Calendars, Cycles, and Cool Coincidences (Part II)

This is my second post on the alignment of Thanksgiving and Hanukkah. Go back and read the first post, if you haven’t done so.

When compared to the Julian or Gregorian calendar, the Hebrew calendar is a different animal entirely. First of all, it is not a solar calendar, but is rather a lunisolar calendar. This means that while the years are kept in alignment with the solar year, the months are reckoned according to the motion of the moon. In ancient days, the start of the month was tied to the sighting of the new moon. Eventually, the Jewish people (and more specifically, the rabbis) realized that it would be better for the calendar to rely more on mathematical principles. Credit typically goes to Hillel II, who lived in the 300s CE. In the description that follows, I will be using Dershowitz and Reingold’s Calendrical Calculations as my primary source, with assistance from Tracy Rich’s Jew FAQ page.

The typical Jewish year contains 12 months of 29 or 30 days each, and is often 354 days long. (See how I worded that? It matters.) Clearly, this is significantly shorter than the solar year, so some adjustments are necessary. Specifically, there is a leap year for 7 of every 19 years. But instead of adding a leap day, the Hebrew calendar goes right ahead and adds an entire month (Adar II), which adds 30 days to the length of the year. Mathematically, you can figure out if year y is a leap year by calculating (7y+1) mod 19—if the answer is < 7, then y is a leap year. In the current year, 5774, the calculation is 7*5774+1 = 40419 = 6 (mod 19), so it’s a leap year. With just this fact, the average length of the year appears to be 365.053—about 4 1/2 hours fast. At a minimum, the leap months explain how Jewish holidays move through the Gregorian calendar: since the typical year is 354 days, a holiday will move earlier and earlier each year, until a leap month occurs, at which point it will snap back to a later date. (Next year, Hanukkah will be on 17 December.)

But it’s not as simple as all that. Owing to the lunar origins of the Hebrew calendar, the beginning of the new year is determined by the occurrence of the new moon (called the molad) in the month of Tishrei (the Jewish New Year, Rosh Hashanah, is on 1 Tishrei). Owing to the calendar reforms of Hillel II, this has become a purely mathematical process. Basically, you take a previously calculated molad and use the average length of the moon’s cycle to calculate the molad for any future month. Adding a wrinkle to this calculation is the fact that the ancient Jews used a timekeeping system in which the day had 24 hours and each hour was divided into 1080 “parts”. (So, one part = 3 ¹/₃ seconds.) In this system, the average length of a lunar cycle is estimated as 29d 12h 793p. While this estimate is many centuries old, it is incredibly accurate—the average synodic period of the moon is 29d 12h 792.86688p, a difference of less than half a second.

Once the molad of Tishrei has been calculated, there are 4 postponement rules, called the dechiyot, which add another layer to the calculation:

1. If the molad occurs late in the day (12pm or 6pm depending on your source) Rosh Hashanah is postponed by a day.
2. Rosh Hashanah cannot occur on a Sunday, Wednesday, or Friday. If so, it gets postponed by a day.
3. The year is only allowed to be 353-355 days long (or 383-385 days in a leap year). The calculations for year y can have the effect of making year y+1 too long, in which case Rosh Hashanah in year y will get postponed to avoid this problem.
4. If year y-1 is a leap year, and Rosh Hashanah for year y is on a Monday, the year y-1 may be too short. Rosh Hashanah for year y needs to get postponed a day.

As someone who’s relatively new to the Hebrew calendar, all of this was very confusing to me. For one thing, it’s not clear that rules 3 and 4 will really keep the length of the year in the correct range. For another, it’s not clear what you’d do with the “extra” days that are inserted or removed. Here’s how I think of it: the years in the Hebrew calendar don’t live in arithmetical isolation, but are designed to be elastic. You can stretch or shrink adjacent years by a day or two so that the start of each year begins on an allowable day. When a year needs to be stretched, a leap day is included at the end of the month of Cheshvan. When a year needs to be shrunk, an “un-leap” day is removed from the end of Kislev.

Now here’s the question my mathematician’s soul wants to answer: How long is the period for the Hebrew calendar? This might seem an impossible question in light of all the postponement rules, but it turns out that each block of 19 years will have exactly the same length: 6939d 16h 595p, or 991 weeks with a remainder of 69,715 parts. As with the Julian calendar, the days of the week don’t match from block to block, so we need to use the length of a week (181,440 parts) and find the least common multiple. Using parts as the basic unit of measurement, we have:

lcm(69715, 181440) = 2,529,817,920 parts ≈ 689,472 years.

Wow! We can also calculate the “combined period” of the Hebrew and Gregorian calendars, to see how frequently they will align exactly. Writing the average year lengths as fractions, the calculation is:

lcm(689472*(365+24311/98496), 400*(365+97/400)) = 5,255,890,855,047 days = 14,390,140,400 Gregorian years = 14,389, 970,112 Hebrew years.

For comparison, the age of the universe is about 13,730,000,000 years. So while particular dates can align more frequently (for instance, Thanksgivukkah last occurred in 1888), the calendars as a whole won’t ever realign again. However, I suppose that claim depends on your view of the expansion of the universe!

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…

The Evolution of Plane Curves

This is the third post in the “Evolution Of…” series; the first and second posts can be found here and here.

This time around, we’ll explore some of the words that have come to be used for various plane curves. First of all, a disclaimer: often, the names of the curves existed many centuries before the development of modern algebra and the Cartesian coordinate system. As a consequence, the original names for the curves are more geometric in origin (imagine one of the ancient Greeks saying “umm, well, it looks like a flower… so let’s call it the flower curve“).

While reviewing the curves listed in Schwartzman’s book, I noticed that most of them can be classified into four major groups: the conics, the chrones, the trixes, and the oids.

1. Conics. You’ve probably heard of them—circle, ellipse, parabola, and hyperbola. The first one has its origins in the Latin word circus, which means “ring” or “hoop.” The other three are Greek, with their original meanings reflecting the Greeks’ use of conic sections. Ellipse comes from en (meaning “in”) and leipein (meaning “to leave out”). For the other two, note that –bola comes from ballein which means “to throw” or “to cast”. So hyperbola means “to cast over” and parabola means “to cast alongside”. (If you check out this image from Wikipedia, it may start to make more sense.)
2. Chrones. The two curves I have in mind here are brachistochrone and tautochrone. In Greek, chrone means “time”. The prefixes come from brakhus and tauto-, which mean “short” and “same”, respectively. So these curves’ names are really “short time” and “same time.” Naturally enough, the brachistochrone is the curve on which a ball will take the least amount of time to roll down, while the tautochrone is the curve on which the time to roll down is the same regardless of the ball’s starting point. Finding equations for these curves occupied the time of many scientists and mathematicians in the 17th century, including a controversy between the brothers Jakob and Johann Bernoulli. You could say they had a “chronic” case of sibling rivalry. 
   
3. Trixes. I am a fan of the feminine suffix –trix because it also provides us with the modern word obstetrics (literally, “the woman who gets in the way”—i.e., a midwife). We don’t use this suffix very much anymore, though aviatrix comes to mind. The algebraic curves in this category are trisectrix (“cut into three”) and tractrix (“the one that pulls”), along with the parabola-related term directrix (“the one that directs”). Interestingly, the masculine form of tractrix gives us the English word tractor.
4. Oids. These were the most fun for me. The suffix is Greek, originating in oeides, which means “form” (though in modern English, “like” might be more appropriate). Here are a some examples: astroid, cardioid, cissoid, cochleoid, cycloid, ramphoid, strophoid. Here are their original Greek/Latin meanings: “star-like”, “heart-like”, “ivy-like”, “snail-like”, “circle-like”, “like a bird’s beak”, “(having the) form of turning”. I’ve provided images of each one below—see if you can match the name to the curve!

Don’t worry, there are plenty more word origins coming later! However, I’ll need a break to recharge my etymology batteries. Expect an “intermission” post in the next few weeks.