Free Iteration in Continued Fraction Approximation for Calendar
Dear Calendar People
Suppose we a looking for a calendar whose leap years are spread as smoothly as possible so that the proportion of leap years approximates some value.
One could use an approximation that arises from continued fractions. I’ve defined a measure of structural complexity of calendar cycles whose leap years are spread as smoothly as possible, which I have defined and explained in some earlier
notes. Each iteration of continued fractions increases the structural complexity by at most 1. Sometimes an iteration leaves the structural complexity unchanged. I call this a free iteration, because it increases accuracy without the cost of increased structural
One example is month-based lunar calendars.
Continued fraction iterations go:
29, 30, 59/2, 434/15, 502/17, 945/32, 1447/49, … which have fractional parts
So the change from 29-day months to 30-day months is a free iteration (complexity 0), the change from 7/15 to 8/17 (15-month yerms to 17-month yerms) is another free iteration (complexity 2) and the change from 945/32 to 1447/49 is yet
another free iteration and one that improves accuracy significantly.
A free iteration only occurs if the corresponding integer in the continued fraction is 1, but the occurrence of 1 does not ensure a free iteration. The case of the golden ratio, all the continued fraction integers are 1 and every other
iteration is a free iteration.
The convergents of continued fractions have the property that for any two consecutive convergents a/b & c/d, ad & bc differ by 1. If this step from a/b to c/d is a free iteration, then a/b can be omitted and the sequence will still have
this property. The length of the shortest such sequence defines the structural complexity.
I wonder if any calendar person with knowledge of mathematics can say more about free iterations or related issues.
In particular if one were to define continued fractions to allow a minus sign as well as a plus sign and do the steps by rounding to nearest integer instead of rounding down, would one still get a sequence of convergents with the above-mentioned
property and would it have minimal length and so have no free iterations?