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Maya Mathematics and World Ages:
Frequently Asked Questions
Lloyd B. Anderson, Ecological Lingusitics, August 2008
0. Overview: Mayan Non-Cyclical Calendrics (summary of findings) Click here.
Claims for Mayan cycles of World Ages are not supported by available information.
1. Introduction, and How the Maya calendar and long count work (continue below on this page)
2. Notations 22.214.171.124.0 in the Maya long count – 3114 BCE vs. 2012 of our era
and Tortuguero Monument 6 Click here.
For a paper by Steve Houston "What will not happen in 2012", please click here.
3. 20 not 13 Baktuns in a Pictun. Why there is no expressed notation 0.0.0.0.1 Click here.
Metaphorical Uses of "13" and long strings of "13"s
4. World Ages in Mesoamerica more generally Click here.
5. Planetary Movements around Winter Solstice 2012 Click here.
6. Maya celebrations of "period endings", and when period endings are not the main point. Not yet.
7. Distinct Concepts of Zero -- a placeholder for no units in a place notation system, Click here.
or a number (absolute zero, nothing), or ...? And a paper by John Justeson
8. Distance Numbers across the Era Date. Long reckonings and "ring numbers". Click here.
The Maya substitute for subtraction. Counts from just a bit later than a period ending
9. Numerology and Cycles Click here.
10. Matching Maya dates with equivalent European dates (the "correlation constant") Not yet.
Some links: See lower on this page (other links on other pages where appropriate)
References: See lower on this page
There is increased discussion of Mayan time reckoning and world ages and the concept of "zero" now as we approach our year 2012, when at or close to the winter solstice a major Maya time period turns over in the largest place of the long count. It is perhaps much like the excitement in the European world around the year 1000, and in our own time to some degree also around the year 2000 (more than just the possible failure of some computer programs and databases).
There are really interesting questions, and we should use the excitement to advance our understanding both of the Maya and of ourselves. That means taking the questions seriously, and helping all those interested to discover what we do and do not currently know about Mayan and Mesoamerican time reckoning, what kinds of tools were used to establish what we now know, and what tools we can use in learning more. Surely we can develop some new ones.
At the same time, there are many unfounded speculations, and the wildest claims making the rounds. Did the Aztecs in fact predict a change of magnetic poles in 2012? One person took it for granted that this is so, and when I said that as far as I knew, there is no basis for such a claim, she was incredulous "Where did it come from, then?". I think the answer must be: from someone's wishful thinking. Rumors and nonsense can easily get taken for granted as established fact.
And back on the other hand again, it is all too easy to forget that many important ideas which we take for granted as valid today, and do not question, were inconceivable and totally rejected by authorities of the time when they were first proposed. Valid ideas can come from anywhere and anyone. The essence of science is being able to throw light on questions, investigate and find out more. Science cannot always provide the right answer. And some valid statements (along with many more which are not valid) must remain in the category of <completely untestable at present> because we lack needed information.
The first five pages listed below are central to discussions of what kind of concept of "World Ages" the Maya had, including whether it was one like that of central Mexico. Remaining pages are on related questions.
These web pages are possible only because of contributions by many people on the Aztlan email list and in previous research. Many contributors will be named and credited explicitly, whenever possible. See below.
Maya Day Names and their spellings: (see Thompson 1961 p.89)
Yucatec names, traditionally used when reading the day glyphs
Imix, Ik, Akbal, Kan, Chicchan, Cimi, Manik, Lamat, Muluc, Oc,
Chuen, Eb, Ben, Ix, Men, Cib, Caban, Etz'nab, Cauac, Ahau
More recent spellings of the same names, using the alphabet approved by the Academia de las Lenguas Mayas:
Imix, Ik', Ak'bal, K'an, Chikchan, Kimi, Manik', Lamat, Muluk, Ok,
Chuwen, Eb, Ben, Ix, Men, Kib, Kaban, Etz'nab, Kawak, Ajaw
Maya Month Names and their spellings: (see Thompson 1961 p.106)
Yucatec names, traditionally used when reading the month glyphs
Pop Uo Zip Zots', Zec (Tzec), Xul, Yaxkin, Mol, Ch'en,
Yax, Zac, Ceh, Mac, Kankin, Muan, Pax, Kayab, Cumku, Uayeb
More recent spellings of the same names, using the alphabet approved by the Academia de las Lenguas Mayas:
Pohp, Wo, Sip, Sotz', Sek, Xul, Yaxk'in, Mol, Ch'e'n,
Yax, Sak, Keh, Mak, K'ankín, Muwaan, Pax, K'ayab, Kumk'u, Wayeb
Names gradually becoming more common, in part reflecting Ch'olan and Kanjobalan and Pokomchi forms and using clues from parts of the glyphic names to choose which names to use:
K'an-jalam, Ik'-(k')at, Chac-(k')at, Sotz', Sek or Kasew, Xul, Yaxk'ín, Mol, Ch'e'n or Ik' Sihom,
Yax Sihom, Sak Sihom, Chak Sihom, Mak, Uniw, Muwaan, Pax, K'an-asiiy, Cumk'u, Wayeb
The Final Days, by Benjamin Anastas (New York Times Magazine, 7/1/07 NO LONGER ACCESSIBLE)
Five Years: 2012 and The End of the World As We Know It, by Tom King (Lawrence.com, 12/10/07)
John Hoopes's article on "Mayanism" is here:
A good example of what John Hoopes has said to the press is here:
Two more links which are less focused (like most blogs and discussion transcripts)
One UTMesoamerica discussion of the Tortuguero Box prophecy for 126.96.36.199.0 is here:
The most important items are David Stuart's view of what we know of the text on the Tortuguero Box (but that is also included here on the page "Notations 188.8.131.52.0" linked to at the top of this page), and the notice that Marcus Eberl and Christian Prager have published some analyses of the divinity name Bolon (Y)okte' K'uh.
Another UTMesoamerica discussion of the Return of Quetzalcoatl is here:
Anders, Ferdinand and Maarten Jansen 1996. Códice Vaticano A. Fondo de Cultura Economica (México)
and Akademische Druck- und Velagsanstalt (Graz, Austria)
Bierhorst, John. 1992. History and Mythology of the Aztecs. The Codex Chimalpopoca. Univ. of Arizona Press
Closs, Michael P. ed. 1986. Native American Mathematics. University of Texas Press
Lounsbury, Floyd 1989. "A Palenque King and the Planet Jupiter". pp.246-259 in Anthony Aveni ed.
World Archaeoastronomy. Cambridge University Press
Miller, Mary and Karl Taube 1993. The Gods and Symbols of Ancient Mexico and the Maya. Thames & Hudson
Schele, Linda 1986, 1987, 1988 Notebook for the Maya Hieroglyphic Writing Workshop at Texas
[similar content is available from The Maya Meetings in later reprints as
A Palenque Triad 2nd edition 1999, with updated readings by Bob Wald]
Stuart, David. 2005. The Inscriptions from the Temple XIX at Palenque
Thompson, J. Eric S. 1960. Maya Hieroglyphic Writing. University of Oklahoma Press
Thompson, J. Eric S. 1972. A Commentary on the Dresden Codex. American Philosophical Society
Substantial contributions were received in recent discussions on Aztlan from Anna Blume, Erik Boot, Carl Callaway, Michael Carrasco, David Hixson, Sid Hollander, John Major Jenkins, John Justeson (an essay he wrote some years ago, pointed to by Randa Marhenke), Jorge Perez de Lara, Mark van Stone
1. How the Maya calendar and long count work
Starting from an idea sketched by Sandy Noble, we can present the Mayan dates just before and after the Winter Solstice of the year 2012. While our own year count is now expressed with a four-digit number such as 2008 or 2012, the Maya long count is usually expressed as a five-digit number, and includes what are loosely called "months" (of 20 days each) and days as well as TUUNs, loosely called "years" (of 360 days) and higher multiples of 20 and of 400 TUUNs.
Some clarifying notes: (a) The "lunar series" in many Maya inscriptions contain an indication of true months counted as mostly alternating 29 and 30 days, giving an average value near 29.5 days. We use the same name "months" for units of the long count of 20 days each without implying that the Maya had the same name for these. When we are more precise, we refer to such a unit of the Long Count as a "WINAL".
(b) The glyph traditionally read TUN or TUUN may in fact have been read HAAB though in long counts still with the value of 360 rather than 365 days. This is a matter of some controversy at present.
(c) Long counts with still larger units are discussed later.
In summary, for 20 December 2012, Mayan long count 184.108.40.206.19 we have:
12 19 19 17 19 (3 Kawak 2 K'ank'in)
Tzolk'in + Haab
Baktuns K'atuns TUUNs WINALs K'INs (Calendar Round)
Range from 0 to 13 0 to 19 0 to 19 0 to 17 0 to 19
or from 0 to 19
In days: 144,000 7200 360 20 1
("year") ("month") "day"
400 TUUNs 20 TUUNs TUUN
The "Calendar Round" consists of the Tzolk'in and the Haab cycles. The Tzolk'in is a cycle of 260 days in which the numbers cycle from 1 to 13 and back to 1 again, the days cycle through a set of 20 names, the same as the length of a WINAL. A particular day name in the Tzolk'in and the last digit of the long count will always correspond, as here the day Kawak always goes along with a "19" in the last position of the long count (and important for period endings, the day Ajaw always goes along with a "0" in the last position of the long count . The Haab is a cycle of 360 days, 18 months with days numbered 0 to 19, plus 5 days at the end numbered 0 to 4, for a total of 365 days. This is the "vague year". Because there was no leap year, the vague year slipped along the seasons, returning to its starting position again in 1507 true years of 365.2422 days = 1508 x 365 = 550420 days (to the nearest 1/100th of a day).
Here is a sequence of dates from just before 2012 to some time after it.
220.127.116.11.19 = 20 December 2012
+ 1 Add 1 day
13. 0. 0. 0. 0 = 21 December 2012 Calendar Round 4 Ajaw 3 K'ank'in
+ 1 Add 1 day
13. 0. 0. 0. 1 = 22 December 2012
+ 1 Add 1 day
13. 0. 0. 0. 2 = 23 December 2012
+ 18 2 days + 18 days = 1 WINAL
13. 0. 0. 1. 0 = 10 January 2013
+17. 0 1 WINAL + 17 Winals = 1 TUUN
13. 0. 1. 0. 0 = 16 December 2013
+ 19. 0. 0 1 TUUN + 19 TUUNs = 1 K'ATUN
13. 1. 0. 0. 0 = 7 September 2032
+ 19. 0. 0. 0 1 K'ATUN + 19 K'ATUNS = 1 BAKTUN
14. 0. 0. 0. 0 = 26 March 2407
+ 1 Add 1 day
14. 0. 0. 0. 1 = 27 March 2407
In the date 18.104.22.168.19, each of the last four digits of the long count starts at its maximum before turning over to zero and carrying one to the next higher unit. In the dates presented after that, starting with 22.214.171.124.0, a progression to each higher unit can be seen separately.
In the first addition, one day added to the first date makes a complete WINAL of 20 days, so we increase the K'INs from 19 to 0 and carry a one to the Winals position. One added to 17 is 18, but 18 WINALS = one TUUN so we increase the WINALs from 17 to 0, and carry a one to the TUUNs position. We increase the TUUNs from 19 to 0 and carry a one to the K'ATUNs position. Finally, we increase the K'ATUNs from 19 to 0 and carry a one to the BAKTUNs position. This last is the first digit of the most common form of long count expression. One added to 12 is 13, so we end up with an even 13 BAKTUNs and no additional KATUNs, TUUNs, WINALs, or K'INs, a date written 126.96.36.199.0. Whether this is simply a normal expression of a date or whether it is something else, with symbolic and metaphorical meanings, is one question being addressed in these FAQ pages. If it is simply an expression of a date, then a further 20 K'ATUNs = one BAKTUN can be added, and we end up at a date written 188.8.131.52.0. But we have no example (?) of such a date being written.
The most contentious point of debate seems to be whether the date 184.108.40.206.0 is a point in the future at which the Maya expected anything significant to happen, at which they expected a long count to begin again, or some world cataclysm to occur, either in a physical sense or in a cosmological sense. Considering the first part of this, have we written the dates 220.127.116.11.0 and those that follow it correctly above as the Maya would have? Or would they have written 18.104.22.168.1 for that same date, because a count had begun again after 22.214.171.124.0 (or a new count had begun, however we should think of it)? Unfortunately, we simply don't know. There is only a single document, a box from the site of Tortuguero, which uses a long count 126.96.36.199.0 with the Calendar round 4 Ajaw 3 K'ank'in confirming that it refers to this date in the year 2012. It does not record dates after that, so we cannot learn from it how the scribes of the time would have recorded such dates which for them were a long distance into the future.
Since we have no texts directly telling us what the Maya thought of the date in 2012 AD at which their calendar would reach 188.8.131.52.0, and only one reference to it confirmed by the Calendar Round 4 Ajaw 3 K'ank'in, we must use indirect methods to discover their thinking regarding this. That is one of the purposes of these discussion pages. (The glyph for the WINAL which in Yucatec was called K'ank'in was probably pronounced UNIW, as shown by its glyphic form matching that pronunciation and several Mayan languages having that name for this WINAL.)
We also have very few texts referring to the date 4 Ajaw 8 Kumk'u which explicitly give for it a long count 184.108.40.206.0. There are many references to the Calendar Round 4 Ajaw 8 Kumk'u, but most of those take for granted its long count position, and use it only to anchor some other event in time. Some examples which do refer to 13 Baktuns explicitly are:
Palenque Temple of the Cross, C3-D3-C4-D4-C5, inserted between two events linked by DN
"Then [verb] 4 Ajaw 8 Kumk'u completion of 13 Baktuns"
To discover what the Maya thought even about the era event 220.127.116.11.0 4 Ajaw 8 Kumk'u, we need an integrated study of all 4 Ajaw 8 Kumk'u dates which are a focus in their texts, and about which something significant is said (omitting ones which are merely a chronological framework on which to hang other dates). Here are a few.
Vase of the 7 Gods Kerr no. K2796
Vase of the 11 Gods Kerr no. K7750
Quiriguá Stela C.
Palenque Temple of the Cross, "deerhoof" verb at C3 followed by 4 Ajaw 8 Cumk'u and
then "there came to completion 13 Baktuns" or "...the 13 Baktuns" in D4-C5.
This date and event are referred to again with the marked anterior form of the event
"Crossed-Stones"-jiiy and probably glyphs C6b through C7 (??-?? Ti'-CHAN-na),
counting 1.9.2 from that until (?) OCH-ta-CHAN-na Hunal-Ye GI on 13 Ik' 20 Mol
(presumably GI "entered the sky". This second event (18.104.22.168.2) is referred to again
with a different verb at D15-C16-D16 (possibly a standstill? check this).
Palenque Temple of the Sun, the "standstill" expression ocurs again at C9-D9, at 22.214.171.124.6
which was also the initial series date. The same "Crossed-Stones" clause as on the
Temple of the Cross occurs at D16-N1-O1-N2 with "4 Ajaw 8 Cumk'u".