the approximating quasi-Helios cycles are also the most complex in Helios Hell. Karl 16(02(09 |
Dear Irv, Helios & Calendar People Quadratic mathematics reveals that the fractional part of the year in Helios Hell, which the Helios and quasi-Helios cycles from Hell approximate is equal to 1/(1 + sqrt(3)) = 0.36602540378443864676372317075294… The continued fraction value is 1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + 1/(1 + etc. The continued fraction convergents are 1/2, 1/3, 3/8, 4/11, 11/30, 15/41, … which give the Helios and quasi-Helios cycles from Hell, which I have listed below.
Therefore one cannot avoid these structurally complicated cycles, by just giving up the symmetry, when approximating a mean year whose fractional part is very near 1/(1 + sqrt(3)). One must also increase the
jitter by having leap years not spread as smoothly as possible. If the mean synodic month were very near 29.53050211698203655389697693… days, then an arithmetic
twelve-month lunar calendar would not be a good idea. Karl 16(02(10 From: Palmen,
Karl (STFC,RAL,ISIS) Dear Irv, Helios and Calendar People Here I list the Helios Cycles I think have the greatest structural complexity and so have the greatest case for simplifying at the expense of greater jitter, if that jitter can be tolerated. I defined a Helios cycle as a cycle of two types of ‘year’ which can be called ‘common’ or ‘leap’, such that the ‘leap years’ are spread as smoothly as possible in the cycle and the cycle is symmetrical in the sense that the first year
is of the same type as the last year, the second year is of the same type as the penultimate year, etc.. I’ve called such cycles Helios cycles, because Helios produced so many examples of them. A single Helios cycle must have an odd number of years in order to have the symmetry defined here. I next list a series of Helios cycles, each of which is the shortest cycle of its level of structural complexity.
Here ‘c’ indicates a ‘common year’ and ‘L’ a ‘leap year’. Leap years are assumed to be a minority: 1 year 0 leap ‘c’ 3 years 1 leap ‘cLc’ 11 years 4 leap ‘cLccLcLccLc’ 41 years 15 leap ‘cLccLcLccLccLccLcLccLccLcLccLccLccLcLccLc’ Here I show them again but with ‘|’ in the middle of every run of an even number of common years within the cycle: 1 year 0 leap ‘c’ 3 years 1 leap ‘cLc’ 11 years 4 leap ‘cLc|cLcLc|cLc’ 41 years 15 leap ‘cLc|cLcLc|cLc|cLc|cLcLc|cLc|cLcLc|cLc|cLc|cLcLc|cLc’ Now I can see that each is form from the previous by replacing each ‘c’ with a ‘cLc’ and each ‘L’ with a ‘cLcLc’. There the number of years and leap years, which are sufficient to determine the Helios cycle are: 1 year 0 leap 3 years 1 leap (1/3 = 0.333333…) 11 years 4 leap (4/11 = 0.363636…) 41 years 15 leap (15/41 = 0.3658536585…) 153 years 56 leap (56/153 = 0.36601307…) 571 years 209 leap (209/571 = 0.36602452…) 2131 years 780 leap (780/2131 = 0.36602534…) 7953 years 2911 leap (2911/7953 = 0.36602540…) …. Y years L leap 3Y+2L years Y+L leap The ratio of leap years converges from below to a constant about 0.3660254, which is the fractional part of the year in Helios Hell, whose approximating Helios cycles are the most complex possible. This fraction is near the fractional part of the number of lunar months in a year about 12.3683 and the number of days in a 12-month lunar year about 354.367 . This explains why their cycles are structurally complicated, when the leap years
are spaced as smoothly as possible. The same can be done with the quasi-Helios cycles, which have an even number of years and are symmetrical, except for the two middle years then starting with the 2 year cycle with one leap year we get 2 years 1 leap 8 years 3 leap (3/8 = 0.375) Octaeteris 30 years 11 leap (11/30 = 0.366666…) Tabular Islamic 30-year cycle
112 years 41 leap (41/112 = 0.3660714…) 418 years 153 leap (153/418 = 0.3660287…) 1560 years 571 leap (571/1560 = 0.36602564…) 5822 years 2131 leap (2131/5822 = 0.36602542…) This converges to the same number, but from above instead of below also the number of leap years equals the number of years in the series of fully symmetrical Helios cycles. And so the approximating quasi-Helios cycles are also the most complex in Helios Hell. Karl 16(02(09 |
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