Maya Numbers

Our number system is based on 10. Ancient Mayans, like the Welsh, had a number system based on 20 (known as the vigesimal system). Here’s a look at the difference between the two systems:

Where we have 10, the ancient Maya had 20.
Where we have 100 (10 times 10) the ancient Maya had 400 (20 times 20).
Where we have 1,000 (100 times 10), the ancient Maya had 8,000 (400 times 20).
Where we have 10,000 (1,000 times 10), the ancient Maya had 160,000 (8,000 times 20).

This continues on for as long as there are numbers that can be counted to.


Number Glyphs

The Glyph for Zero
Different kinds of glyphs were used for 0 during the Classic period. This changed to the stylized shell in the Postclassic period.




Glyphs for Numbers One through Four
The glyph for number 1 is a dot, the glyph for 2 is represented by 2 dots, 3 is written with 3 dots and 4 dots represents the number 4.



The Glyph for Number Five
The glyph for number 5 is a bar. 10 is represented through a pair of bars, and 15 is a group of 3 bars.




Combining Dots and Bars
Dots and bars can be used to represent numbers 1 through 19. For example, 6 is represented by a bar and a dot. For larger numbers, the ancient Maya used a place notation system.




Ancient Mayan Place Notation System
Imagine a column divided horizontally at regular intervals. The lowest section is the 1s space, the second place is the 20s space, the third place is the 400s space, the fourth place is the 8,000s place, the fifth one is the 160,000s place. This is how the Maya wrote large numbers, instead of horizontally like we do.

Here is an example of how it works: envision the 8,000s space having a horizontal bar. That placement makes 40,000 because 8,000 times 5 equals 40,000.

For another example, envision the 1s space has a bar and the 20s space has a dot. That makes the number 25 because 1 x 5 + 20 x 1 = 25.

A third example: imagine the 1s space has 4 dots, the 20s space has two bars and a dot and the 400s space has 4 dots. This makes 1,824 because

400 x 4 = 1,600
20 x 11 = 220
1 x 4 = 4

Add these numbers together and you get 1,824.




References:

"The Ancient Maya"; Robert J. Sharer, Loa P. Traxler; 2006

University of Nevada, Las Vegas: Marjorie Barrick Museum: Mayan Math

Codices: The Books of the Maya

The Maya crafted accordion-style books on what is called copo (or amatl in the Nahuatl language) -- paper created from the inside bark of fig trees -- whose pages they thinly coated with lime plaster. Instead of printing, Maya scribes would draw information and pictures of activities ranging from religious duties of the priests to crop production. Today, these books are called codices.

There are not many codices around now however. Due to the tropical climate of some areas of the Maya world, codices tended to rot away after a time -- though fragments of codices have been found in tombs. Also, after the Conquest, Franciscan “missionaries” such as Bishop Diego de Landa ordered Maya codices to be burned. Despite the climate
and the burn order, so far four codices have been brought to light.

These four codices are the Dresden Codex, the Paris Codex, the Madrid Codex and the Grolier Codex. This last codex was discovered fairly recently, in 1971. These surviving codices contain information on Maya rituals, divination and astronomy but do not contain much historical information.

University of Arizona Libraries: Mayan Codex Facsimiles

“The Ancient Maya”; Robert J. Sharer, Loa P. Traxler; 2006

Palomar College: Wayne's Word: Stranglers and Banyans

Puuc Architecture

Puuc is a term used for one of the various styles of architecture that Maya developed before the arrival of the Spanish. Puuc -- or “low range of hills” -- was in vogue in the Yucatan Peninsula from around 770 to 1000 AD, (as around 1000 AD the Maya had abandoned the Puuc style sites.) Sites known for their Puuc style architecture include the cities of Sayil, Kabah, Uxmal and Labna.

Basic Setup
The Puuc style takes a building made from stones held together by a lime-based concrete and uses it as a “core” for the building. Over this core, thin pieces of cut stone are set to form a decorative veneer. Usually, only the top half of Puuc style buildings are decorated with the veneer, leaving the bottom half un-embellished.

Veneer Features
Veneer designs used for Puuc buildings include lattices made of “x” shaped stones and stepped frets -- a series of connected blocky spirals -- and also Chac masks -- which themselves are made of several pieces. Another mosaic piece this style makes use of is the engaged column, which isn’t a real column, but has a flat back that is attached to the wall. These designs would be combined in various ways to form the ultimate veneer.

Corbel Arch
The corbel arch is another part of Puuc architecture. It is made with “boot-shaped” stones to create an arch with a stepped appearance. Not to be confused with a “true” arch, the stones of the corbel arch do not use each other to hold each other up (and therefore, the corbel arch does not need a keystone.)

Alignment Features
Puuc style buildings tend to be aligned so that they all face the central plaza that was used for ritual purposes. The ceremonial centers in Puuc style sites are aligned to be 14 degrees to the right (east) of north.

Hybridization
The Puuc style wasn’t always used by itself. Sites exist that are a blend of Puuc style and a style that could be a northern Mexican style. One example is Chichen Itza, which was controlled by the Toltec people for a time. Chichen Itza contains both older buildings of purely Puuc (“Classic Puuc”) architecture and newer buildings of Puuc hybrid architecture. The Classic Puuc architecture is found in the southern area of the city, while the newer, hybrid buildings are found in the northern area of the city. However, it is possible that some of what looks like it came from northern Mexico could have just been invented in Chichen Itza.

References:
University of Idaho: Non-Western Architecture: Puuc Style

University of Idaho: Non-Western Architecture: Puuc Style Facades

University of Idaho: Non-Western Architecture: Puuc Style Facades II

University of Idaho: Non-Western Architecture: Puuc Style Vaults

University of Idaho: Non-Western Architecture: Puuc/Mexican Style

"The Ancient Maya"; Robert J. Sharer, Loa P. Traxler; 2006

Calendar Series 6 of 6 -- Other Calendars

Other than the Calendar Round and Long Count, the ancient Maya used other systems for guiding their culture, which archaeologists understand to varying degrees. These “calendars” include the cycle of Venus, the Lords of the Night sequence, the 819-day count, the cycle of the moon and possibly the cycle of Mars.

Venus
Ancient Maya astronomers appear to have calculated Venus’s cycle this way: it rose as the Morning Star for 236 days, disappeared after this for 90 days, rose as the Evening Star for 250 days and disappeared for 8 days before becoming the Morning Star again. This makes a total of 576 days.

However the way the ancient Maya calculated Venus’s cycle does not accurately reflect what happens. Venus actually goes through a cycle that varies from 580, 587, 583, 583 and 587 days for an average of 583.92 days per cycle. It is the Morning Star for around 240 days, disappears for about 90 days, is the Evening Star for around 240 days, disappears again for 14 days and finally is the Morning Star again.

Moon
The ancient Maya kept track of the moon, calculating lunar months. As the lunar month equals about 29.53059 days, the ancient Maya tried to figure out a way to calculate the lunar month in terms of whole numbers and succeeded. They found that 4,400 days is very close to 149 lunar months (4,440/149 = about 29.53020). Seven pages of the Dresden Codex talk about lunar months. You can see records of the moon in inscriptions of the Long Count.

K’awiil 819-Day Cycle
Mayan gods could have multiple forms or “aspects”. K’awiil (also known as God K to archaeologists) is a god believed to be a lightning god and possibly a patron god of kings. It is also understood that he had four aspects, each connected to a color and direction: red is paired with west, white with north, black with west and yellow with south.

The 819-day cycle -- which may have begun at Palenque -- associated with these four aspects. When expressed, the 819-day count goes like this: a glyph could be a verb comes first, then a number in the 819-day cycle, followed by a reference to K’awiil, then a cardinal direction and then a color. The purpose of this “calendar” is not fully understood, It is possible that it was used for rituals.

Nine Lords of the Night
This “calendar” is understood to be nine-day cycle of the Nine Lords of the Night -- the gods of Xibalba, the Maya underworld. Each of the nine gods took his turn being connected to a day. Though not entirely clear, in Long Count inscriptions there are two glyphs that they think are connected to the sequence of the Lords of the Night. The Lords of the Night glyphs are numbered G1 through G9.

Mars
In the Dresden Codex there is a table that talks about numbers that are multiples of 78. Mars has a synodical period of approximately 780 days, so it is possible that the ancient Maya kept track of Mars cycle in the sky.


References:

“The Ancient Maya”; Robert J. Sharer, Loa P. Traxler; 2006


“Understanding Maya Inscriptions”; John Ferguson Harris, Stephen K. Stearns; 1997

"Handbook to Life in the Ancient Maya World"; Lynn V. Foster, Peter Mathews; 2005

"Yaxchilan: The Design of a Ceremonial City";Caroline Elaine Tate; 1992

Calendar Series 5 of 6 -- Correlation

The day on which the current baktun of the Maya Long Count will end is -- as said before -- still something of a matter of debate. Depending on the book, the current baktun will end on December 21, 2012, on December 23, 2012 or on December 22, 2012. Why don’t these books agree?

Matching up -- or correlating -- the Maya’s Long Count and Calendar Round with our calendar is a study filled with a variety of gathered clues, disagreeing factions and even a plethora of mathematical programs used to aid calculation. Various theories are still employed for correlation and lines have been drawn as people are still trying to refine the match between our calendar and those of the ancient Maya.

The Main Correlations
Currently, two main correlations exist for matching up our calendar with the ancient Mayans’ calendars. These are the Spinden Correlation (also called the 11.16 Correlation) and the Thompson Correlation (also called the 12.9 Correlation, the Goodman-Martinez-Thompson Correlation or the GMT).

The Spinden Correlation fits very well with what archaeologists have found in the northern area of the Yucatan -- but nowhere else. The Thompson Correlation however – which places dates 260 years later than the Spinden Correlation -- fits better with what archaeologists have found everywhere else. Of the two correlations, the Thompson Correlation is more accepted in Maya studies.

However, a third notable correlation exists, created by George Vaillant. It is known as the 11.3 correlation. Some people who study the Maya keep an open mind about this correlation.

Short Count Correlation Clue
So far, it is not possible to directly make a correlation of the Long Count with our calendar. Instead, direct correlation is made with the Short Count, a shorter version of the Long Count, found in such places as books written after contact with the Spanish, such as the books of Chilam Balam. One such date correlation found in these books matches the Short Count date Katun 13 Ahau with January 6, 1542 -- the day the city of Merida was founded.

This is not a perfect clue however. There is still the issue of making the Long Count match the Short Count -- and the fact that a person using the Short Count dates assumes that it is still in sync with the Long Count.

Astronomical Clue
Post-Spanish contact information isn’t the only kind of clue found in old artifacts. Lunar eclipse information in the Dresden Codex is another piece of information people use to correlate the Mayan calendars to ours. People use this information by calculating backwards when lunar eclipses would have occurred.

Correlation Clues via Carbon Dating
Archaeologists have been collecting clues for date correlation while in the field. An example of this is the carbon dating of sapodilla lintels at Tikal. The first time they did this, the Spinden Correlation seemed to be right. However, when they tested the lintels again the Thompson Correlation seemed the most right, matching with 10 of the 12 wood samples they used.


Other Considerations
When making the attempt to correlate the calendars, there are several other things that a person must consider. One of these considerations is the ancient Maya concept of a day – which is understood to have gone from sunset to sunset and not midnight to midnight like with today. Also unlike our concept of a day, the Maya did not give the day its number until it was over.




References:

"The Ancient Maya", 5th Edition; Michael C. Coe;

Dartmouth College: Izapa: Birthplace of Time: Correlation

“The Ancient Maya”; Robert J. Sharer, Loa P. Traxler; 2006

Calendar Series 4 of 6 -- Long Count

The Long Count is a linear timekeeping method of the Maya. Unlike the Haab and Tzolkin the Long Count does not start over. Instead, it has a starting date -- the date of the current world’s creation -- and keeps increasing forever. In a way it is like how we count the years, although we do it in terms of AD and BC.

By using the Long Count, the ancient Maya were able to keep track of dates in a larger context than they could with just the Calendar Round. This makes them different from other cultures that just used the Calendar Round, such as the Aztecs.

History
The oldest Long Count dates thus far discovered in the Maya world are located in the southern area and date from the Late Preclassic period. From this time, it seems that use of the Long Count spread to the Maya lowlands by the Classic period.

Set Up of the Long Count
Parts of the Long Count include:

Kin -- this is equal to one day.
Uinal -- this is equal to 20 kin. 18 uinals made a tun.
Tun -- a 360-day period made of 18 uinals (or uinics**).
Katun -- a period of 20 tuns which equals 7,200 days.
Baktun -- a period of 20 katuns or 144,000 days.

The Maya also had larger periods time, but they were not used in the Long Count. These are:

Pictun -- a period of 20 baktuns.
Calabtun -- a period of 20 pictuns.
Cinchiltun -- a period of 20 calabtuns.
Alawtun -- a period of 20 cinchiltuns.

Most of the names are not the original names the Maya used. They are names that they were given in more recent times.

Transliterating the Long Count
The Long Count is written in our number system as a series of numbers with periods in between them For example: 8.0.0.0.0 means 8 baktuns, 0 katuns, 0 tuns, 0 uinals and 0 kin. 1.3.3.7 is 1 katuns, 3 tuns, 3 uinals and 7 kins.

Long Count in Terms of Inscriptions
The Long Count was used at the beginning of inscriptions. Dates introducing inscriptions went like this: first came an introductory glyph that was larger than the other glyphs. In the middle of this glyph was a glyph of the patron god of the Haab month in which the Long Count date took place.

Then came the Long Count date, after which came the Tzolkin date. Following the Tzolkin date was a glyph possibly depicting a Lord of the Night, then a glyph that could be a title of the Lord of the Night and 6 glyphs on the moon date in connection to the Long Count date. Finishing up this long date sequence was the Haab date.

Cultural Significance
The ancient Maya believed that the current world began on the last day of the last baktun: 13.0.0.0.0, a date on which 4 Ahau of the Tzolkin and 8 Cumku of the Haab fell. People have tried to figure out when this was, and the most commonly accepted date is 11, 12, or 13 August 3114 BC. A common projection for the finishing of the current baktun is December 21, 2012.

In the Maya culture, the completion of a baktun was a significant event -- the completion of a “great cycle”. It may have even been a time to celebrate for the people who saw it completed. In this way it is like how the year 2000 was for us.


**In Classic Mayan writings, either word was used. However, more often they used uinic. In Colonial times, writers used uinal more.


References:
“A Forest of Kings”; Linda Schele, David Friedel; 1990

Southern Polytechnic State University: Mayan Myths -- Mayan Calendar

“The Ancient Maya”; Robert J. Sharer, Loa P. Traxler; 2006

Maya Calendar Series 3 of 6 -- Calendar Round

As mentioned in the overview article, the Calendar Round is the combination of the cyclical Tzolkin and Haab calendars that starts over every 52 years. Like with the Haab and Tzolkin calendars by themselves, a common way to visually explain the Calendar Round is by representing the calendars as interlocking gears.

With other Mesoamerican cultures, the Calendar Round was the longest length of time they recorded. This was not the case with the Maya, however, who also used the non-repeating Long Count -- which could be used to count beyond the 52 years of Calendar Round.

How Calendar Round Dates are Expressed
The Haab and Tzolkin dates are written today as a pair of dates. For example, the day on which the Tzolkin date 13 Ahau and the Haab date 8 Xul both fall is simply referred to as 13 Ahau 8 Xul. And once it occurs, a pairing of dates won’t occur again until 52 years have gone by.

Repititions
Within the 52-year cycle, the calendars complete their own cycles at rates independent of each other. The Haab repeats 52 times and the Tzolkin repeats 73 times before the two calendars can meet up again to begin a new Calendar Round.

Year Bearer
The Tzolkin date that fell on 0 Pop of the Haab was called the “year bearer.” The “characteristics” of a Tzolkin date that occurred on 0 Pop were thought to influence all the days of the Haab.

With the way that the calendars were set up, only 4 named days out of the 20 named days of the Tzolkin could ever occur on 0 Pop. In the Classic period, the 4 days -- known as “year bearers” -- were Akbal, Lamat, Ben and Etz’nab*. Taking into account the 13 numbers that are part of the Tzolkin, 52 unique Tzolkin dates could occur as “year bearers” (13 x 4 = 52). After 52 years, this sequence of year bearers started over.

Here’s a sample of the pattern of year bearer dates as they occurred: 1 Akbal, 2 Lamat, 3 Ben, 4 Etz’nab, 5 Akbal, 6 Lamat, 7 Ben, 8 Etz’nab, 9 Akbal, 10 Lamat, 11 Ben, 12 Etz’nab, 13 Akbal, 1 Lamat…

*These four days sometimes varied by region, as in the center of the Maya Lowlands, the days were Ik, Manik, Eb and Caban in the Classic Period. By the time of the Conquistadors, the days had become Kan, Muluc, Ix and Cauac.


“A Forest of Kings”; Linda Schele, David Friedel; 1990

“The Ancient Maya”; Robert J. Sharer, Loa P. Traxler; 2006

“Yaxchilan: The Design of an Maya Ceremonial City” Carolyn Elaine Tate; 1992

Maya Calendar Series 2 of 6 -- Haab

Author’s note: I will be putting on pictures of the glyphs soon.


The word Haab (or “vague year”), is the modern name for the ancient Maya’s sun-based calendar used for knowing when to plant crops. This calendar is made of 20 “months” (uinals) of 18 days (18 times 20 is 360), plus a group -- or “month”-- of 5 days at the end of the year. Altogether, this makes the Haab a 365-day calendar.

The “Months”
Months are referred to by their names in Yucatec Mayan: Pop, Uo, Zip, Zotz’, Zec, Xul, Yaxkin, Mol, Ch’en, Yax, Zac, Ceh, Mac, Kankin, Muan, Pax, Kayab, Cumku and finally Uayeb -- the 5 day period whose name may mean “resting of the year”. Each month was ruled by a deity who was its patron.

How The Haab Worked
Unlike our calendar, instead of each month being numbered 1 through 30, 1 through 31 or 1 through 28 (or 29), the days of each Haab month were numbered 0 through 19. For example, the days of month of Uo would be written 0 Uo, 1 Uo, 2 Uo and so on through 19 Uo -- and then the next month would begin (0 Zip). The same process was used for Uayeb, in that the 5 days were numbered 0 through 4.

The first day of a month could be referred to as the “seating” of the month -- so, for example, 0 Cumku is the “seating” of Cumku. The particular influence of a month began on this day of “seating”. This also means that although 0 Pop was the beginning of Pop -- the first month of the Haab -- 1 Pop was still the Haab’s new year’s day.

Uayeb had a special consideration to the ancient Maya. It was considered an unlucky period of days.

Consideration: Drift
As is known, the year is actually about one-quarter of a day (6 hours) longer than 365 days. People working in the Maya field of archaeology have found that some ancient Maya seem to know this fact, though no one seemed to have adjusted the Haab to express this.

References:
“A Forest of Kings; Linda Schele, David Friedel; 1990

“The Ancient Maya”; Robert J. Sharer, Loa P. Traxler; 2006

Maya Calendar Series 1 of 6 -- Tzolkin

Author’s note: I will be putting on pictures of the glyphs soon.

The sacred calendar used by the Maya -- today called the Tzolkin -- is a calendar composed of 20 days and the numbers 1 through 13. Each day had a name and was paired with a number in an interlocking cycle that repeated every 260 days (13 x 20 = 260).

So far as it is known, the Tzolkin is not based on any natural occurrences, like the sun-based Haab. Various ideas exist about it though -- there is an idea that it is the length of time it takes for a baby to form and be born.

Day Names
The 20 named days for the Tzolkin are Imix (water lily), Ik (wind), Akbal (night), Kan (corn), Chicchan (snake), Cimi (death), Manik (hand), Lamat (Venus), Muluc (water), Oc (dog), Chuen (monkey), Eb (tooth), Ben (reed), Ix (jaguar), Men (eagle), Cib (soul), Caban (earth), Etz’nab (flint), Cauac (storm cloud) and Ahau (lord).

Each day of the Tzolkin was represented by a glyph wrapped around a “cartouche”. People studying Mayan glyphs have found that the Maya would use different glyphs for the same day.

How it Works
The Tzolkin starts 1 Imix, 2 Ik, 3 Akbal, 4 Kan -- and so on until you get to 13 Ben. Then the numbers start over: 1 Ix, 2 Men and so on until 7 Ahau -- then the day names start over but the numbers keep going: 8 Imix, 9 Ik, 10 Akbal, 11 Kan, 12 Chiccan, 13 Cimi -- then the numbers start over again with 1 Manik, 2 Lamat, 3 Muluc, etc... This cycle of repeating numbers and days eventually starts over after 260 days-- beginning another Tzolkin.

Today, sometimes people explain this calendar, like the Haab and Calendar Round, with an image of inter-locking gears. The numbers 1 through 13 make one gear and the day names make up the other gear.

Use
And what was the Tzolkin used for? It was used for prophecy making and as a calendar of ceremonies. One Maya people -- the Kaqchikel Maya -- once used the Tzolkin to name their children.

References:
"The Ancient Maya"; Robert J. Sharer, Loa P. Traxler; 2006

"A Forest of Kings"; Linda Schele, David Friedel; 1990

"Hands-On History World History Activities"; Garth Sundem, Kristi A, Pikiewicz; 2006

Maya Calendar Series -- Overview

Author’s Note: This is part one of a series on the Maya calendars. Each section in this post will eventually have a separate post which I will hyperlink to its respective header.

In Mesoamerica -- possibly in Izapa during the 1300s BC -- two calendars developed, a sacred calendar and a calendar based on the sun. These calendars spread to peoples including the Mixtecs, Teotihuacanos and Zapotecs as well as the ancient Maya, who used the two calendars in an interlocking system.

Of the peoples that used it, the ancient Maya are believed to be the people who refined the system the most. Other than the two calendars, they also kept other “calendars”.

The Sacred Calendar: Tzolkin
The sacred calendar -- which was used for religious purposes -- is known today as the Tzolkin. This calendar has an interlocked cycle of 13 numbers and 20 days that started over after 260 days (13 x 20 = 260). Each of the 20 days had a name.

The Sun-Based Calendar: Haab
The sun-based calendar -- which was used for more everyday purposes -- is known today as the vague year or the Haab, and was like our calendar. Made up in total of 365 days, the Haab contains 18 months, each having 20 days, plus a group of 5 days (18 x 20 + 5 = 365). Each month of the Haab, like our calendar (the Gregorian calendar), has a name.

Calendar Round: The Interlocking System
The Calendar Round is the name that people who study the Maya use to describe the interlocking system of the Tzolkin and the Haab. The cycle produced by combining these two calendars repeats every 52 years.

Long Count
The Long Count is a system of units for keeping track of how many days had passed since the creation of the current world. Something only the Maya used, unlike the Haab and Tzolkin, the Long Count does not repeat, but continues to go up in number. Its smallest unit was a day from which it progressed to larger and larger units. Currently, the Long Count’s start date is thought to be sometime in 3114 BC.

Other Calendars
The ancient Maya also used some other cycles, and worked them into their timekeeping. Some of these cycles are the cycle of Venus, an 819-day count centered on the god K’awiil and the cycle of the moon as well as the cycle of the Nine Lords of Night.


Correlation
Through the years, various people tried to find a date or dates to match our calendar with the ancient Mayan Calendar Round and Long Count. So far, a match known as the Gordon-Martinez-Thompson correlation (or GMT correlation) is the most commonly accepted as the right match. However, not everyone thinks it is as accurate as it ought to be.


References:
"A Forest of Kings"; Linda Schele, David Friedel; 1990

Dartmouth College: Chapter 6 The Long Count: Astronomical Precision

"The Ancient Maya"; Robert J. Sharer, Loa P. Traxler; 2006