Mayan Mathematics
Today, we’re traveling back more than a thousand years to explore a remarkable mathematical system developed deep in the jungles of Mesoamerica. Before calculus, before Newton, before Europe fully embraced the idea of zero, there was the Maya. Their number system, calendar, and understanding of zero rivaled and in some ways exceeded their contemporaries around the world. We’re talking about Mayan Mathematics. We’ll explore how their base-20 system worked, how zero was conceptualized centuries before it became mainstream, and how modern scholars like Ernst Förstemann helped us rediscover this knowledge. We are going to decode the brilliance of the Maya, one dot, bar, and shell at a time.
The Maya, Masters of Time and the Cosmos
The Maya civilization thrived in what is now southeastern Mexico, all of Guatemala and Belize, and parts of Honduras and El Salvador. At its peak, from roughly 250 to 900 CE, known as the Classic Period, the Maya developed intricate city-states like Tikal, Palenque, Copán, and Chichén Itzá. These were not only political and economic centers, but also cultural and scientific hubs, brimming with artisans, priests, architects, and astronomers.
From these jungle metropolises, the Maya observed the skies with astonishing dedication. They tracked solar cycles, lunar eclipses, and the movements of Venus and Mars. Time wasn’t just measured, it was revered. Every ritual, harvest, and political event was synced with the cosmos. Their concept of time was cyclical, not linear, and that cyclical rhythm formed the very backbone of their culture. The past, present, and future all echoed one another through calendar cycles.
To make sense of these vast cycles, some lasting thousands of years, the Maya needed mathematics. But not just tally marks or trade-based counting. They needed a system that could measure not only the number of cacao beans in a bag, but the number of days since the creation of the world.
So they built calendars that could do both. Their temples and pyramids weren’t just religious structures, they were astronomical instruments. At Chichén Itzá, the famous pyramid of El Castillo casts a serpent-like shadow during the equinoxes. At Uxmal, buildings are oriented with the rising and setting of Venus, which was central to both warfare and prophecy. These alignments weren’t accidental. They were deliberate, math-driven expressions of cosmology in stone.
But to accomplish this, the Maya needed a number system that could handle immense time spans and cyclical logic. They needed more than counting, they needed structure, scale, and zero.
The Vigesimal System – Base 20
Unlike our base-10 decimal system, which likely stems from counting on ten fingers, the Maya used a base-20, or vigesimal, system. Why base-20? One theory suggests they counted on both fingers and toes. Another suggests the base-20 system could more easily accommodate complex calendar math.
In the Mayan system:
The numbers 0 through 19 are the building blocks.
The number 20 is like our 10; it starts a new positional level.
400 is 20×20; 8,000 is 20×20×20, and so on.
Each position increases by powers of 20, except in the calendar system, which we’ll get to later.
The genius of this system is its positional structure. Just like in our own decimal system where the digit “3” means something different in 30 vs. 300, the position of symbols in the Mayan system changes their value.
The Maya managed large-scale calculations with extraordinary economy of space. For example, a large number like 2,482 in Mayan numerals would be represented with just a few glyphs across three levels.
The Symbols: Dots, Bars, and Shells
The Maya had a beautifully simple way of representing numbers:
A dot (•) = 1
A bar (,) = 5
A shell = 0
So:
1 = •
5 = ,
6 = , •
10 = , ,
13 = , , •••
20 = new place value (second level up)
The Mayan number system was vertical in structure, a striking contrast to the horizontal layout we’re used to in modern numerals. Think of it as a layered stack, where each level holds a specific value based on powers of 20. At the bottom level is the units place, representing 1s. The next level up is the 20s place, that’s 20 times the value below. Above that is the 400s place (20 × 20), then the 8,000s place (20 × 20 × 20), and so on.
Let’s look at an example:
Imagine a vertical stack with:
2 dots at the third level → that’s 2 × 400 = 800
1 dot at the second level → 1 × 20 = 20
3 dots at the bottom → 3 × 1 = 3
Altogether, this number equals 823. That’s it, just three glyphs, neatly stacked. No separate symbols for hundreds or thousands. Just dots, bars, and shells arranged in vertical layers.
What makes this system so elegant is that it’s positional and exponential, like our own base-10 system. A dot in one level doesn’t carry the same value as a dot in another. This allowed the Maya to write large, complex numbers efficiently and compactly, a necessity when carving into stone or painting on delicate codex pages.
But this wasn’t just about mathematics, it was about meaning.
Mayan numbers weren’t hidden in ledgers or buried in calculations. They were proudly and artistically displayed on temple steps, stelae, ceramic vessels, and city walls. On public monuments, large dates from the Long Count calendar were recorded in these vertical stacks of glyphs, often surrounded by images of kings, gods, or cosmic symbols. They gave weight to political power and religious legitimacy by anchoring a ruler’s deeds within the context of time and celestial order.
Even within sacred texts like the Dresden Codex, the numerical stacks served not just to track astronomical events but to guide rituals, forecast eclipses, and calculate offerings. The glyphs were rendered with such care that numbers themselves became aesthetic and spiritual elements.
The vertical format wasn’t just a style, it reflected the Mayan worldview, where time, space, and hierarchy were experienced in layers. Just as their pyramids rose in stacked platforms toward the sky, so too did their numbers build meaning upward. Math wasn’t only functional. It was visual, ritualistic, and cosmic.
The Invention of Zero
One of the Maya’s most extraordinary contributions to the world of math is their early use of zero.
Zero was not merely a placeholder in Mayan math, it was treated as a full digit. They represented it with a shell-like glyph. This allowed them to express quantities with place value clearly, such as 20 (1 in the second level and 0 in the base) or 400 (1 in the third level, 0 in the second, 0 in the base).
While zero was also independently developed in ancient India, the Maya were one of the earliest civilizations to use zero in a positional system, dating at least to the 4th century CE. This gave them a functional edge over many ancient cultures that lacked such a concept.
Historian Georges Ifrah calls the Mayan use of zero “one of the most striking inventions ever to emerge in a mathematical culture isolated from the Old World.”
Timekeepers of the Universe – Calendars and Astronomy
The Maya developed multiple calendars:
Tzolk’in: A 260-day ritual calendar
Haab’: A 365-day solar calendar
Long Count: Used to track vast historical and mythological spans of time
The Long Count is especially interesting from a mathematical perspective. It used a modified vigesimal system:
1 kin = 1 day
1 uinal = 20 days
1 tun = 18 uinals = 360 days
1 katun = 20 tuns = 7,200 days
1 baktun = 20 katuns = 144,000 days
Why did they use 18 instead of 20 for the tun? Likely to approximate the solar year (360 + 5 extra days = 365).
This calendar allowed them to date events thousands of years in the past and future. The famous 13.0.0.0.0 Long Count date corresponds to December 21, 2012, not the end of the world, just the end of a great cycle. It marked the completion of 5,125 years since their mythological creation date of 3114 BCE.
Their calendars were tied to astronomy, particularly the cycles of Venus. The Maya tracked Venus’s synodic period of 584 days with extraordinary precision, enough to create eclipse-prediction tables. Their astronomical observatories, such as those at Copán and Uxmal, aligned with celestial events.
Mayan Math in Daily Life
Math wasn’t just for the stars and pyramids. The Maya used it in trade, architecture, taxation, and agriculture. Their marketplaces functioned with currency-like cacao beans, obsidian blades, and woven textiles.
Their irrigation and planting systems required calculating seasonal cycles, rainfall expectations, and field rotations. Markets needed pricing systems. Tribute records were recorded with clear numeric glyphs on monuments, murals, and codices.
For example, murals at Bonampak show tribute lists using dots and bars to represent payments in cloth, food, or cacao.
Mathematics was woven into the social and political fabric, too. Rulers often claimed their authority through grand calendar-based inscriptions, situating themselves in cosmic cycles as divine actors.
Discovery of the Dresden Codex
The Dresden Codex is not just a book, it’s a time capsule. It is the most complete and best-preserved of the four surviving Maya codices, and it’s nothing short of a miracle that it still exists. Handmade from amatl, a type of bark paper, and painted with vivid natural pigments, the codex is folded accordion-style into 39 double-sided pages. When unfolded, it stretches over 3.5 meters (about 11.5 feet) long.
Scholars believe it was created in the Postclassic Period, likely between the 11th and 13th centuries CE, though its contents almost certainly draw from mathematical and astronomical knowledge handed down from the Classic Period (250–900 CE). The scribe who composed the Dresden Codex was not just a writer but a mathematician, astronomer, and ritual specialist, perhaps even a priest.
Its pages are filled with tables predicting eclipses, charts tracking the synodic cycle of Venus, calendar calculations, and ritual schedules. There are gods, serpents, and symbols of fire, maize, and rain, each connected to a moment in time and a mathematical value. What makes the codex truly remarkable is that it weaves mathematical precision with spiritual cosmology.
But how did this ancient Mesoamerican manuscript end up in Saxony, Germany?
The exact chain of custody is lost to time, but it likely came to Europe during the early 1700s, perhaps taken from the Yucatán Peninsula by a Spanish colonial official, merchant, or missionary. It eventually entered the collection of Johann Christian Götze, a theologian and librarian, who facilitated its acquisition by the Royal Library in Dresden in 1739. At the time, it was admired for its exotic artwork and the novelty of its unknown script, but its content remained a mystery.
For over a century, the codex sat largely misunderstood. European scholars in the 18th and early 19th centuries had no frame of reference for its complex glyphs and numeric tables. Some believed it was simply an astrological almanac. Others assumed it was a work of mythology or fiction. Its pages were studied more as art than as a scientific or mathematical record.
That began to change in the late 19th century, thanks to the efforts of a German scholar named Ernst Förstemann, the librarian who would become the first person to unlock the mathematical system of the codex. Förstemann discovered that the dots and bars weren’t decorations, they were numbers. He realized that the codex contained base-20 calculations, tables of Venus’s position in the sky, and precise calendrical cycles stretching thousands of years.
Ernst Förstemann: Decipherer of Mayan Math
Ernst Förstemann (1822–1906), a German historian, librarian, and philologist, became the chief librarian at the Royal Library in Dresden. In the late 1800s, he began closely examining the Dresden Codex.
Förstemann’s great breakthrough came when he realized that the repetitive symbols weren’t decorations or astrological symbols but numbers, written in dots and bars, structured vertically. He recognized the base-20 system and saw how dates and astronomical intervals related to real celestial events.
Most crucially, he identified the shell glyph as a true zero, not a decorative flourish, but a number. This was groundbreaking. It was the first time a European scholar had acknowledged the depth of Mayan numerical sophistication.
In 1901, he published his seminal work: Commentary on the Maya Manuscript in the Royal Public Library of Dresden, which mapped out the structure of the Long Count calendar and many of the astronomical tables, especially those related to Venus.
His analysis laid the groundwork for modern Mayan mathematics scholarship.
But the codex’s journey didn’t end there.
During World War II, the city of Dresden was heavily bombed in 1945 by Allied forces. The Royal Library, and much of the historic city, was devastated. The Dresden Codex, stored at the time in fireproof archives, survived the bombings but suffered water damage from firefighting efforts. Some pigment was blurred, and a few sections were nearly lost. But remarkably, the majority of the codex remained intact.
Today, it is one of the crown jewels of the Saxon State and University Library (SLUB) in Dresden. It has been fully digitized and made freely accessible online, offering scholars, teachers, students, and the public a chance to engage directly with one of the greatest surviving records of Maya scientific thought.
20th Century Breakthroughs in Decipherment
Following Förstemann’s work, interest in the Maya grew. But it wasn’t until the mid-20th century that Mayan epigraphy took off:
Tatiana Proskouriakoff demonstrated that stelae were not just mythological, they were historical records.
Yuri Knorozov, a Russian linguist, showed that Mayan glyphs were phonetic, not ideographic as long assumed.
David Stuart, starting as a prodigious teenager, cracked dozens of glyphs, tying them to real names, dates, and dynasties.
These breakthroughs connected Mayan math not just to astronomy but to language, dynastic history, and ritual life. Today, both Indigenous and academic scholars work together to decode and revive this rich heritage.
Why Mayan Math Matters Today
Mayan mathematics offers a parallel development to classical traditions, demonstrating that complex systems can arise independently. It challenges the Eurocentric timeline of mathematical progress.
Their innovations in zero, place value, and base-20 logic reflect a profound understanding of time and space. Rediscovering this knowledge not only honors Indigenous intellectual heritage but also reframes how we think about the history of mathematics.
By preserving and celebrating Mayan math, we gain more than numbers, we gain insight into a worldview that saw math as sacred, cyclical, and woven into the structure of the universe.
The Maya didn’t just do mathematics, they lived it. It was carved into stone, painted into sacred books, echoed in temple alignments, and inscribed into the very rhythm of daily life. Their math was not abstract and detached; it was sensory, celestial, and sacred. To study Mayan mathematics is to step into a civilization where numbers had personalities, calendars had souls, and time itself was a living, breathing force. In recognizing the legacy of Mayan mathematical genius, we’re not merely filling a gap in the historical record, we’re expanding the map of human imagination. We’re remembering that there are many ways to understand the universe, and the Maya, through their shells, dots, bars, and stars, offered us one of the most beautiful.
Some of the following sources are affiliate links:
Ifrah, Georges. The Universal History of Numbers. Wiley, 2000.
Coe, Michael D. and Houston, Stephen. The Maya, 9th Edition. Thames & Hudson, 2015.
The Dresden Codex, the great Maya book of the stars, By Carlos Rosado van der Gracht. Yucatan Magazine.
SLUB Dresden Digital Collection: https://digital.slub-dresden.de/werkansicht/dlf/29691/1