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Algorithms Calendar Calculations: Easter Is a Quasicrystal

Revealing the divine mathematics of a holiday

MATHEMATICAL RECREATIONS by Ian Stewart

Scientific American, March 2001

Ten years ago my first Mathematical Recreations column was about Fermat’s Christmas Theorem. With the Lenten season upon us, it seems only fitting to devote this 96th, and my final, column to Easter.

Christmas always falls on December 25, but Easter is quite another matter. The holiday can fall on any date between March 22 and April 25, a five-week window. The date changes from year to year for a number of reasons. First, the date has to be a Sunday because the crucifixion occurred on a Friday and the resurrection on a Sunday. Second, the New Testament says the crucifixion took place during the Jewish holiday of Passover, which is celebrated for eight days following the first full moon of spring.

The date of Easter is thus linked to several astronomical cycles, and it is here that the difficulties arise. The lunar month is currently about 29.53 days long, and the solar year about 365.24 days long. This leads to 12.37 lunar months per year, an inconvenient relationship because it is not an integer. It so happens that 235 lunar months are very close to 19 solar years, and the church’s system for assigning a date to Easter exploits this coincidence. In A.D. 325 at the Council of Nicaea, church leaders decided that Easter should fall on the first Sunday after the first full moon occurring on, or after, the spring equinox (the date in March on which day and night have equal length).

The year was based on the Julian calendar back then, with one leap year in every four. The dates of full moons were assumed to repeat every 19 Julian years; a bit of juggling with the calendar made this period equal to 235 lunar months. The 19-year period was called the lunar cycle, and each year’s position in this cycle was indicated by its Golden Number, which ran from 1 to 19. The entire cycle of the Julian calendar would repeat every 76 years—after four lunar cycles of 19 years each, the pattern of leap years would repeat. The mathematical principle here is that the length of a cycle is equal to the lowest common multiple of the lengths of its constituent subcycles (76 is the lowest common multiple of 19 and 4).

Easter dates repeated in a 532-year cycle, because 532 is the lowest common multiple of 76 (the Julian calendar’s cycle) and 7 (the cycle of days in the week). It was a tidy system, but unfortunately it did not accurately account for the true lengths of the lunar month and solar year. As the centuries passed, the calendar started to slip relative to the seasons. (Dante, the medieval Italian poet, pointed out that eventually January would cease to be part of winter.) Finally, in 1582 Pope Gregory XIII reformed the calendar by omitting the leap day in all years that are multiples of 100, except for years that are multiples of 400 (such as the year 2000, for example). To correct for the previous slippage, the Gregorian calendar for 1582 skipped the 10 days between October 4 and 15.

The church’s new procedure for calculating the date of Easter assigned each year a number called the Epact, an integer between 0 and 29. This number reveals the phase of the moon on January 1 of each year, with 0 indicating a new moon and 15 a full moon. Every so often the cycle of Epacts must be revised to account for the fact that 235 lunar months do not exactly equal 19 solar years. The last such correction occurred in 1900, and the next will be needed in 2200.

Using the Golden Number and Epact for any given year, one can calculate the date of the first full moon of spring. To determine the date of Easter—the first Sunday after the full moon—the church assigned each year a dominical letter, from A to G, indicating the date of that year’s first Sunday: A for January 1, B for January 2, and so on. Every leap year has two dominical letters, one for January and February and the other for the remaining months.

The system has its flaws. The church considers March 21 to be the perennial date of the vernal equinox, but the real astronomical equinox can occur as early as March 19 (as will happen in 2096). Also, the moon does not slavishly follow ecclesiastical conventions. In 1845 and 1923 the first full moon of spring occurred after Easter Sunday in the world’s easterly longitudes.

In 1800 German mathematician Carl Friedrich Gauss invented a simple algorithm that incorporated the church’s rules for calculating Easter’s date. Unfortunately, Gauss’s work contained a minor oversight: it gives April 13 for the year 4200 when the correct date should be April 20. He corrected this error by hand in his own copy of the published paper. The first flawless algorithm was presented in 1876 in the journal Nature by an anonymous American. In 1965 Thomas H. O’Beirne of Glasgow University published two such procedures in his book Puzzles and Paradoxes (Oxford University Press). O’Beirne’s method puts the various cycles and adjustments into an arithmetical scheme.

In general terms, the date of Easter slips back by about eight days each year until it hops forward again. The pattern looks irregular but actually follows the arithmetical procedure just described. In 1990 Alan Mackay, a crystallographer at the University of London, realized that this near-regular slippage ought to show up in a graph that compared the date of Easter with the number of the year. The result is approximately a regular lattice, like the arrangement of atoms in a crystal.

The peculiarities of the calendar, however, make the dates vary slightly as compared with the lattice. The graph more closely resembles a quasicrystal, a molecular structure built for the first time in the early 1980s. Quasicrystals are not as regular as crystals, but their arrangement of atoms is by no means random. The structure is similar to a curious class of tilings discovered by University of Oxford physicist Roger Penrose; these tilings cover the plane without repeating the same pattern periodically. The atoms of quasicrystals have the same near regularity, as do the dates of Easter. The holiday is a quasicrystal in time rather than space.

Under the rules of the Gregorian calendar, the cycle of Easter dates repeats exactly after 5,700,000 years. Long before the first repeat, though, the rules will have slipped relative to astronomical realities: the lengths of the month and day are slowly changing, mainly because of tidal friction. Just for the fun of it, though, try to calculate Easter’s date for the year 1000000.

 Choose any year of the Gregorian calendar and call it x. To determine the date of Eas-ter, carry out the following 10 calculations (it’s easy to program them on a computer): 1. Divide x by 19 to get a quotient (which we ignore) and a remainder A. This is the year’s position in the 19-year lunar cycle. (A + 1 is the year’s Golden Number.) 2. Divide x by 100 to get a quotient B and a remainder C. 3. Divide B by 4 to get a quotient D and a remainder E. 4. Divide 8B + 13 by 25 to get a quotient G and a remainder (which we ignore). 5. Divide 19A + B – D – G + 15 by 30 to get a quotient (which we ignore) and a remainder H. (The year’s Epact is 23 – H when H is less than 24 and 53 – H otherwise.) 6. Divide A + 11H by 319 to get a quotient M and a remainder (which we ignore). 7. Divide C by 4 to get a quotient J and a remainder K. 8. Divide 2E + 2J – K – H + M + 32 by 7 to get a quotient (which we ignore) and a remainder L. 9. Divide H – M + L + 90 by 25 to get a quotient N and a remainder (which we ignore). 10. Divide H – M + L + N + 19 by 32 to get a quotient (which we ignore) and a remainder P.Easter Sunday is the Pth day of the Nth month (N can be either 3 for March or 4 for April). The year’s dominical letter can be found by dividing 2E + 2J – K by 7 and taking the remainder (a remainder of 0 is equivalent to the letter A, 1 is equivalent to B, and so on). Let’s try this method for x = 2001: (1) A = 6; (2) B = 20, C = 1; (3) D = 5, E = 0; (4) G = 6; (5) H = 18; (6) M = 0; (7) J = 0, K = 1; (8) L = 6; (9) N = 4; (10) P = 15. So Easter 2001 is April 15.