Two Metapatterns Across Cultural Timekeeping

Ryan Cuddy
New York University

Citation information: Cuddy, Ryan . 2017. Two Metapatterns Across Cultural Timekeeping. The NYU Student Journal of Metapatterns, volume 1, issue 1. Available at:


While we live on Earth, inventing a patterned, structurally cyclical calendar that permanently and accurately records the passing of days is impossible. Across world history, cultures have attempted to invent repetitive calendar systems that nest solar days into solar years. So far, a perfect system does not exist; however, increasingly better systems have gradually evolved since civilization began. This paper is about the practice of inventing a timekeeping strategy, and has nothing to do with physics. I explore two patterns that are my own observations. The first is what I call strange time: a solitary, recurrent segment of a calendar system that is unlike any other segment in length. The second is what I call leap cycles: the largest unit of time in any calendar that both repeats itself infinitely, and is identical in length during each repetition. I’ll also be calling on what Volk’s metapatterns to describe some structures and patterns in calendars. To conclude, I will reinforce the failure and impossibility of all solar calendars, and propose a few alternatives.


Days and years have a fundamental incompatibility that all solar calendars must face. The “solar year” is the length of time between summer solstices. Right now, the average solar year is approximately 365 days, 5 hours, 48 minutes, and 45 seconds. We experience our lives in whole solar days, but since the number of solar days in each solar year is not a whole number, each year of a calendar cannot be the exact same number of days. If every year were the same whole number of days, like 365 or 366, then eventually the calendar accumulate enough error so as to be inaccurate. Leap years do not solve this either, because the solar year is ~365.242, not 365.25, solar days long. The complexity of the .242 fraction makes creating a permanently accurate, repetitive calendar very difficult. Cultures around the world have all had to come up with seemingly counterintuitive practices in order to make astronomically accurate calendars based on days. So far, solar calendars have only approached perfection.

Background — Calendar Segmentation

Before getting into the two main concepts of this paper, I must acknowledge that yearly calendars are always segmented. Since the moon cycle is 29.53 days long, twelve is the average number of new moons in each solar year, making it a reasonable number for historic calendars divide the year into. The calendars used in Zoroastrian Persia, Sumeria, Zhou-era China, Classical Greece, and Imperial Rome all segmented their years into twelve segments.

There were calendars based on numbers other than twelve. The beloved Mayan calendar had multiple cycles based on different numbers, but its solar calendar was divided into eighteenths, as was the Aztec calendar. Pre-Zhou China’s calendar was divided into fifths. Hellenistic Rome, leading up to the lifetime of Julius Caesar, had ten segments.

Volk describes the metapattern of holons and clonons. Colons are identical, interchangeable units that compose a body of units, and holons are each unique parts of a body. He draws the wonderful example of humans: working in a factory, we are clonons, each identically doing the other’s job; whereas seated around a dinner table, we are holons, each a unique contribution to the collected company.

Figure 1. This figure shows two wholes divided into segments. The left wholeis divided into identical clonons, the right whole into unique holons.

It is easy to think of year segments, months, as holons, since each has a unique name with associated holidays and climate. Calendar segments develop cultural significance: the Aztecs celebrated the occurrence of each; the pre-Zhou Chinese associated chakras with each; we associate festivity with December, etc. However, year segments are functionally clonons. Each does the same job as the other: segment 1, segment 2, segment 3,…segment 12. Calendars are mathematically segmented into clonons of identical or near-identical length.

Figure 2. Our own year segmentation. Blue months have 31 days. Green 28-29 days, and red 30, a few holons repeating themselves.

The Strange Time

Calendars are segmented into clonons, except for one holon: the strange time. Strange time is the occurrence of a calendar segment that is different from all other calendar segments. Sometimes, the strange time rounds out the solar year, tacking a few extra days onto the end in order to make the segmented calendar 365 days long, a difficult number to divide evenly.1 Sometimes, the strange time only occurs once every n number of years to make up for accumulated error. When a strange time occurs every nth year, it’s called an intercalation.

Figure 3. These two “calendars” each show four years, in which each year has 5 segments. The calendar on the left has a short strange time at the end of every year. The one on the right has an intercalary strange time every two years.

An easy example to understand the strange time is the Aztec calendar, very similar to the Mayan. In the Aztec solar calendar, Xiuhpohualli, eighteen twenty-day periods occur back to back over the course of the year for a total of 360 days in the year. The Aztecs knew the year was about 365 days long, though, and employed a five-day period at the end of their 360-day year to round out the calendar. Eighteen identical periods followed by one strange period: the strange time. Recalling Volk’s clonons and holons, the strange time is a single holon at the end of a series of clonons.2

Figure 4. Xiuhpohualli, one year with eighteen 20-day segments and a single 5-day segment.

The Julian Calendar, where every fourth February gets an extra day, has an intercalary strange time every four years. February 29th is not the strange time: the entire 366-day leap year is the strange time. A strange time can be longer or shorter than the other segments, but it is always different in length than the other periods. In Figure 3, I only highlighted red the inserted biannual, intercalary period. This was just for visual simplicity: in fact, the entire year during which the intercalary period exists is a strange time. In a biannually intercalary calendar, either the year with the inserted period of the year without the inserted period could technically be considered a strange time, but not both.

Figure 5. The Julian and Gregorian calendar, where blue years have 365 days and red years have 366 days. 2016 was a strange time for its length, among other things.

All reasonably accurate solar calendars have a holonistic strange time at some point. I also want to stress that strange times are different from intercalations in that they sometimes occur every year.

Let’s look at some neat examples. The Attic Calendar, used in Athens in between the lifetimes of Homer and Plato,3 had twelve months with 29 or 30 days in each for a total of 354 days. Every other year, an intercalary, month-long strange time was added, making every other year technically a strange time. The Sumerian Calendar, with 12 lunar-based segments totaling 354 days like the Attic, added a 62-day long, thirteenth month every six years. The Sumerian strange time, then, is this 416-day intercalary year. The (geographically) nearby early Persian calendar also added a thirteenth calendar segment every six years to accommodate their too-short calendar. At the end of pre-Zhou China’s five phases, each with 72 days for a total of 360 days, a five day strange time called gatha had to be added every year.

Figure 6. Each of these calendars shows six years. The Attic on the left has three strange times in six years, the middle Summerican has one strange time every sixth year, and the pre-Zhou on the right has one strange time every year. Since the pre-Zhou calendar’s strange time occurs annually, the calendar shown here is divided into “months” (its year’s segments).

According to Íslendingabók, written ~1130 CE, the 10th century CE Icelandic parliament had to meet once a year, or every 52 weeks (which had 7 days each). Their 52-week year gave the Icelandic calendar 364 days. Íslendingabók notes that in a 364-day calendar, the occurrence of summer “receded back toward the spring,” meaning the solstice was occurring earlier and earlier in the too-short calendar. To fix this, the newly week-based society decided to add an extra week every seventh summer, inaugurating an intercalary 53-week strange time every seven years.

My favorite example is the Julian Calendar, used in Europe until our modern Gregorian calendar replaced it in 1582 CE. Before Roman ruler Numa Pompilous invented two new months, Ianuarius and Februarius (January and February), an unnamed winter strange time took place in Rome after ten named months passed. The ten months began with the Spring, and lasted roughly until the winter solstice, after which the unnamed, unsegmented strange time, varying in length, lasted until the Spring equinox. Ianuarius and Februarius divided that strange time into two months, eliminating it altogether and creating a 12-month calendar year. With no strange time, the calendar year fell out of cycle with the motion of the sun.

Realizing this, Julius Caesar simply re-started the calendar at a new point, January 1st, in an attempt to reset the calendar with the sun’s cycle. However, without a strange time, this measure would only delay future misalignment. After Julius’s famous death, Augustus invented the leap year, placing a 1-day strange time at the end of Pompilous’s ‘new’ month, February, every four years. Thus, the 365-365-365-366 pattern was born.

If the solar year cycle took 365.25 days, instead of ~365.242 days, Augustus’s solution would be perfect- useful for the rest of history. It would take over 1,500 years for Pope Gregory XIII to implement a better one.

Leap Cycles

My second concept is the leap cycle, borrowing “leap” from the Julian “leap year.” I’ll start off by explaining the simple Julian Calendar’s leap cycle, but all calendars that have strange times also have leap cycles. A calendar’s leap cycle is the largest unit of time, infinitely recurring, that is always identical in length to other units of the same name. An example will help here.

In the Julian Calendar, which adds a 366th day to every fourth year, the leap cycle is four years long. Since the length of the word “year” changes every fourth year, it is more accurate to say that the Julian leap cycle is 1,461 days long [(3)(365)+366]. A Julian Calendar’s year is not always identical to other units of the same name, since year A may have 365 days and year B 366. However, the Julian Calendar's 4-year leap cycle always contains 1,461 days. There is no larger unit of time in the Julian Calendar; identical leap cycles recur forever in time’s one dimensional march.

Figure 7. One Julian leap cycle. Blue segments are years with 365 days, red segments are years with 366 days.

Figure 8. Three Julian leap cycles: Years 1-4 are one cycle, 5-8 another, and 9-12 another. Not all years are equally long, but each of these leap cycles has 1,461 days.

The Gregorian Leap Cycle

The Gregorian leap cycle is the most complex I have found in my research. It lasts 146,097 days, First, some history is necessary to explain the difference between the Gregorian and Julian calendars.

When Pope Gregory XIII issued the Inter gravissimas in 1582, he made two adjustments to Augustus’s Julian Calendar that ended up shrinking the average year length by 10 minutes and 48 seconds. First, he abolished the leap year- the occurrence of February 29th- during years that are divisible by 100, such as 1600, 1700, 1800, etc. Second, he blocked his own abolition during years that are divisible by 400. So…

1600: Divisible by 100, also divisible by 400- February 29th occurs
1700: Divisible by 100, not divisible by 400- February 29th does not occur 1800: Divisible by 100, not divisible by 400- February 29th does not occur 1900: Divisible by 100, not divisible by 400- February 29th does not occur 2000: Divisible by 100, also divisible by 400- February 29th occurs
2100: Divisible by 100, not divisible by 400- February 29th does not occur

So, given Gregory’s change, what is the Gregorian leap cycle? What is the largest unit of time that is always identical to other units at the same scale, or denoted by the same word? In our Gregorian calendar, it is 400 years. Since “years” are not all the same length, let me be clear: the Gregorian leap cycle is 146,097 days long. It is always 146,097 days long.

I’m talking about the clumping of days into identically long units of multiple days.4 Consider the units of days in the Gregorian Calendar as they are clumped, and whether or not they are equal in number of days:

  • Months are unequal: February (28-29) < April (30) < May (31)
  • Years are unequal: 1999 (365) < 2000 (366) > 2001 (365)
  • Leap year cycles (Julian leap cycles) are unequal: (1896 through 1899) > (1900 through 1903) < (1904 through 1907)5
  • Even the centuries are not equal: The 1600’s (36,525) > the 1700’s (36,524); the 2000’s will be longer than the 1900’s were.

The leap cycles, which reset in our Gregorian calendar every time a leap year is divisible by the number 400, however, are always equal- 146,097 days long. The Gregorian year 1582 had to skip ten days in order to accommodate the Julian Calendar’s overly long years, so I think it’s reasonable to consider January 1, 1583 the beginning of the first Gregorian leap cycle and December 31, 1983 the end, placing us at the beginning of the second.6

Before we more briefly identify some other leap cycles, I want to stress a point. Leap cycles are a pattern of timekeeping. A leap cycle is not an inherent unit of time; a leap cycle is the length of time it takes different calendars to reset. All calendars are inherently cyclical, and most contain multiple cycles within cycles. A leap cycle is the longest cycle of any given calendar.

Some leap cycles are just one year long. I say “one year” instead of the number of days, because a “year” can be different lengths in different calendars. Recall the Maya and Aztec calendars that, each year, put a 5-day strange time after eighteen 20-day “months”. The Maya strange time in this case functions as the beginning of a new leap cycle; the leap cycle is one year long. Though the months are all the same length, the strange time’s abnormal length is the reset point. The pre-Zhou China, too, had 1-year leap cycles.

After five 72-day months, a strange time ended the year, beginning the five months over again. Therefore, the pre-Zhou calendar had a 1-year leap cycle. Hellenistic Rome, with its 10 months totaling 304 days and subsequent winter strange time, also had a 1-year leap cycle. In these three cases, no cycle within the calendar is longer than one year.

Recall 10th century Iceland's strange time: the addition of a 53rd week of the year every 7th year. In this calendar, the leap cycle is seven years long. In this case, it’s better to say the leap cycle is 2,555 days long [(6)(52 weeks)+53 weeks]. Not all years are equally long, so the occurrence of year A may record a different amount of time than the occurrence of year B; however, the occurrence of leap cycle A is identical in length to leap cycle B.

Recall the 12-month Attic Calendar, which alternated between 29 and 30-day months in an A,B,A,B,… pattern, totaling 354 days. Every other year, the interjection of a 13th month, a strange time, kept the calendar reasonably in line with the solar year. The Attic Calendar’s strange time wasn’t always the same number of days, though, as timekeepers just added however many days were necessary to align the cycle with the summer solstice7 ( This is a wonderful example because we know something the Athenians didn’t. To them, their leap cycle would have been every two years with the coming of the strange time.

The Sumerian calendar, with its additional month every six years, are just the same concept as the Icelandic and Attic examples. Their leap cycle begins on the first day of the intercalary strange time, and ends the day before the next intercalary strange time. For the Sumerians and early Persians, this would have been seven years.

Deeper Conclusions and the Next Calendar

All leap cycles contain one strange time. This is obvious in calendars like the Sumerian, where a strange time comes every six years to reset accumulated error, inaugurating a new leap cycle. It’s not so obvious in the Gregorian Calendar. In fact, I have not yet explicitly stated the Gregorian strange time in this paper.

Recall the Gregorian leap cycle is 400 years, or 146,097 days, long. The Gregorian strange time therefore occurs once every 400 years, but how long does it last? It occurs during the special period when Pope Gregory XIII breaks his own rule: when a year divisible by 100 doesn’t skip its leap year like it’s supposed to. This has only happened in 1600, and 2000. In both of those years, a leap cycle began. The Gregorian strange time is exactly 36,525 days, or 100 years and one day, long. Let’s break it down:

  • A leap year occurs in 1600. From 1600-1699, 25 leap years occur.
  • A leap year is skipped in 1700. From 1700-1799, 24 leap years occur.
  • A leap year is skipped in 1800. From 1800-1899, 24 leap years occur.
  • A leap year is skipped in 1900. From 1900-1999, 24 leap years occur.
  • A leap year occurs in 2000, restarting the cycle. From 2000-2099, 25 leap years will occur

Every fourth century that has one more day than the other centuries. The Julian strange time occurs when every fourth year gains a day, and the Gregorian strange time analogously occurs when
every fourth //year ending with “00” // gains a day. This leads me to a realization: we are currently living in a Gregorian strange time. The 21st century has one more day than the three centuries before or after it will.

Figure 9. Two Gregorian leap cycles. The lengths of the centuries are different- the 1600’s and 2000’s have one more day than the others, making them strange times.. However, the length of any four concurrent centuries is always equal to the length of any other four concurrent centuries — 146,097 days.

The Gregorian Calendar will eventually be replaced. The Gregorian system is still not quite perfect, despite being highly reliable over the span of a human lifetime. Every Gregorian leap cycle has 97 leap years, meaning the average Gregorian year is 365 + (97/400) days long, or 365 days, 5 hours, 49 minutes, 12 seconds. Meanwhile, the average solar year is as many days and hours, 48 minutes, 45 seconds. The difference is 27 seconds: our year is 27 seconds too long. At this rate, it would take 1,600 years for the Gregorian calendar to overshoot by a half-day, indicating the wrong date. 435 years have elapsed since this countdown began — 1165 years to go. This is fancifully arbitrary in our lives, but sooner or later, the humans of the future will have to replace the Gregorian Calendar.

Pope Cuddy’s suggestion: every year that is divisible by 3,200, scratch out Pope Gregory’s second rule. He cancelled leap year in years divisible by 100, but canceled the cancellation in years divisible by 400. If we cancel the cancellation of the cancellation every 3,200 years, we will subtract one day from Gregory’s 27-second over calculation. I say every 3,200 years because we’re currently adding a half day every 1,600 years. If >.5 days gets added, the rounded date will be wrong. I said earlier that this will happen 1600 years from the Gregorian beginning, which is the year 3182 CE. Near perfect! The new Cuddaic leap cycle would be 3,200 years long. Since Gregory’s is simply 1/8th of the Cuddaic minus 1, let’s do the math. [8(146,097)-1] = 1,168,775-day Cuddaic leap cycle.

Presuming my suggestion is not adopted, here’s one other possibility. If humanity expands to other planets, establishing a universal calendar based on cosmology will be difficult. Expanding away from Earth would free us from the impossibility of the solar calendar, since solar years would be different lengths on various planets. A repetitive calendar with nested segments could be kept track of by a computer for all time. We wouldn’t even have to give up our 7-day week lifestyles if we didn’t want to: 7 days make a week, 4 weeks make 28-day months, thirteen 28-day months make 364-day years for all humanity, regardless of galactic location. The perfect system could land us right back where we started!

Author’s Note

I am not a physicist or horologist. I have done research into ancient calendar systems, but have not researched each extensively. My work in this paper was about the overarching patterns of calendars and their inevitability, not any specific calendar or culture in particular. Some ancient calendars are very complex, and I have done my best to represent them simply.

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