Skip to main content

Calendars and continued fractions

Calendars are based on three frequencies: the rotation of the Earth on its axis, the rotation of the moon around the Earth, and the rotation of the Earth around the sun. Calendars are complicated because none of these periods is a simple multiple of the other. The ratios are certainly not integers, but they’re not even fractions with a small denominator.

As I wrote about the other day in the context of rational approximations for π, the most economical rational approximations of an irrational number come from convergents of continued fractions. The book Calendrical Calculations applies continued fractions to various kinds of calendars.

Ratio of years to days

The continued fraction for the number of days in a year is as follows.

365.2424177 = 365 + \cfrac{1}{4 + \cfrac{1}{7+ \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{5 + \ldots }}}}}}

This means that the best approximations for the length of a year are 365 days plus a fraction of 1/4, or 7/29, or 8/33, or 23/95, etc.

You could have one leap day every four years. More accurate would be 7 leap days every 29 years, etc. The Gregorian calendar has 97 leap days every 400 years.

Ratio of years to months

If we express the ratio of the length of the year to the length of the month, we get

\frac{365.24244}{29.53059} = 12 +\cfrac{1}{2 + \cfrac{1}{1+ \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{18 + \cfrac{1}{3 + \ldots }}}}}}}

By taking various convergents we get 37/3, 99/8, 136/11, etc. So if you want to design successively more accurate lunisolar calendars, you’d have 37 lunar months every 3 years, 99 lunar months every 8 years, etc.

Related posts



from John D. Cook https://ift.tt/2xuMhvf
via IFTTT
Labels:

Comments

Post a Comment

Popular posts from this blog

Using RStudio and LaTeX

(This article was first published on r – Experimental Behaviour , and kindly contributed to R-bloggers) This post will explain how to integrate RStudio and LaTeX, especially the inclusion of well-formatted tables and nice-looking graphs and figures produced in RStudio and imported to LaTeX. To follow along you will need RStudio, MS Excel and LaTeX. Using tikzdevice to insert R Graphs into LaTeX I am a very visual thinker. If I want to understand a concept I usually and subconsciously try to visualise it. Therefore, more my PhD I tried to transport a lot of empirical insights by means of  visualization . These range from histograms, or violin plots to show distributions, over bargraphs including error bars to compare means, to interaction- or conditional effects of regression models. For quite a while it was very tedious to include such graphs in LaTeX documents. I tried several ways, like saving them as pdf and then including them in LaTeX as pdf, or any other file ...
Post a Comment
Read more

Explaining models with Triplot, part 1

[This article was first published on R in ResponsibleML on Medium , and kindly contributed to R-bloggers ]. (You can report issue about the content on this page here ) Want to share your content on R-bloggers? click here if you have a blog, or here if you don't. Explaining models with triplot, part 1 tl;dr Explaining black box models built on correlated features may prove difficult and provide misleading results. R package triplot , part of the DrWhy.AI project, is aiming at facilitating the process of explaining the importance of the whole group of variables, thus solving the problem of correlated features. Calculating the importance of explanatory variables is one of the main tasks of explainable artificial intelligence (XAI). There are a lot of tools at our disposal that helps us with that, like Feature Importance or Shapley values, to name a few. All these methods calculate individual feature importance for each variable separately. The problem arises when features used ...
Post a Comment
Read more