Calendars, Cycles, and Cool Coincidences (Part I)

You might have heard that Hanukkah and Thanksgiving coincide this year. More specifically, you may have heard that the first day of Hanukkah (25 Kislev in the Hebrew calendar) coincides with 28 November, which just happens to be the fourth Thursday of the month. Somewhere along the way, a few clever marketers dubbed this day “Thanksgivukkah”, and America has responded: the LA Times has a recipe for “turbrisket”, kids in South Florida have been designing “menurkeys”, and Zazzle.com has a line of Thanksgivnukkah greeting cards. Christine Byrne has assembled an entire Thanksgivukkah menu. I, for one, am enjoying all of the portmanteaus. (Speaking of which, have you heard of Franksgiving?)

But in addition to being a fan of portmanteaus, I’m also a fan of calendars. Some weeks ago, I began to hear from various sources that Hanukkah won’t line up with Thanksgiving for another 70,000 years or so. This got me curious, so I started researching the question myself. It turns out that the relationship between the two holidays has been examined on at least three blogs over the past few years. The first were the Lansey brothers in 2010, followed by  Stephen Morse in 2012 and Jonathan Mizrahi in January of this year. Morse’s post includes a “When Did?” page with a Javascript calendar program, and Eli Lansey kindly includes a Mathematica notebook to help the math-inclined to do the computations themselves.

Morse reports that Thanksgiving will again occur on the first day of Hanukkah in the year 79,043, while Mizrahi says it’ll be in the year 79,811. Mizrahi, by his own admission, is being cute with this number:

In all honesty, though, all of these dates are unfathomably far in the future, which was really the point [of the post].

In this post, I won’t go into exactly how the ≈79,000 number was unearthed. I will, however, sketch out some of the major features of both the Gregorian and Hebrew calendars, and how they have given rise to this strange, new holiday of Thanksgivukkah. In the end, we will find that the year 79,811 is not nearly as unfathomable as we can get.

First of all, the Western calendar as we’ve come to know it began its life as the Egyptian calendar. After the Canopus Decree in c. 238 BCE, each year in the Egyptian calendar was 365 days long, with an additional day added every 4 years. There were twelve 30-day months and five (or six) epagomenal days—days with no year or month assigned to them—to celebrate the coming of the new year. I think Pharaoh Ptolemy III said it best:

This festival is to be celebrated for 5 days: placing wreaths of flowers on their head, and placing things on the altar, and executing the sacrifices and all ceremonies ordered to be done. But that these feast days shall be celebrated in definite seasons for them to keep for ever … one day as feast of Benevolent Gods be from this day after every 4 years added to the 5 epagomenae before the new year, whereby all men shall learn, that what was a little defective in the order as regards the seasons and the year, as also the opinions which are contained in the rules of the learned on the heavenly orbits, are now corrected and improved by the Benevolent Gods.

The Egyptian model came to Rome with Julius Caesar’s calendar reforms in 46 BCE, which fixed the seriously messed up Roman calendar. It all went pretty well for the first several centuries, but there was a tiny fly in the ointment. The average length of a year in the Julian calendar is 365.25 days, while the solar year is approximately 365.24219 days long. So the Julian calendar ran slow—about 11.25 minutes per year—for 1600 years until this problem was fixed by the Gregorian calendar reforms of 1582. More specifically, Pope Gregory XIII issued a papal bull, Inter gravissimas, in which he declared that leap years would continue to occur by 4 would be leap years, except that years divisible by 100 but not by 400 would no longer be leap years. So, 1900 was not a leap year, but 2000 was. This provides an average length of 365.2425, which is only 0.00031 days (about 27 seconds) longer than the solar year. In the 431 years that have passed since the birth of the Gregorian calendar, this error has only accumulated to 3.2 hours. While the calendar isn’t perfect, it’s really quite good, especially considering that the solution amounted merely to omitting 3 leap days every 400 years. 

The key concept I’m interested in here is periodicity. In mathematics, a function is said to be periodic if it exactly repeats its values in regular intervals (or, periods). The sine function is an example of this: sin(x) = sin(x+2π) = sin(x+4π) = …, for any value of x in the interval [0,2π]. It’s very important to distinguish between a periodic function and a function that just happens to repeat some of its values. For example, the function f(x) = 1 – x² repeats itself since f(-1) and f(1) are both equal to zero, but that doesn’t mean the function repeats itself exactly on an interval. Loosely speaking, a function is periodic when the entire curve repeats itself, not just a few select points.

We can transfer this idea to a given calendar without too much trouble:

• A calendar’s cycle is amount of time it takes for the calendar to repeat itself exactly.
• A calendar’s period is the amount of time it takes for the calendar to repeat itself exactly, while also taking the days of the week into account.

For consistency, it’s best to measure both the cycle and period in days, but sometimes I’ll divide by the average length of a year. For example, the Julian calendar has a cycle of 1461 days, and dividing by 365.25 gives a result of 4 years. To get the period, we need to remember that since there are 52 weeks plus 1 or 2 days in any given year, the days of the week won’t line up every 4 years. So we have to take the least common multiple to get the period: lcm(1461, 7) = 10,227 days = 28 years. For the Gregorian calendar, the cycle is 146,097 days (400 years) and the period is lcm(146097, 7) = 146097 days = 400 years—this is because 146097 happens to be a multiple of 7.

400 years is a long time, and this post has gotten pretty long, too. So I’ve broken it into two parts. Come back soon for Part II, where we will examine the mathematical labyrinth that is the Hebrew calendar…