Mayan arithmetic

The long count is an odometer: five dials, one bent gear, and no top end in sight. ← Calendar Contraptions

Most calendars you have met count within a year: months, weeks, the day after the solstice. The Maya Long Count does not care about the year at all. It is a pure day counter — an odometer that has been running since a creation date in 3114 BCE and simply names today by how many teeth the machine has advanced since then. Five dials, each twenty times the one before it — except one gear, bent on purpose, so the third dial lands on a number the sun can live with. Below: a working machine, and the story of its parts.

Five gears, one bent

The long count is usually called vigesimal — base-20 — and that is almost true. In a pure base-20 system every gear would turn twenty times the one before it. Here, four of the five gears do. The exception is the second gear from the right: the winal, which holds not twenty days but exactly 20, and the tun above it counts 18 winals, not 20. Everything else is clean twenties. The reason is the same reason your own calendar has leap years: twenty twenties would make a "year" of 400 days, which overshoots the sun by five weeks, while 18 × 20 = 360 undershoots it by five days — close enough to stay tethered to the seasons, and the tun really did mean "year". A pure base-20 machine with the same five gears would roll 20,000 days per top-tooth; the Maya got 144,000 — and a fifth gear that turns once every 394-odd years instead of every 30.

Written on stone, the numbers were dots (1) and bars (5) stacked in columns, one column per gear — so a scribe needed a symbol for "this gear is empty", and that is where the zero comes in (below). The gear names after bʼakʼtun — piktun, kalabtun, kʼinchiltun, alautun — are modern coinages by Mayanist scholars, not words the Classic Maya are known to have used.

Zero, drawn as a shell

Place notation is a pact: the meaning of a digit depends on which column it sits in. That only works if an empty column can say so — without a zero, 1.0.13 and 1.13 are the same string of bars and dots with different meanings and no way to tell them apart. The Maya wrote the zero as a shell glyph, and the long count is one of the earliest known uses of a zero as a place-holder anywhere in the world — older than every surviving European zero, contemporary with the earliest Indian ones. The machine on this page needs it for exactly the same reason a stela did: the dials most days are not all zero, but the day the count rolls over, four of the five must be written as nothing. The creation date itself — four zero gears and a fifth at 13 — is mostly zeros, and it was carved on stone over and over.

The Calendar Round, and why the long count exists

Day to day, the Maya ran on two interlocking wheels. The tzolkʼin runs a 13-tooth number wheel against a 20-tooth name wheel — 13 × 20 = 260 unique days, from 1 Imix to 13 Bʼen, then 1 Ix through 7 Ajaw and back to 8 Imix. The haabʼ is a 365-day vague year: eighteen months of twenty days (0 Pop, 1 Pop … 19 Pop, then 0 Woʼ …) plus five closing days called Wayebʼ, an unlucky stretch when the portals to the underworld stood open and one did not wash or comb one's hair. Both wheels turn every day; any given pairing of the two — a Calendar Round date like 4 Ajaw 8 Kumkʼu — repeats after the least common multiple of 260 and 365: 18,980 days, which is 73 tzolkʼin years and 52 haabʼ years, about 52 of our solar years. "Roughly once each lifetime," as the archaeologists put it — long enough to feel unique, short enough that a 60-year-old has seen every date twice. A history written in Calendar Round dates gets ambiguous after 52 years, and so the long count exists: it is the unambiguous, linear counter laid under the wheels.

The 52-year drift between the two systems has its own arithmetic hook. A 365-day haabʼ is 28 weeks of 13 plus 1, so the tzolkʼin number of each new year's first day advances by exactly 1 from year to year; it is 18 twenties plus 5, so the tzolkʼin name advances by five. Neither wheel ever resyncs with the seasons — the haabʼ drifts against the sun like the Egyptian year — and the Calendar Round simply doesn't care. It is a 52-year clock with no year numbers at all.

The correlation question

Here is the strange part: we can read every digit on every stela, but matching their day count to ours took centuries of argument, because the people who could have told us stopped carving long counts about 1,200 years ago. The working answer is the GMT correlation — Goodman, Martínez, Thompson — a fixed offset of 584,283 days: Julian Day 584,283, the day the count reads 13.0.0.0.0 4 Ajaw 8 Kumkʼu (the stelae write 13 where a calculator writes 0), is 11 August 3114 BCE in the proleptic Gregorian calendar, which fell on a Monday. A respected minority correlation, Lounsbury's, puts the same day two days later — the evidence runs through colonial documents (Landa's notes, the Oxcutzcab chronicle, the books of Chilam Balam), lunar data carved beside long count dates, and the fact that highland Guatemala communities still keep the tzolkʼin today, all of which pin down the GMT count. The stakes are not abstract: under GMT the 13th bʼakʼtun completed on 21 December 2012; under Lounsbury, 23 December. Every date on this page uses GMT — call every output honest to within two days of a centuries-old argument.

The odometer that doesn't stop

In 2012 the machine completed its 13th bʼakʼtun — 13.0.0.0.0, Friday 21 December in the GMT correlation — and a modern industry of end-times books read "rollover" as "end of world". The stones say otherwise. The bʼakʼtun gear runs to twenty, not thirteen: the current era's dates simply continue 13.0.0.0.0 → 14.0.0.0.0 → … → 19.0.0.0.0, and a monument at Palenque goes further, recording the completion of the first piktun — twenty bʼakʼtuns, all five gears flipping to zero at once — on 13 October 4772. The west panel of Palenque's Temple of the Inscriptions counts from king Kʼinich Janaabʼ Pakal's birth (9.8.9.13.0, 24 March 603) by a distance number of 10.11.10.5.8 to land on 21 October 4772 — the 80th Calendar Round anniversary of his accession (5 Lamat 1 Mol, both times), eight days after that piktun completion. A king's courtiers, in the 7th century, scheduled a commemoration past the year 4772. The full odometer has no known top: Stela 1 at Coba writes the creation date with twenty 13s stacked above the bʼakʼtun gear — a count one scholar called "the starting point of a huge odometer of time" — good for about 2.687 × 10²⁸ years, some two quintillion times the age of the universe. As for the rollover of 2012 itself, the curator Susan Milbrath put it plainly: "We have no record or knowledge that they would think the world would come to an end." The Popol Vuh, the colonial Kʼicheʼ creation epic, does say the current world is the fourth — but its previous worlds lasted 13 bʼakʼtuns by the count, and the wheels this time kept turning past 13 without ceremony.

What this page doesn't do

All dates come out on the proleptic Gregorian calendar — the one nobody was using before 1582 (see ten days that never happened) — under the GMT correlation only; a Lounsbury believer reads every date on this page two days differently, and the correlation question above is why. These are calendar dates, not instants: nothing depends on your time zone, and "today" means your device's own date. The stela box accepts bʼakʼtuns 0–19; higher gears (piktun and up) are history's footnote, not a mode. The page does not touch the lunar series, the lords of the night, or the short count — each of which is its own contraption, and some are probably worth their own page.