Endless progress of time: The Mayan Long Count
I conceive the endless progress of time as the supreme mystery of Maya religion, a subject which pervaded Maya thought to an extent without parallel in the history of mankind. — Eric Thompson, 1950
Brief summary
This excerpt from Richard Elwes' book Huge numbers investigates some fascinating features of the Mayan calendar.
The Maya have lived in the Yucatán Peninsula in Central America, since 2000 BCE or even earlier. Several million Mayan people live in the region today, and these people’s ancient forebears were the citizens of the classic Mayan period, from around 250 CE until a so-called collapse around 900 CE saw much of the southern region abandoned.
The Mayan Base Twenty Numeral System
Today our usual way of writing numbers is a place value system: in "333" the rightmost symbol means "three", while the symbol in the middle means "three tens" and the left most symbol means "three hundreds". The Mayan system was like this too, but their columns were not units, tens, hundred, etc., but units, twenties, four-hundreds, and so on, each standing for twenty of the previous.
The classical Maya had a base twenty place value written numeral system (see right), and three overlapping dating systems they used simultaneously: a Tzolkin year of 260 days for their religious calendar, a Haab year of 365 days for"tracking the seasons, and a third system, the astonishing Long Count Calendar, used for creating the dates we find on stone monuments throughout the region. Its usage defines the period known as "classical Mayan" But they counted their days from a much earlier date — a mythical day zero which today's scholars put at 11 August 3114 BCE.
The Mayan Long Count
The long count was based on yet another notion of a year: the 360-day Tun. This was divided into eighteen 20-day months or Winals. The appearance of eighteen here introduces an awkward asymmetry into the system. At every other level a new unit is formed as twenty of the previous one, which matches the base-20 numeral system, but not in the third position. When long-count dates are written out, rather than being a pure base-20 system, the third entry represents a quantity of eighteens. Instead of units of 1, 20, 400, 8000, 160,000 days — each twenty of the last — we find units of 1, 20, 360 (i.e. eighteen twenties), 7200 (twenty 360s), 144,000 days (twenty 7200s), and so on, respectively known as the Kin (day), Winal (month), Tun (year), Katun, and Baktun, with the Katun representing approximately 20 solar years and the Baktun 394.
The Baktun was the Mayan way of marking the current historical era and referencing earlier ones. With these units, dates could be given as a sequence. For instance, the date 9.13.10.0.0 represented the start of a new Tun year: the tenth Tun of the thirteenth Katun of the ninth Baktun. Starting the count on the zero day of 11 August 3114 BCE, this works out as 26 January 702 CE in our calendar.
The Maya saw great symbolism in the beginnings and ends of these periods, moments to be marked with suitable rites, perhaps including the sacrifice of captured enemies. Popular interest in these time cycles outlasted the classic Mayan civilisation and saw a huge spike in the run up to 21 December 2012— a date calculated to be the end of the thirteenth Baktun. While the end of any Baktun would be a major moment in the calendar, the thirteenth was seen as especially momentous due to one surprising fact: The Maya did not record their zero day as one might expect, as 0.0.0.0.0, but instead as 13.0.0.0.0. Day zero, yes, but seemingly of the thirteenth Baktun of some previous epoch.
Why not zero Baktuns, and in particular, why thirteen? Mayan experts have been asking this question for years — we shall come back to it in a moment — but one possibility is that our world was theorised to have been born just as another ended, when that earlier realm had completed its thirteenth Baktun. So the arrival of our world at that same date, 13.0.0.0.0, was certainly a portentous moment.
This solemn occasion was marked with ceremonies at ancient temple sites in Mexico and Guatemala, with thousands of Mayan people attending, along with crowds of tourists and VIPs including the presidents of Guatemala and Costa Rica. Further afield, this date caught the imagination of all manner of new-age mystics and free spirits, and in particularly excitable circles it started to be interpreted as the date of Armageddon, or some other global transfiguration. While the date did hold a special significance, there is no evidence that the ancient Maya foresaw it as Doomsday. (And if they had, they would have been just as misguided as the clusters of deluded souls who spent 21 December 2012 huddling together against the imminent arrival of alien invaders. The Maya are too often forgotten in Eurocentric histories of science, but let's not overcorrect and ascribe them supernatural powers.)
The Mayan Grand Long Count
One piece of evidence that the Maya did not see 13.0.0.0.0 as the end of the world is that we know they referred to dates beyond this time. Twenty Baktuns make up a longer unit — one Piktun of 2,880,000 days or around 7885 years. The Temple of the Inscriptions is one of the most spectacular of Mayan archaeological sites, in the ancient city of Palenque in Mexico. The temple is the tomb of King Pakal, who ruled Palenque for sixty-eight years after ascending to the throne at age twelve. Carvings at the site mention the date 1.0.0.0.0.8, one digit longer than the five-place dates we usually see. This day marks an anniversary of exactly eighty calendar rounds since Pakal’s accession, one Piktun plus eight days after day zero—in our calendar October 21 in the year 4772 CE. The exact significance of this anniversary is not clear, perhaps foreseen as a date for the glorious leader’s return.
Still the Maya did not stop. Longer time periods have been found for which scholars have provided names: One Kalabtun is twenty Piktuns (57,600,000 days or over 150,000 years), one Kinchiltun is twenty Kalabtuns (at 1,152,000,000 days, over 3 million years), an Alautun is twenty Kinchil tuns (23,040,000,000 days, over 60 million years), and a Hablatun twenty Alautuns (460,800,000,000 days or over 1 billion years). These higher levels appear rarely, but when they do it is mostly in one particular and rather puzzling context. In Yaxchilan, another famous Mayan ruin in southern Mexico, there are two flights of stone steps heavily decorated with stunning Mayan hieroglyphs. On the seventh step of the second staircase, we find the date 13.13.13.13.13.13.13.13.13.13.9.15.13.6.9.
At first glance this would seem to be a day in the unimaginably distant future. But putting it alongside other more conventional dates written nearby, it is clear that it refers to the date when the carving was inaugurated, 9.15.13.6.9 (15 October 744 CE). But it is written preceded by ten 13s. This adds further mystery to the question of the Mayan date of creation. The Yaxchilan stairway implicitly puts the start of history not at 0.0.0.0.0 nor even as 13.0.0.0.0 but as 13.13.13.13.13.13.13.13.13.13.13.0.0.0.0. This indicates an earlier world—or series of earlier worlds—dating back more than 55 trillion years (2×1019) before 11 August 3114 BCE. Modern science puts the age of our universe at around 13 billion years old, less than one four-thousandth of the age implied by this date.
At least, it would do, if this were the full extent of it. But on three stelae (stone monuments) in the ancient city of Coba, another archaeological site in modern-day Mexico, we find the largest Mayan number discovered so far. They rewrite day zero as: 13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.0.0.0.0
Such inscriptions put the 3114 BCE dawn of our world "more than 28 octillion years after the true initial base date in the incomprehensible past", according to Mayan expert David Stuart, in his 2011 book, The Order of Days. He argues this date is the basis of a "Grand Long Count", the fullest expression of the Mayan calendar which we usually see only in abbreviated form. The time for this whole grand long count to reset would be a cycle of over 15 nonillion days (1.5×1031) or 43 octillion years. This reset date would take us to a point long beyond the demise not only of the sun but of the Milky Way galaxy itself, whose stars will long since have been snuffed out, and whose freezing remnants will have been consumed by a supermassive black hole or ejected into the cosmic vacuum. So if the Grand Long Count reset is imagined to represent the end of the world, the Maya would appear to have overestimated by some distance.
Other Mayanists disagree that the Coba stele figure should be interpreted literally as a date, arguing that the string of twenty 13s plays a symbolic role, somehow providing emphasis to this crucially important date.
Regrettably, we will probably never be able to resolve this argument; the evidence which might have settled it has gone up in smoke. Either way, though, the fact remains that the classical Maya took the time and trouble to think about numbers of this size, to write them out fully, and indeed to carve them on major public monuments. Whether intended literally or metaphorically, numbers up to the scale of nonillions played an important role in Mayan public life—not something we can say today.
Did they devise a powerful positional numeral system to be able to express the gigantic numbers they were already contemplating? Unlikely—the chain of causation probably ran the other way around—their wonderful numeral system, perhaps the most elegant ever devised, opened the door to regions of numbers they had never previously imagined, which they then weaved into their cultural myths and rituals.
We will probably never know whether the Maya really imagined a world which exists on such immense timescales. But on the other side of the world, there were people who certainly did—and indeed continue to do so.
This is an extract from Huge Numbers (A Story of Counting Ambitiously, From 4½ to Fish 7) by Richard Elwes (Basic Books, April 2026).
“Humanity has always been entranced by big numbers — the bigger the better. This fascinating exploration of the giants of the mathematical world is clear, informative, and immensely readable. Wonderful!” – Ian Stewart
Richard Elwes is an associate professor of mathematics at University of Leeds in the UK, where he has taught subjects including mathematical logic and the history of mathematics. He is the author of six popular maths books including Huge Numbers. The book is available to order in the UK and in the USA.