Introduction to Mesoamerican Mathematics

[00:00:00] Hello, I’m Anthony Aveni, a professor of astronomy and anthropology, and today I’d like to take you on an exciting tour through the world of Maya numbers, Maya mathematics. You know, writing and numeration are two of the outstanding hallmarks of the ancient Maya culture, [which set] that culture apart from the other cultures of Mesoamerica, at least the way we think about it. I think that writing and numeration are part of the reason why we dare to use the word “Classical” to describe these ancient Maya, [why] we dare to compare them with our precious Greeks and Romans. We don’t know whether writing and numeration first came together at the same time in Mesoamerica, although I think they actually did. We do know that the first numbers seem to appear carved in stone back in the sixth century B.C., but they’re not Maya. They are in fact from the Zapotecan highlands near the city of Oaxaca.

This carving in stone shows one of the early examples, not the earliest by any means, but one of the early examples of numeration, and it isn’t Maya. It actually comes from the Zapotecan highlands of Oaxaca, far from Mayaland, and clearly the Maya must have imported this, as I believe all of the cultures of Mesoamerica imported it. There was a single system of Mesoamerican mathematics with all of the properties, the general properties, I am describing. But it seems to have been the Maya who developed this to a high art, that is, to great extremes, so we should be clear about that.

Gestures and Writing

[01:31:00] Counting and writing began with gestures. If I were to count for you, “one-two-three,” with the tips of my fingers, [and] if I had no material medium through which to express my numbers, it is logical that the tips of my fingers might very well become dots as I transform them into a stone tablet. If I continued to count, “four-five,” my extended hand—abstractly—could represent a bar, and this is where our bar-and-dot notation of the ancient Maya actually came from. The examples that you see in my first pictures, the number on the right—two bars and three dots—would be five and five and three, that’s thirteen. The number on the left—a bar and four dots—would be a nine. Maya numbers have a way of coming to life. (In fact the numbers that I am showing you now are not even yet Maya; these particular ones are from the ruins of Xochicalco in highland Mexico.) When I say numbers seem to come alive, here we see two dates being drawn together. They are being drawn together by a rope, held by a hand—the binding of the years, the binding of two different dates together. You can even see the taut stress around that rope, as if one of the dates seems to be buckling under the pressure.

Living Numbers

[02:50:00] When the Maya acquired numeration somewhat later than the sixth century B.C., they brought their numbers to life. We don’t often think of numbers as having a life of their own. For example, if I were to say to you the number “three,” you would immediately respond to me, “Well, ‘three’ what?” We use numbers to describe the quantity of things—three cacao beans or three cactus plants. But the Maya, although they used numbers to describe things, also indicated that the numbers had a life of their own, as you can see exemplified in this stone carving, a stela, from the ruins of Guatemala (Quiriguá, Guatemala). Look at the number fifteen (the third one down from the top), pulling a date, pulling a load of time on a tumpline. It is as though “he” is carrying the burden of time. But he is to be distinguished from the number before him, the one above, which is the number nine. So every number had a god, and every number had its own personality. I suppose [that] if we [of European descent] could think back to our ancient times, there was a time when seven was lucky and thirteen was unlucky. We still know of many hotels and airlines that do not have “thirteen” numbered floors of their hotel, or rooms numbered thirteen, or seats numbered thirteen. So, maybe numbers really do have a life of their own.

Numbers and Time

[04:06:00] Above all, the Mayas used numbers to describe time and, in fact, there is scarcely an example that exists in the Maya world where numbers do not relate to time. Let me show you one example, [but] first let me tell you that there are two different media—two primary media—through which the Maya express their numbers. They are carved in stone, where they seem to be referring to dynastic history—the exploits of the lords and ladies who ruled the Maya world. And then we see them expressed in books. I am going to begin by talking about the monumental inscriptions—the carved inscriptions we see on monuments. And then, a little bit later, we will fly up to the highest realm of Maya mathematics and talk about those books, many of which have not been decoded.

My first example is not a carving in stone; it is a carving on a jade plaque. I chose it because it is easy to decipher, and [easy] to get across to you what these numbers actually mean. They are colossally huge numbers that stagger our imagination and make us really wonder what the Maya were up to. If you have been around the ruins of Tikal or Copan or any of the other sites, you [have seen] these carved stones standing in the open plazas, in front of pyramids. I remember the first time I saw them at Tikal. [I thought they looked] rather like gravestones. Well, they do, like a gravestone, mark the important dates in the life of the person to whom they are dedicated. And the jade plaque I am showing you does the same thing. Now that we know how to read Maya dots and bars, look at the number from the top down. You can see the first entry, an eight (a bar and three dots), followed by two bars and four dots (that is a fourteen), and then a three and then a one and then a twelve: eight, fourteen, three, one, twelve. Now, if we were to read that number 8-14-3-1-12 in our digital system, it wouldn’t make much sense. But the Maya used a vigesimal system, that is, a base-twenty system. Can you guess why?

Vigesimal versus Decimal

[06:08:00] Our digital system comes from the ten fingers on our hands. For example, when I write the number 584, I know I mean four 1s, eight 10s, and five 10s x 10s, although we don’t consciously think of that. When the Maya wrote 584, they meant something entirely different. If they were counting cacao beans, they meant four 1s, eight 20s, and five 20s x 20s in that base-twenty system. Since we know people counted on their fingers and toes in the tropics, it is perfectly logical that they would develop a vigesimal, rather than a decimal, system. Well, this count of 1s, 20s, 400s, 8,000s is changed as you see it on the Leyden plaque because time is being expressed, and when time is being expressed, the count goes [as follows]: 1s, 20s, 360s, 20 × 360s, and so on. That is, the third place in the order of the numeration is changed. I think the Maya did that to equate that third place with the approximate number of days in a year, which is about 360.

If you think that’s quirky to use a different notation system when you describe time, just have a look at your wristwatch. Well, right now it’s eleven minutes after ten, and when this hour ends, it will be fifty-nine minutes after ten, and then the minute after 10:59 will not be 10:60, 10:61, 10:62; it will be 11:00. We change over to the next unit at the fifty-ninth minute. I think we do that for reasons that have to do with the Babylonian calendar and, in fact, they stem from the same reason that the Maya did it—to make the units of time more nearly equal to natural time cycles like the day and the year.

Base-Twenty Notation

[07:51:00] Let us read that number, 8-14-3-1-12, in base-twenty notation, as a time unit, rather than as a count of articles. We have twelve “quin” on the bottom. The word kin is the Maya word for day and sun and time—a very interesting word that suggests to us that the keeping of time was of vital importance to the Maya. The next number up says one, that’s one 20, and the 20s are winals (uinals), as they were called. We might think of them as units very much like our month. Of course, our month has thirty days, sometimes thirty-one. The Mesoamerican month, which was celebrated throughout all of Mesoamerica, not just Mayaland, was the_“winal_, a twenty-day unit.

Then there are the tuns. There are three of those. That’s the third entry, three units of the 360 days. The tun is the word for year. And then above that, 14, 20 × 360. We might think of that as score years, [e.g.] “Four score and seven years ago.” Those are katuns. And the overturning of a katun was really a big deal, almost parallel to our overturning of a millennium, when the numbers go to zero every twenty years in this case. Then the number at the top, the eight, which refers to the number of batuns, and that would be the number of 20 × 20 × 360-day units. Well, if you are not staggered by that number, let me translate it to our decimal system: it is 1,253,912.

Numbers and History

[09:22:00] Well, 912 what? It is 1,253,912 days after creation—that is, the putative creation of the recent epoch of the many cyclic epochs that the Aztecs and the Maya and other Mesoamerican people believed in. That is the date that begins with August the 12th, 3113 B.C., which is the zero point from which the Maya pegged the current creation. By the way, that odometer is going to turn over on December the 23rd, 2012, so if you’re a Maya worshipper, you’d better be ready.

All of these days in the so-called “long count,” then, are counted from creation. And if we translate that date into our time, by using the most widely accepted correlation between the Maya and the Christian calendars, we get September the 15th, 320 A.D., so that’s probably the date that plaque was carved, if it isn’t the date commemorating the event that took place. Notice in that image of the Leyden plaque, now housed in the museum in the University of Leiden, in Holland, that that date takes up literally half of the space of that document. It takes up a whole side. So timekeeping and numbers were certainly very, very important to the ancient Maya.

What’s the date about? Turn over to the flip side and have a look at what is being commemorated, and you’ll see a ruler. He looks like a male, garbed in what looks like hundreds of pounds of regalia. I think he probably weighs ninety pounds soaking wet. He’s got a huge headdress, a bauble coming out of his nose. He’s holding a staff, and he is stomping on a rather diminutive looking victim, probably the person he conquered. This event refers to a capture, the defeat of an enemy by the ruling lord. From the hole punched in the top of that little jade plaque—about that high [motions]—we can see that it probably was worn around the neck. So I fancy that there was a proud ruler, celebrating, commemorating the day of the victory of some great battle.

Well, I think, as you can see from my example of Maya mathematics carved in stone and on jade, Maya numeration [or] Maya timekeeping, at least through that medium, is all about the perceived history of these ruling dynasties—history as they perceived it. Why peg the date of your capture, or for that matter your marriage, birth, death, or an alliance to a count that goes back to 3113 B.C.? Well, it’s probably because you believe that your lineage is descended from the ancestor gods who in fact are responsible for your being here. So, this history, then, is duly recorded in documents carved in stone all over the Maya world.

The Maya Books

[12:05:00] When we turn to the Maya books, we get insight into what I call the “stratosphere” of the Maya intellect, that is, the highest point, the most complicated computations, the most precise predictions, the most accurate numbers, the biggest numbers you’ll ever see in Mayaland.

Here in my next image we see a couple of pages of a facsimile of the Madrid Codex, so called after the library in Madrid where it now resides. How did it get there? Well, we know that the first Inquisitors who came to the New World were horrified when they saw these books. And I think we even today are taken aback by the exceedingly grotesque-looking imagery of anthropomorphic and zoomorphic figures. Of course the dots and the bars were all over the page as well. So we know that this has a lot to do with mathematics. Bishop Diego de Landa, when he first encountered these books, says, in his Relación de las cosas de Yucatán (his book about the things of Yucatan), “we found among these people a great number of books, which were written with their characters. But as they contained the lies of the devil, we burned them all, over which they grieved.” Well, I suppose if aliens came here and burned all of our libraries and all of our computer disks on which we store all of our knowledge, our literature, our poetry, our records, we would grieve, too.

You are looking at one of four books that was probably snatched out of the fire. It is all that remains, all that is extant of the Maya written record in books, saved from that great conflagration in the year 1555 in Maní, Yucatan, where a big bonfire was built by Landa and his cohorts to destroy these books. I imagine a conquistador rolled it up, put it in his boot, took it back to Spain, sold it—perhaps made some money with it. We do not know. But we do know that in the nineteenth century this fragment of Maya writing turned up in a janitor’s trash heap—this piece of codex, as it is called—and its decipherment has only taken place in the last century. The first step in that decipherment was the recognition of the numbers and the calendrical cycles. [These] were the easiest parts to decipher. The glyphs were much harder to decipher.

What I find fascinating about this book as a scientist, is that it is all about astronomy; it is all about mathematics. In fact, four out of four Maya books [are] basically math books; they are almanacs that keep track of cycles of time. Think of that: if one book were found [after] our holocaust in Cincinnati, another in Seattle, another in Florida, we would begin to come to the conclusion that whoever wrote these books had mathematics and numeration on their minds, and that certainly was the case.

Venus and Mathematics

[14:47:50] I want to show you an example of what the late and great Maya epigrapher Sir Eric Thompson called a “subtle and mechanically beautiful product of the Maya mentality.” It is a page from an almanac of the Dresden Codex (because it is in the library of Dresden that is where it ended up), which depicts the movement of Venus. And the reason why it causes me almost to take deep breaths every time I look at it, is that there is a table following the movement of the planet Venus accurate to one day in five hundred years. And we have to wonder what drove the Maya to such precision and such accurate sky-watching and the computation of such large numbers. Why, we could use this table today, with a suitable correction table accompanying it, to predict the movement of Venus, even this year or next year.

[The page] features the Maya Venus god (Quetzalcoatl, as he was called in Central Mexico, Kukulkan in the Maya area), a long-nosed, hook-nosed god always wearing a serpent headdress. You can see him flinging spears at a victim who lies below, and that victim is being impaled by a spear. The picture up at the top shows some kind of augury, some kind of incense-burning or offering—perhaps an offering of blood to this deity to propitiate him.

The Anales de Cuauhtitlan, a book from highland Mexico, written right around the time of the Conquest, tells us that this was their god, this Quetzalcoatl, as they called him. And they say that this god, this feather-serpent deity, died and went into the underworld. And there he resided for eight days. It was not until eight days passed that he was resurrected and brought into the sky as the morning star, as the resurrected god. So those of us who think that Christianity has any kind of patent on resurrection, really ought to think twice, because the Maya had their own concept of resurrection. You may have heard, in other guises, about the wanderings of Quetzalcoatl, who went out and then was exiled and then came back. This is the same myth of return, departure and return, very much the way Venus returns. It is a marvelous celestial metaphor for that wandering god, because Venus is the only planet in the sky, save for Mercury, [that] always stays close to the sun.

The Heliacal Rise

[17:12:00] Perhaps some of us are used to thinking that these esoteric books have little to do with reality, but I want to show you, in connection with Venus, what actually happens in the sky. And I want to show you the event that’s being referred to, via photographs in this case, in this particular document. The picture shows you three frames. The frame at the left, taken shortly before dawn, shows a bright star very close to the horizon. Well, that star is the planet Venus. It is making its first appearance, what the astronomers call the “heliacal rise” (“Helios” being the Greek word for sun), the rising relative to the sun, after it has been absent for several days, owing to its proximity to the sun. And there you see it making its first appearance. On that very day it makes its appearance; it comes into the sky where it had not been.

In the second frame, taken a few days later, Venus is higher in the sky, visible for a little bit more in the predawn light. And then in the third morning-twilight frame, Venus is still higher. The key is the first frame. That’s the event—the pivotal point in time—when the phenomenon of the return of the dying Kukulkan [Quetzalcoatl], if you will, is visible back in the sky. And that is what is being charted, in this particular document, in the Dresden Codex. Ah, but how they charted it.

Charting Venus

[18:33:00] They had [Venus’s journey] split into intervals, as you see them here written in dot-and-bar notation, into four subintervals. There is the first interval, the time period when Venus is visible in the sky as a morning star. Then there is an interval of disappearance when Venus is lost in the light of the sun. And the third number refers to its appearance as the evening star. And, finally, there is the final disappearance before Venus returns. So Venus has a four-cycle motion in the sky.

Let’s try to read those numbers now that we know dots and bars. The red number on the left marked “A1” is 12, well actually it is 11, two bars and one dot. Then three bars and one dot; [together] that would be 11,16, and we would translate that [as] sixteen 1s, eleven 20s—236 in our lingo. The next number 4,10 would be 90, then 12,10 which is 250. The last number, however, is really interesting for two reasons: 0, 8. Eight is the number of days when Kukulcan [Quetzalcoatl] is dead in the underworld. And there is that number, that important number that precedes the picture, that eight-day absence before Venus returns to the morning sky. The other reason why it is important is because it has the zero in it, and I wanted to show you the Maya zero.

The Number Zero

[19:53:00] Now the Maya were unique in the New World in using that zero, and of course when you have numeration by position with a zero in place, the zero enables you to do rather sophisticated mathematical computations. If you don’t believe that, try doing what the Romans would have had to do in contemporary times, add and subtract and multiply by using Roman numerals—no zero there, very difficult.

Zero [among the Maya] looks like a seashell. It actually represents a clenched fist. (You remember my gestural origins of numeration?) The clenched fist means completion, and what else do you do when you complete one of the orders of numbers, but show it with your hand that it is now complete. And, if I turn my fist sideways, doesn’t it resemble a stylized conch shell? And that is really where the shell symbol for zero comes from.

Sometimes the Maya are very playful in the codices. You’ll even see, in one of the pages of the Dresden table not shown, the zero represented as a little snail, with his little head and little feelers protruding out of the shell. So the Maya really were, you could almost say, were making their numbers, as I suggested earlier, making them come to life.

The Number 584

[21:01:00] If you add up these four intervals—and if you are doing this in Maya notation—you get 584. That is the time it takes Venus to go through a full cycle of its phases—that is, from morning star to evening star back to morning star again, 584, a recognized period by modern astronomers who can follow it. What’s interesting about the 584, and I think it is why the Maya seized upon Venus, is that it “commensurates,” as the mathematician would say, with our seasonal year of 365. It harmonizes with 365 in the perfect ratio of 8 to 5—that is, 584 is to 365 as 8 is to 5 (584:365::8:5). It all sounds so Pythagorean, doesn’t it?

Ah, that means in effect that the movements and phases and apparitions of the great star, the resurrected god of creativity, in effect are perfectly cyclical with the seasonal year. For example, if the heliacal rising of Venus were to take place on the 1st of January, 2002, we can be guaranteed to see another one on the 1st of January, 2010, but for a minor correction—in other words, exactly eight years later.


[22:13:00] Getting these cycles of the objects in heaven to fuse with the cycles of agriculture, fertility, birth and death, and so on, on earth was the key to the Maya calendar. It was a very, very real concern, not some abstract, esoteric belief that was being pursued by elitists way out on a limb. And if we read the glyphs in between the pictures, they tell us something very different from highfalutin mathematical computations. That inscription, for example, reads, “He is seen in the east, this great star Venus; woe to the pregnant female, woe to the second sowing of maize.” That is striking because it means that what these predictions are all about is astrology—astrology, which we relegate to the back pages of our newspaper.

It was very important to the Maya to peek around into the future, through the crack of time into the future, to know what was going to happen in the heavens. Not necessarily what was fatalistically going to befall them, but what was going to happen so they would know how to deal with it, so that they might know how to influence the gods to avert it. So, I don’t see the Maya at all as these pessimistic, hangdog fatalistic people, waiting for lightning to strike from the cloud of gloom that hung over their heads. This isn’t the picture I get at all.


[23:33:00] What are we to make of this Maya fascination with numbers, this obsession with time? On the one hand, as a scientist, I am attracted to them. I feel they are like me because they were talking to the universe through the lingua franca of science—that is, mathematics. They really thought that the universe spoke to them in mathematical terms, and that is something that we derive from our ancient Greek heritage, through the Renaissance, and we share [this] with them [the Maya] so that they are like us. That is why I think we admire them.

On the other hand, I am taken aback when I read those omens. I am almost shocked to think that they would develop such a sophisticated intellectual process, this high mathematics, for the sake of religion. And that causes me to think twice about who I am. And, in effect, studying Maya mathematics is an exercise in holding a mirror to my own face to see who I am, to see why I practice what I do, why my culture is the way it is. And I think in my deepest moments of thought, our incursion into Maya mathematics really causes us to think about what it means to be human.