Welcome to Math Science History.
Today, we are going to look at how
Mayan math was practically used from the stars
to the markets.
And we'll walk through the story of how
the Dresden Codex was found, preserved and decoded.
I'm your host, Gabrielle Birchak.
I have a background in math, science and
journalism.
And by the time you're done listening to
today's podcast, you're going to know a great
deal more about the amazing and advanced mathematics
of the Mayan culture.
In the year 415, the infamous philosopher and
mathematician Hypatia of Alexandria, Egypt was savagely murdered
by church monks.
This murder shocked the Roman community and its
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I weave in the details of her education,
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There is truly more to Hypatia's life than
her death.
Hypatia, the sum of her life, written by
me, Gabrielle Birchak, is now on sale on
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Buy your copy today.
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're going to explore how their base 20
system worked, how zero was conceptualized centuries before
it became mainstream and how modern scholars like
Ernst Forstmann helped us rediscover this knowledge.
We are going to decode the brilliance of
the Maya.
One dot, one bar and one shell at
a time.
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, Copan
and Chichen Itza.
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, the present and the future all
echoed one another through calendar cycles.
To make sense of these vast cycles, some
lasting thousands of years, the Maya needed mathematics.
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 Chichen Itza, the famous pyramid of El
Castillo casts a serpent-like shadow during the
equinoxes.
At Uxmal, the 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.
Unlike our base 10 decimal system, which likely
stems from counting on 10 fingers, the Maya
used a base 20 or a vigesimal system.
So why base 20?
Well, one theory suggests they counted on both
fingers and toes.
Another theory suggests that 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 times 20, 8000 is 20
times 20 times 20, and so on.
So that number 20 starts at the basic
positional level, kind of like where we put
our 10s.
Each position increases by the power of 20,
except in the calendar system, which I'll go
into later.
The genius of the system is its positional
structure.
Like in our own decimal system, where the
digit 3 means something different in 30 versus
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 2482 in
Mayan numerals would be represented with just a
few glyphs across three levels.
So the Maya had a really beautiful, simple
way of representing the numbers.
The number 1 was represented by a dot.
The number 5 was represented by a bar
or like a hyphen.
And then the number 0 was represented by
a shell.
Basically, it was an ellipse that they would
put a design within, so it looked like
a shell.
So number 1 would be a dot, 5
would be a bar or a hyphen, 6
would be a bar and a dot, because
that's 5 and 6, 10 would be two
bars next to each other, 13 would be
a bar with three dots, and then the
number 20 would be a new place value
being the second level up.
So for example, in the second level, number
20 would be represented by a dot, because
1 times 20 is 20.
The Mayan number system was vertical in structure,
which was a striking contrast to the horizontal
layout we are used to in modern numbers.
So think of it as a layered stack
where each level holds a specific value based
on the powers of 20.
At the bottom level is the units place
representing the ones.
The next level up is the 20s place,
and then that's 20 times the value below.
And above that is the 400s place, because
20 times 20 is 400.
And then the 8000s place, which is 20
times 20 times 20, would be the level
above that.
So I'm going to provide an example on
my website at mathsciencehistory.com.
And as a side note, while you're there,
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Your support means the world to me.
Anyhow, back to the Mayan number system.
So imagine a vertical stack with two dots
at the third level, one dot at the
second level and three dots at the bottom.
So the two dots at the top level,
the third level, that would be two times
400, which is 800.
In the second level, that's one dot.
That would be one times 20.
Then at the very bottom, we have three
dots.
Three times one is three.
So the number equals 823.
That's it.
It's that simple.
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 they
were carving in 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.
We'll be right back after a quick word
from my advertisers.
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.
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
values clearly, such as 20, 1 in the
second level or 0 in the base, or
400, 1 in the third level or 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, quote, one of the most striking inventions
ever to emerge in a mathematical culture from
the old world.
The Maya developed multiple calendars.
They were basically timekeepers of the universe.
They developed a 260-day ritual calendar called
the dzulkin.
They developed a 365-day solar calendar called
the hab.
And they used another calendar called long count,
which was used to track vast historical and
mythological spans of time.
The long count is especially interesting from a
mathematical perspective.
It used a modified the decimal system.
For example, one kin equaled one day.
One wino equaled 20 days.
One tun equaled 18 winos, which was 360
days.
One katun equaled 20 tuns, which was 720
days.
And then one baktun equaled 20 katuns, which
was 144,000 days.
What's interesting for the tun, which was 18
winos or 360 days, they used 18 instead
of 20 because it was likely to approximate
the solar year, which was 360 plus five
extra days.
This calendar allowed them to date events thousands
of years in the past and in the
future.
The famous 13.0.0.0.0 long
count date corresponds to December 21st, 2012, which
is not the end of the world, just
the end of a great cycle.
It marked the completion of 5125 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.
It was so extraordinary, it was enough to
create eclipse prediction tables.
Their astronomical observatories, such as those at Copan
and Uxmal, aligned with these celestial events.
However, math wasn't just for the stars and
the 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 numerical 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.
Now on to the Dresden Codex.
It's not just a book.
It's a time capsule.
It is the most complete, best preserved of
the four surviving Maya codices.
And it's nothing short of a miracle that
it still exists.
Handmade from a model, which is a type
of bark paper, and painted with vivid natural
pigments, the Codex is folded accordion-style into
39 double-sided pages.
When you unfold it, it stretches over 3
.5 meters long.
That's incredible.
Scholars believe it was created in the post
-classic period, likely between the 11th and 13th
century CE.
Though its contents almost certainly draw from mathematical
and astronomical knowledge handed down from the classic
period, which was 250 to 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 monument
in time and a mathematical value.
What makes the Codex truly remarkable is that
it weaves mathematical precision with spiritual cosmology.
So, how did this ancient Mesoamerican manuscript end
up in Saxony, Germany?
Well, the exact chain of custody is lost
to time, but it likely came to Europe
during the early 1700s.
Perhaps taken from the Yucatan Peninsula by a
Spanish colonial official, a merchant, or a missionary,
it eventually entered the collection of Johann Christian
Götz, a theologian and a 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 still remained a mystery.
So for over a year, 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 scientific or mathematical record, because it was
very beautiful.
But that began to change in the late
19th century thanks to the efforts of a
German scholar named Ernst Forstmann, the librarian who
had become the first person to unlock the
mathematical system of the Codex.
Forstmann discovered that the dots and bars weren't
decorations, they were numbers.
He realized that the Codex contained base 20
calculations, tables of Venus' position in the sky,
and precise calendrical cycles stretching thousands of years.
So here's a little bit about Ernst Forstmann.
I bet you've never heard the name.
And his story is actually pretty interesting.
He was a German historian, a librarian, and
a philologist.
He became the chief librarian at the Royal
Library in Dresden.
In the late 1800s, he began closely examining
the Dresden Code.
Forstmann'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 called
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 and 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 it suffered
water damage from the 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 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.
We'll be right back after a quick word
from my advertisers.
Following Forstmann's work, interest in the Maya grew.
But it wasn't until the mid-20th century
that Mayan epigraphy took off.
During this time, great strides in understanding Mayan
mathematics were made.
Tatyana Proskouriakoff, a Russian-American Mayanist scholar and
archaeologist, demonstrated that stelae were not just mythological,
they were historical records.
Yuri Norozov, a Russian linguist, showed that Mayan
glyphs were phonetic, not ideographic as long assumed.
And then David Stewart, 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 beautiful and rich
heritage.
So why does Mayan mathematics matter today?
Well, 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 it also reframes how we think
about the history of math.
By preserving and celebrating Mayan mathematics, 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 just didn't 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 are reminded that there are so many
ways to understand the universe, and the Maya,
through their shells, dots, bars, and stars, offered
us one of the most beautiful examples of
how math resides within us and around us.
Thank you for tuning in to Math Science
History, and until next time, Carpe Diem.
Thank you for tuning in to Math Science
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