The new Mayan glyphs from Xultun, Guatemala: lots of spectacular calendar calculations, but nothing whatsoever about 2012 AD
The crop-circle phenomenon began in earnest around 1990, with fifty or more field pictures in several dozen countries around the world. Quite often these crop pictures show long-forgotten Mayan themes such as a “feathered serpent”, an “Ahau calendar”, or “planetary systems” which match a date of December 23, 2012 when the Mayan Long Count calendar ends. Thus it would be of considerable interest to learn beforehand, whether anything important will happen on that date?
Recently a series of ancient Mayan glyphs were discovered in Guatemala, as wall murals dating back to 800 AD. Those important glyphs remain only partly understood (see “Ancient Maya Astronomical Tables from Xultun, Guatemala”, W.A. Saturno et al., Science 336, 714-717, 2012). Nevertheless, our popular press has been quick to tell people, based on an incomplete understanding, that nothing important will happen at the end of 2012 AD (see www.abc.net.au or www.telegraph.co.uk).
Perhaps something important will happen in December of 2012, or perhaps not? Yet we should not lie about the facts. Here I will explain what those new glyphs tell us, in terms so simple that even a high-school student should be able to understand.
Summary of what the new Mayan glyphs from Guatemala tell us
The small temple or house which was excavated shows three different kinds of mathematical calculation: (i) a lunar phase calendar, (ii) a long-term match of their lunar phase and eclipse calendars, and (iii) a long-term match of their Calendar Round to four other calendars with 360, 819, 1209 or 1677 days per cycle. None of these glyphs say anything about an upcoming date of December 23, 2012 AD. After providing a brief lesson on how to read Mayan numbers and dates, I will explain each of those three ancient wall calculations in turn.
How to read Mayan numbers and dates
The Mayans developed many different kinds of calendar, then used those calendars to record important historical dates on standing stone structures known as “stelas”. By analogy, our modern culture might commission a statue in Washington D.C. to honour President John F. Kennedy, saying that he “died on November 22, 1963”. The ancient Mayans in a similar fashion erected many stelas to mark important royal occasions, such as the birth or accession to the throne of their kings.
Part of a famous stela from La Mojarra in Veracruz is shown below. This slide gives an explanation of the most basic Mayan numbering system, and how to count by it:
On the left we can see a series of five numbers, reading from top to bottom 8-5-16-9-7. In order to find a date, first we have to multiply all of those numbers as follows:
(8 x 144,000) + (5 x 7200) + (16 x 360) + (9 x 20) + (7 x 1) = 1,193,947 days
Then we must add 1,193,947 in days to a “starting date” of August 13, 3114 BC, in order to get the stela date of July 13, 156 A.D
Their Long Count numbering system did not use base-10 as we use today. Instead they chose maximal values for each digit of 12-19-19-17-19, so that the next number (after adding one) becomes 184.108.40.206.0. Thus they used base-13, base-20, base-20, base-18 and base-20 for five successive digits of 144,000, 7200, 360, 20 or 1 in days.
A second example is shown below from Tres Zapotes in Veracruz. This example shows a stela date of 7-16-6-16-18:
Once again, we can find a total number of days by multiplying as:
(7 x 144,000) + (16 x 7200) + (6 x 360) + (16 x 20) + (18 x 1) = 1,125,698 days
Then when we add 1,125,698 days to a starting date of August 13, 3114 BC, we find September 3, 32 BC.
In our modern culture, we count dates from the birth of Jesus Christ in Bethlehem 2000 years ago. Our current date of May 16, 2012 AD lies (2012 years x 365.25) + (5 months x 30.5) + (16 x 1) days past the chosen starting date in our culture. Perhaps the ancient Mayans knew something about the date of August 13, 3114 BC which we have since forgotten?
Close to the bottom of this slide, we can see a second symbol (marked in blue) which translates to “6 Etznab”. That is a date from another of their calendars called the “Tzolkin”, which uses a 260-day cycle. By combining their Tzolkin with a “Haab” or 365-day solar calendar, the Mayans were able to set up an even longer cycle called the “Calendar Round”, which lasted for 18,980 days or close to 52 solar years. When they carved stelas, they would often add symbols for both the Long Count and the Calendar Round, so that both sets of dates would mutually confirm one another.
Without further ado, let us get on with it, and examine three new kinds of glyph as found on wall murals at Xultun in Guatemala, dating back to 800 AD!
A generalized lunar phase calendar
The first set of glyphs found at Xultun was a “generalized lunar calendar”. We can understand what those lunar glyphs tell us, by reference to the Long Count numbering system just explained above. If we work out the numerical value of each glyph in a Long Count numbering system, we find that each glyph differs from its neighbour by 177 or 178 days, equivalent to six lunar months of 29.53 days:
Three different kinds of “Moon symbol” (wide-eyed, frowning or smiling) were drawn above each column of numbers in the order 1-2-3-1-2-3, to suggest that we are concerned here with three successive six-month periods of our Moon, lasting for 531 or 532 days (equal to eighteen lunar months in total).
The first glyph shows 0-8-17 in a Long Count numbering system, amounting to (8 x 20) + (17 x 1) = 177 days. The second glyph shows 0-17-14, amounting to (17 x 20) + (14 x 1) = 354 = 2 x 177 days. The third glyph shows 1-8-11 for 531 = 3 x 177 days, while the fourth glyph shows 1-17-8 for 708 = 4 x 177 days.
This sequential process continues across the wall, until the last four glyphs show 11-14-12 for 4252 days, 12-5-9 for 4429 days, 12-14-6 for 4606 days, or 13-5-4 for (13 x 360) + (5 x 20) + (4 x 1) = 4784 days. Their last value of 4784 matches closely 162 lunar months of 29.53 days, or equivalently 27 six-month lunar periods of 177.2 days.
A modern value would be 4783.96 days (over approximately 13 solar years). Pretty close to 4784! In summary, this lunar phase calendar seems quite precise and also quite general, since it does not refer to any particular date in history.
The lunar “ring number”
Nearby on the same wall, we can see a “ring number” of 4-15-5-14 in the Long Count numbering system, That “ring” symbol was used when the Mayans wished to subtract some value from their end date of 13-0-0-0-0. By subtracting, we find (13-0-0-0-0 minus 0-4-15-5-14) = 12-15-4-12-6 in the Long Count calendar, which lies 34,314 days before their end date.
In order to confirm that calculation, some ancient Mayan scholar added at the top of his glyph another symbol called “10 Kimi” from the 260-day Tzolkin calendar. By checking the subtracted date of 12-15-4-12-6 with an on-line computer program, we may confirm that it does match “10 Kimi” in the Tzolkin (see www.pauahtun.org or www.pauahtun.org).
Why did an ancient Mayan scholar draw that particular “ring number” here, directly next to his “lunar phase calendar”? We can figure out that puzzle quite easily!
If we divide 34,314 by a monthly lunar period of 29.5306 days, we find 1162 (1161.98) lunar months. Next if we divide 34,314 by an “eclipse year” of 346.6201 days, we find 99 (98.99) years of solar or lunar eclipses. In other words, the “ring number” of 34,314 days marks a long-term match between their lunar-phase and eclipse-year calendars:
34,314 days = 1162 x 29.53 days (lunar phases) = 99 x 346.62 days (eclipses)
Modern values would be 34,314.6 days for 1162 lunar phases, or 34,315.4 days for 99 eclipse years. In summary, this “ring number” again shows a general astronomical value, and not anything specific to some historical date.
Four very large numbers on the opposite wall
A separate set of glyphs on the opposite wall show four very large numbers, given in the Long Count numbering system. The first two of those large numbers as 18 x 18,980 or 63 x 18,980 are well known, from the study of many other Mayan inscriptions or codices. The latter two numbers as 93 x 18,980 or 129 x 18,980 are not yet known:
There has been widespread speculation in the popular press, that such large numbers might relate in some way to December 23, 2012 AD, when our current cycle of the Mayan Long Count calendar ends! Such speculations are not based on any kind of fact however, and seem highly doubtful.
All four of those large numbers seem to represent general calculations, which tell when the Mayan Calendar Round of 18,980 days coincides with four other calendars in use around 800 AD. The cycle lengths of those four other calendars would apparently be 360, 819, 1209 or 1677 days.
First large number of 341,640 days = 18 x 18,980 days (2-7-9-0-0 in the Long Count)
This first large number is quite well-known from past studies of Mayan inscriptions or codices. It represents a long-term match between their Calendar Round of 18,980 days, and their Long Count calendar of 360 days:
341,640 days = 18 cycles of 18,980 days in the Calendar Round = 18 x (20 x 949)
= 949 cycles of 360 days in the Long Count = 949 x (18 x 20)
Their Calendar Round of 18,980 days was based on a medium-term match between the Haab (solar calendar) of 365 days, and the Tzolkin calendar of 260 days. To be specific, 18,980 = 52 x 365 = 73 x 260 days. There are incidentally 65 Sun-Venus conjunctions of 292 days each in 18,980 days, as a possible astronomical basis for the Calendar Round.
In order to reach a true solar year of 365.25 days, the Mayans added 13 days at the end of any 18,980-day period, where (52 x 0.25) = 13. In our modern culture, we add 1 day at the end of any four-year period, as a “leap year” where (4 x 0.25) = 1.
Unfortunately there can be no match among all three Haab, Tzolkin and Long Count calendars for a single 18,980-day period since:
(18,980 / 360) = 52 + 13/18
leaves 52 + 13/18 as a non-integral remainder. Yet if we multiply by 18 to get 341,640 days then:
(341,640 / 360) = (18 x 52) + 13 = 949
and it does leave 949 as an integral remainder among the Haab, Tzolkin and Long Count. In other words, this particular period of time tells how long it takes for all three calendars to coincide.
For example, a Long Count date of 1-0-0-0-0 matches 3 Ahau, 13 Chen in the Tzolkin or Haab (November 15, 2719 BC). By adding 2-7-9-0-0 or 341,640 days, we can find another Long Count date of 3-7-9-0-0 with the same 3 Ahau, 13 Chen (April 3, 1783 BC) (see www.pauahtun.org).
A related “distance number” from the Dresden Codex shows 1,366,560 days = 72 x 18,980 days as 9-9-16-0-0 in the Long Count. That large number tells how long it takes for the Calendar Round to coincide with the Long Count on four successive occasions.
Second large number of 1,195,740 days = 63 x 18,980 days (8-6-1-9-0 in the Long Count)
The second large number from Xultun, Guatemala is also well-known from previous studies of Mayan inscriptions or codices. It represents a long-term match between their Calendar Round of 18,980 days, and another enigmatic calendar of 819 days which may have been developed for astronomical purposes.
Some people have suggested that their 819-day calendar was derived from the synodic period of Mercury as 116 days, since 819 = 7 x 117. Not a perfect match, but Mercury was assigned a synodic period of 117 days in the Dresden Codex, increased by one day from its true astronomical value of 116. (Venus was likewise assigned there a synodic period of 585 days, increased by one day from its a true astronomical value of 584).
This second large number of 1,195,740 days (close to 3276 years) equals precisely 7/2 the previous value of 341,640 days (close to 936 years):
1,195,740 = (7/2) x 341,640
Furthermore, it is the least-common multiple of their 18,980-day and 819-day calendars:
1,195,740 days = 63 cycles of 18,980 days in the Calendar Round = 63 x (20 x 949)
= 1460 cycles of 819 days = 73 x (20 x 819)
Unfortunately there can be can be no match among all three Haab, Tzolkin and 819-day calendars for a single 18,980-day period since:
(18,980 / 819) = 23 + 11/63
leaves 23 + 11/63 as a non-integral remainder. Yet if we multiply by 63 to get 1,195,740 days then:
(1,195,740 / 819) = (63 x 23) + 11 = 1460
and it does leave 1460 as an integral remainder among the Haab, Tzolkin and 819-day calendars. How clever is that? In other words, this particular period of time tells how long it takes for all three calendars to coincide. For more details, please see “New Mathematical Methods for the 819-day Count” (www.pauahtun.org).
Third large number of 1,765,140 days = 93 x 18,980 days (12-5-3-3-0 in the Long Count)
Fourth large number of 2,448,420 days = 129 x 18,980 days (17-0-1-3-0 in the Long Count)
The third and fourth large numbers from Xultun, Guatemala pose a puzzle, since academic scholars have not encountered them before in surviving Mayan inscriptions or codices. Each shows a precise fractional multiple of 341,640 days:
1,765,140 = (31/6) x 341,640
2,448,420 = (43/6) x 341,640
Could the ancient Mayans have developed two more calendars which we do not currently know about, with periods longer than 360 or 819 days? If so, then their third and fourth large numbers might tell how long it takes for the Calendar Round of 18,980 days to coincide with each of two additional “unknown” calendars.
We can deduce the cycle lengths of those two “unknown” calendars as follows. The wall mural from Guatemela shows four large numbers equalling 18, 63, 93 or 129 x 18,980 in days. The first two of those numbers tell when the Calendar Round will match calendar cycles of 360 or 819 days in an integral fashion:
(18,980 / 360) = (18,980 / 20 x 18) = 52 + 13/18 known cycle of 360 days
(18,980 / 819) = (18,980 / 13 x 63) = 23 + 11/63 known cycle of 819 days
The latter two of those numbers presumably do the same:
(18,980 / x) = (18,980 / 13 x 93) = 15 + 65/93 unknown cycle of x days for n = 13
(18,980 / y) = (18,980 / 13 x 129) = 11 + 41/129 unknown cycle of y days for n = 13
If we divide 18,980 by (n x 93), then we can find an integral remainder of (n / 93) for all of n = 1, 2, 4, 5, 10, 13, 20 or 26. Thus in principle, we can find many possible cycle lengths of x = 93, 186, 372, 465, 930, 1209, 1860 or 2418 days which divide evenly into 1,765,140. Yet once we exclude the trivial results, and include an observed ratio of large numbers as:
93 / 63 = x / 819 = 1209 / 819
then we may suggest x = 1209 days as the most likely candidate, where n = 13 and x = 13 x 93.
Similarly for the fourth large number:
129 / 63 = y / 819 = 1677 / 819
we may suggest y = 1677 days as the most likely candidate, where n = 13 and x = 13 x 129.
Looking again at their wall mural (shown above), we can see that two early-stage calendar cycles of 63 or 18 x 18,980 days were drawn together on the left, while two “unknown” calendar cycles of 129 or 93 x 18,980 days were drawn together on the right. Such an observation suggests perhaps that the novel 1677 and 1209-day cycles were late-stage developments around 800 AD? Further research into this interesting subject may yield more definite conclusions.
Four Tzolkin symbols were drawn on the wall above those four large numbers
Four different symbols from the 260-day Tzolkin calendar as “9 K’an”, “1 Kawak or Kaban”, “? Manik” and “13 Chikchan” were drawn above those four large numbers on the same wall, respectively in terms of increasing cycle length. I looked for some astronomical or calendrical significance to those symbols, for example in terms of the tropical year (365.24 days) versus Calendar Round year (365.00 days), but could find none.
These Tzolkin symbols may be culturally specific, and tell when each long-term cycle begins or ends. For example, the 819-day cycle is thought by many to have begun on a date of “1 Kaban”, three days before “4 Ahau” for 0.0.0.0.0. That is the same Tzolkin symbol which was drawn on the wall above 8-6-1-9-0, for a long-term match between their 819-day calendar and the 18,980-day Calendar Round.
Summary and conclusions: the long-predicted Mayan year of 2012 AD has arrived!
To conclude, it should be emphasized that none of the new Mayan glyphs found at Xultun, Guatemala show any relation whatsoever to the year 2012 AD, when 13-0-0-0-0 will be reached as an end date for the current Mayan Long Count calendar. Indeed, each large number as drawn there may be counted either forward or backward in time, from an unknown starting date.
Many articles in the popular press about this subject were apparently written by journalists who do not understand (for example www.time.com). The original article in Science magazine (cited above) says nothing about the year 2012 AD.
In any case, we should respect the ancient Mayans for being great mathematicians. How many modern people will be able to understand the lucid explanations provided above? Are we watching too much TV, and not learning or thinking as much as we should?
If the ancient Mayans chose to begin their Long Count calendar on August 13, 3114 BC, while ending it on December 23, 2012 AD, perhaps they possessed some kind of knowledge which we today have long forgotten? As the summer of 2012 unfolds, we will look eagerly in the fields to see which new crop pictures might appear there, and also we will wait expectantly for whatever the important date of December 23, 2012 might bring:
Red Collie (Dr. Horace R. Drew, Caltech 1976-81, MRC Laboratory of Molecular Biology 1982-85, CSIRO Australia (1987-2010)