Ernest P. Moyer
PO Box 1206
Hanover, PA 17331

June, 2010
Professor David Stuart
Department of Art & Art History

The University of Texas at Austin
1 University Station D1400
Austin, TX 78712–0340

The Mayan Calendar Design
(Part Two)

Summary of Part One

Maya Fascination with the Number 13

The Maya had a fascination with the number 13. It shows on their monuments, and in their Codices.

The first illustration is from WAYEB NOTES, No. 14, 2004: A PRELIMINARY RULING SEQUENCE OF COBÁ, QUINTANA ROO, Sven Gronemeyer, Institut für Altamerikanistik und Ethnologie (IAE), Rheinische Friedrich-Wilhelms-Universität, Bonn.

The date on the stele is 4 Ajaw 8 Kumku, with 20 "13's."

This graphic shows a sketch of the details of the stele, including the date in the upper left-most segment, columns M, N, rows 1-14. The uppermost part of the stele is eroded; we must guess about the two top glyphs.

Click on the thumbnail to obtain a full picture.

In 1979 Ian Graham published information on Yaxchilán Hieroglyphic Temple Stairway 2, Step VII, Corpus of Maya Hieroglyphic Inscriptions. Vol. 3, Part 2.  There an inscription has 8 "13's": 3 Muluc 17 Mac. This is equivalent to October 19, 744 with an expanded expression of time. See his illustration at (2).jpg

See also The PARI Journal, a quarterly publication of the Pre-Columbian Art Research Institute, Volume IX, No. 3, Winter 2009:

Archaeological and Epigraphic Studies in Pol Box, Quintana Roo, by Octavio Q. Esparza Olquin, Escuela Nacional de Antropología e Historia, and Vania E. Perez Gutierrez.

Click on the thumbnail to obtain a full picture.

See their Figure 8 where Bird Jaguar IV wears the outfit of a ballplayer. A bound captive descends in the form of a sphere (drawing by Ian Graham). The presence of the two elves, a most fascinating insight into Maya culture, are completely ignored by Olguin and Gutierrez.

The clarity of the photograph taken by Jami Dwyer belies the sketchy drawing published by Graham. As Dwyer commented: "Completely unique figures way up high at the Yaxchilan ruins. I didn't see anything else like these little elves at these ruins or any other." Dwyer released this photograph into the public domain 4 November 2006.

In the Dresden Codex, page 52, Section A, column 4 is a sequence of 13 "13's".

This may be downloaded directly as a complete Codex file:

This evidence raises the natural question: What is the meaning of these extraordinary numbers? Linda Schele and cohorts proposed that this number means 41,945,040,000,000,000,000,000,000,000 years into the past, greater than the assumed age of the universe. Even the Hindu's gave their god Bramha an entire life span of only 311 trillion, 40 billion years: 311,040,000,000,000. The Sumerians in the 19th century BC also gave an enormously large 15-place sexigesimal number, 13 22 50 54 59 09 29 58 26 43 17 31 51 06 40, equaling the 20th power of 20, which is 104,857,600,000,000,000,000,000. We are uncertain if the Sumerians intended this as an indication of the age of the universe. See

Click on the thumbnail to obtain a full picture.

Susan Milbrath, in Star Gods of the Maya: Astronomy in Art, Folklore, and Calendars, discusses the eclipse table in the Dresden Codex on pages 51 to 58. Here is an illustration of these pages, containing the 13 13's.

A full photograph may be found at either of these URLs:


Since the number of 13's differs among the several sources a natural question is raised of the intent of providing such large numbers. Were they "real" in the thinking of the Maya, or merely fascinating imagination? Or did they represent an indefinite image of ages, of eternity?

Eric Thompson, in his speculative dissertation, Maya Hieroglyphic Writing, proposes that these numbers should be "20" and not "13." He states that, In any case, the record is positive proof that the Maya thought of the pictun as composed of 20, not 13, baktuns, page 314.

Thompson had a habit of expressing positive proof when he meant mere speculation. The Maya never knew of pictuns, an invention of modern Maya students to make discourse more convenient for themselves, and therefore could not have offered positive proof of any such concept.

Elsewhere, APPENDIX IV, Maya Calculations Far into the Past and into the Future, page 314, he says:

Again, he repeats this assertion:

In other words modern Maya students have invented these terms, and pervasively used them for their own convenience, so that they have taken on a life of their own, but they never existed in the minds of the Maya themselves.

Lloyd Anderson in a web site at

offers interesting discussion on this very problem. He himself believes the "20" was a reality to the Maya; he reviews some of the evidence. But then he resorts to doubt when he says: It is not obvious that many of these "13"s are counting anything at all! That is a fundamental question here, not usually asked. In other words, the "13's" might have been merely a fanciful plaything in the minds of the Maya. If the Maya intended "20's" why write "13's"?

Tzolkin Patterns

A full Long Count from the initial date of August 11, 3114 BC to the fill-up date at December 21, 2012 (give or take a few days in the origin point) shows 1,872,000 days. Division of this number by 260, the Tzolkin cycle period, yields an even 7200 cycles. The Tzolkin does not end in a partial cycle at the end of the Long Count; it ends where it started, at 4 Ahau. In other words the Tzolkin is not only mathematically locked onto the Haab, it is also mathematically related to the Long Count.

Examine how the Tzolkin rotates through its sequence, starting at 4 Ahau, the start date of the Maya Calendar.

Tzolkin Cycles


Glyph Columns


Ahau Imix Ic Akbal --- Oc --- Cuac
4 5 6 7 --- 1 --- 10
11 12 13 1 --- 8 --- 4
5 6 7 8 --- 2 --- 11
12 13 1 2 --- 9 --- 5
6 7 8 9 --- 3 --- 12
13 1 2 3 --- 10 --- 6
7 8 9 10 --- 4 --- 13
1 2 3 4 --- 11 --- 7
8 9 10 11 --- 5 --- 1
2 3 4 5 --- 12 --- 8
9 10 11 12 --- 6 --- 2
3 4 5 6 --- 13 --- 9
10 11 12 13 --- 7 --- 3

I have shown here one full Tzolkin cycle of 260 steps. As we move through the first 20 glyphs of the Tzolkin cycle, we will come next to 11 Ahau, and a new series of 20 glyphs. And so on through the 260 steps. We can see how the Ahau (and all other glyphs) rotate through different tone numbers, because of the manner in which the tones are numbered: 13. 4 goes to 11 Ahau, to 5 to 12 Ahau, and so on, causing the sequence to jump one number as it rotates around the Tzolkin. This pattern produces a 40-day rotation from 4 (+40) to 5 (+40) to 6 Ahau, and so on, jumping to the next sequence at each step and around a full cycle for return to 4 Ahau. This basic design, without a common multiple between the 20 glyphs and the 13 tones, forces the 260 steps. As a consequence this ripple will produce pattern effects in all related sequential numbers, no matter how they are grouped together, Long Count Round Numbers, such as, or any other grouping.

Note that 4 Ahau starts with a 0 position; it will never see anything but a 0 because of the 260 day cycle.

A curiosity exists in that the first tone row, starting at 4 Ahau, will go half way through (10 glyphs), then roll over at 1 Oc, and end at 10 Cuac. Did the designer purposely start at 4 Ahua to cause this numbering to fall in two halves of 10 each, so that the next glyph would start at 11? Did he know something about decimal numbering which is not obvious from the Maya vigesimal numbering system?

The following table shows how the ripple provides a basis for the number 13 to appear in the LC. This pattern is also true for all other singular dates, except that only 13 ends a full LC cycle at 12-21-2012. These are all 4 Ahau (0) anniversary dates. (In contrast, the LC Round Number falls on September 18, 1618, 5 Ahau.)

No. of Days No. of Tzolkin Long Count Date Tzolkin Date
260 1 4-27-3113 BC 4 Ahau
4680 18 6-4-3101 BC 4 Ahau
4940 19 2-19-3100 BC 4 Ahau
93,600 360 11-18-2858 BC 4 Ahau
93,860 361 8-4-2857 BC 4 Ahau
98,280 378 9-10-2845 BC 4 Ahau
98,540 379 5-28-2844 BC 4 Ahau
1,872,000 7200 12-21-2012 AD 4 Ahau

Note that the 13's have a large step from 19 to 360 Tzolkin, but a much larger step from 379 to 7200. This jump comes about because there are no possible intervening stages between 2844 BC and 2012 AD. This is a clue as to why the number 13 was so important to the Maya.

          Long Count Days
12 x 144000 = 1728000
19 x 7200 = 0136800
19 x 360 = 0006840
17 x 20 = 0000340
19 x 1 = 0000019
  Total days = 1871999

Consider how the Long Count is organized. I have arranged this table so that when we add one number it rotates to the next full Long Count. This is the full count that worries so many people.

The Tzolkin place at which this table rests prior to the next increment is 3 Cuac. This is the Tzolkin date prior to 4 Ahau.

This is a modified vigesimal table. Every increment in level is done with a multiplier of 20, except for two levels: level 2 has a multiplier of 18, while level 5 has a multiplier of 13. If the Maya were organizing this mathematical table according to their rules of vigesimal calculations they should have placed a multiplier of 20 at each level. (And this is why Thompson and others suppose a 20 should have come at the end of the LC.) Why did the calendar designer change those two levels? The strange coincidence is that the second level multiplier (18) is the number of time periods in one Haab year (erroneously referred to as months in the scholarly literature). In the top level the multiplier is the number of day counts (13) in the Tzolkin cycle. These two numbers are the peculiar numbers in the formulation of the Haab and Tzolkin aspects of the Mayan calendar.

I have not attempted a mathematical analysis of how these numbers affect the overall structure of the calendar. We can see how it is profound. I leave that for another time.

If we examine the Long Count in terms of its properties what would we find? I use the 4 Ahau anniversary date as a reference point. The Long Count starts at that point; 4 Ahau should inform us of events as we progress through that Count.

If we start at 4 Ahau and advance 553 Tzolkin cycles, as shown in the Long Count Table, we come to at 3 Ahau. But that point comes with a fraction of a Tzolkin cycle remaining in the count, 220. You can see this in the above Tzolkin Cycle Table, where 3 Ahau appears at the 12th Tone Row. Thus one can follow the increments from the bottom of the Tzolkin Table. Back up 40 cells from the end to bring us to 3 Ahau. (260 - 40 = 220). Then continue back another 40 (220 - 40 = 180) to 2 Ahau at And so on to 1, 13, 12, 11, 10, . . . to the end of 4 Ahau at But each Long Count Round Number calendar date always brings us to an Ahau Tzolkin date.
Here is a tabulation of the Long Count Calendar showing how this feature appears as a Tzolkin date:

Long Count Table
Long Count Round Numbers

Number of Tzolkin Cycles

Fraction of Tzolkin remaining in days

Difference in Tzolkin Cycles Calendar
BC to AD
Date 553 220   11/13/2720

3 Ahau 1107 180 554 2/16/2325 2 Ahau 1661 140 554 5/21/1931 1 Ahau 2215 100 554 8/23/1537 13 Ahau 2769 60 554 11/26/1143 12 Ahau 3323 20 554 2/28/748 11 Ahau 3876 240 553 (-) 6/3/354 10 Ahau 4430 200 554 (+)  9/5/41 9 Ahau 4984 160 554 12/9/435 8 Ahau 5538 120 554 3/13/830 7 Ahau 6092 80 554 6/15/1224 6 Ahau 6646 40 554 9/18/1618 5 Ahau 7200 ----- 554 12/21/2012 4 Ahau

The calendar and Tzolkin dates were calculated using software provided by John Kostura, C K Marketing Inc., 3629 NW 10th Street, Oklahoma City, OK 73107. I checked these values against software provided by Ed Barnhart, of the Maya Exploration Center, 7301 Ranch Road 620 N Suite 155 #284, Austin, TX 78726. I found a difference of one year BC in the two published results.

I have arranged this table to illustrate the manner in which different Long Count Rond Numbers provide a count of Tzolkin cycles. As the Tzolkin rolls along it will come to 7200 4-Ahua "0" anniversary dates from beginning to end of the Long Count. But those "0" anniversary Long Count dates fall at intervals that do not coincide with the Round Number anniversaries.

When the Long Count comes to a full Round Number, for example, it does not reach the end of that series at 4 Ahau. Some Tzolkin days remain. Similarly down through the list. Whenever the Long Count moves on to the next Round Number the day name will end on a different Tzolkin Ahau, always one day less than the previous one (1 rotates to 13 at Tzolkins that rotate to these partial Long Count cycles will never end at 4 Ahau. I cannot find a Round Number Long Count that will end at a 4 Ahau anniversary -- except the last one. That is at

These are all Ahau dates, rotating to a new (one less) value as we increase the Long Count Round Number increment.

Now look at the balance of the Tzolkin remaining at each anniversary. At it is 220, or 40 less that the start at 260. It continues to be 40 less at each Round Number Long Count date, 180 at, 140 at, and so on. It even skips from 20 back to 240 at to Again it comes to rest at zero only on This sequence is due to the ripple effect shown in the Tzolkin Cycle Table.

Consider the number of days to reach a Long Count of 144,000 days. (November 13, 2719 BC, from Barnhart's software, with the same 3 Ahau.)  But 144,000 is not an even multiplier of 260. The number is 553.84615384615384615384615384615 . . . And 0.84615384615384615384615384615 X 260 = 220 days. This can be verified by the Table for the Long Count Days above. Each Round Number Long Count is multiplied by 144,000. I repeat the numbers here to make the calculation more apparent.

Long Count Period No. of Days Fraction Remaining No. of Days Remaining
1 144,000 0.8461538461538461538461538461 220
2 288,000 0.6923076923076923076923076923 180
3 432,000 0.5384615384615384615384615385 140
4 576,000 0.3846153846153846153846153846 100
5 720,000 0.2307692307692307692307692308 60
6 864,000 0.0769230769230769230769230769 20
7 1,008,000 0.9230769230769230769230769231 240
8 1,152,000 0.7692307692307692307692307692 200
9 1,296,000 0.6153846153846153846153846154 160
10 1,440,000 0.4615384615384615384615384615 120
11 1,584,000 0.3076923076923076923076923077 80
12 1,728,000 0.1538461538461538461538461538 40
13 1,872,000 0.0 0

Because of the way the Tzolkin is designed the Long Count rotation Round Numbers come back each time to one Ahua less in count. As these rotate through one full Long Count they will end with the Ahau one less in count: up to will yield one less Ahau, 12 to 11. Eventually this process will end with a full cycle of 4 Ahau at

The reason we fill out the Ahau rotation at by coming back to 4 Ahau, the starting point, is the manner in which the factors are organized in the Tzolkin of 260 days: 13 and 20. It takes 7200 Tzolkin cycles to reset the whole system.

Now examine how the Long Count Round Numbers stack up as we increment by single digits. From we will go through 20 days to rotate to one higher level. ( As we continue through the full Tzolkin of 260 days we will continue to add one by one to the next higher level. At the end of the first Tzolkin cycle this will end at 13 on level 2. However, since the value of the next higher level is 18 it will fill up only part way through the second Tzolkin cycle: 13 + 5 to jump to level 3, and so on.

Suppose the Long Count had a shorter count in each step: for example This means the calendar would never reach beyond 11/13/2720 BC. The Maya could not keep track of time for more than 554 years. And the calendar would end with 3 Ahau, instead of 4 Ahau. Therefore the calendar had to increment one step higher to bring a full cycle of Ahau's. This means the anniversary date of 4 Ahau at was a mathematical necessity, not a sentimental notion. It also means that whoever created this system had to know how the Ahau's would cycle through the calendar. That is why the Maya did not bother to show Long Count dates higher than five places. For the inventor of this calendar, and for the Maya user, the end of time -- not the end of the world, but the end of the age -- would come when the full cycle was complete on 4 Ahau, but not before then. Long Count numbers beyond that date did not matter because a new calendar cycle would begin. They expected a transformation in the world that would revamp time. Only modern godless minds cannot penetrate this mathematical destiny -- mathematics determining fate.

We can now understand why the Maya stopped their Long Count at, not at some mysterious 20. It was the point at which there was a complete Tzolkin Ahau rotation that met the 4 Ahau anniversary. In their design this was 7200 Tzolkins. That was the time when all calendar elements would reset to zero -- and not before then.

The top level multiplier of 13 in the Long Count table has surely annoyed a lot of people in the Maya scholarly community. They would prefer 20, a nice round number (in their academically trained minds)? Bur none of the monumental or Codex evidence shows 20. When the baktun fills up it rolls over from 13 to zero. We have not been witness to that event, even though it will occur at the winter solstice at the end of 2012 (or some nearby yearly anniversary). But the Maya knew it would be the end of current time.

Then the World would enter a new age.

The Maya citizen not only had a daily reminder of his place in the flow of time, of fate -- he also knew when that age-ending point would occur. His calendar kept him securely tied to the Gods.

How much has the knowledge of the immanence of destiny, fate from the Gods, now returned to all of world civilization?

This calendar could not have come about through evolutionary happenstance. It had to be designed. By a master mind who worked out all of its ramifications. Furthermore, that design was by an intelligence that we humans can understand. That intelligence would have been an extraordinarily bright human being, or an intelligence that was superhuman. The design had to be imposed upon a primitive society. But that imposition could come only through the persuasive powers of that Intelligence. The primitive society had to have a great, if not immense, respect for that Intelligence. The magnitude of that respect had to approach the worship of a god. We know him from the Maya records as Kukulcan.

Kukulcan, or some personality we remember by that name, knew the prophecy of time. And designed a calendar system that would keep all citizens tied to the Gods.

In 1924 Herbert Spinden, an early researcher into the Maya, wrote the following about the Mayan calendar:

  • The invention of the Central American calendar in the Seventh century before Christ may he described with all propriety as one of the outstanding intellectual achievements in the history of man. This calendar solved with conspicuous success the great problem of measuring and defining time which confronts all civilized nations. Moreover it required the elaboration of one of the four or five original systems of writing the parts of speech in graphic symbols. and it conjoined with this supplementary invention of hieroglyphs the earliest discovery of the device of figures with place values in the notation of numbers. This time machine of ancient America was distinctly a scientific construction, the product of critical scrutiny of various natural phenomena by a master mind among the Mayas. It permitted a school of astronomer-priests to keep accurate records of celestial occurrences over a range of many centuries, with the ultimate reduction of the accumulated data through logical inferences to patterns of truth.  (—Herbert J. Spinden: The Reduction of Mayan Dates 11-9-24)
Indeed this was one of the outstanding intellectual achievements of man. Indeed, it was a scientific (mathematical) construction. As I have shown here all three aspects of the Maya calendar were locked together: the Tzolkin, the Haab, and the Long Count. Such an intricate arrangement could not have been accomplished without tremendous intellectual agility. We had the evidence sitting in front of our eyes without the ability to see through it. I think this is a commentary on our modern intellectual attitudes.

As James Q. Jacobs said:

In my view we are still mostly in the stage of diviners and prognosticators. In a little while we will come up against the abrupt traumatic events of the end of this earth age.