The 3-11-Pik Formula


Barbara MacLeod


An 8660-day interval ( in Maya notation) is discussed by Matt Looper (2002) with respect to a carved bone from Burial 116 at Tikal (Tikal MT 26), and is referenced by a compound ‘11-pik’ (termed “11-Pih” in Matt Looper’s discussion). He also provides most of the extant examples (nine are now known) from the monumental inscriptions. On the bone one finds three phrases in a vertical sequence:  ‘first 11-pik’, ‘second 11-pik’, ’ third 11-pik’. Each statement is preceded by a Calendar Round. The first is ( 6 Ajaw 8 Mak; the second is ( 8 Ajaw 8 Mol, and the third is ( 10 Ajaw 8 Zip.  Looper determined that these CRs are separated by an interval of only 8660 days, and the first is counted from ( 4 Ajaw 8 Kumk'u.


What is also true, but apparently not considered in Looper’s investigation, is that the vastly greater interval of eleven times 144,000 days likewise leads from 4 Alaw 8 Kumk’u to a Calendar Round date 6 Ajaw 8 Mak. Therefore the next 11-pik interval, and the next, will lead to the second and third CR positions noted in the 8660-day interval. So there is a hitherto unrecognized relationship of congruent counts built into what I will call the ‘3-11-Pik Formula’, with the greater cycle counted by eleven-bak’tun leaps from the Creation date of 4 Ajaw 8 Kumk'u.  4 Ajaw 8 Kumk’u  4 Ajaw 8 Kumk’u

 +                                   + 0

________                                     ________  6 Ajaw 8 Mak            1.4 .   6 Ajaw 8 Mak


8660 days mirrors  the product of 11 times 144,000 days, and each is then multiplied in turn by 3—a result to be discussed further below. The ratio between the constituents—that is, eleven bak’tuns or 1,584,000 days divided by 8660 days—yields a divisor of 182.90993. When one multiplies 8660 by 182.90993, one gets 158399999…or in effect 1,584,000. But if one rounds 182.90993 up to 183 and multiplies 8660 by that, the result is 1584780, or 780 days more than 1,584,000.


1,584,000 x 3 = 4,752,000. This completes the formulaic statement of three eleven-piks.

4,752,000 divided by 365 is 13,019.178. If the tropical year or 365.2422 days is used, the quotient is 13,010.544.


It is important to recognize that the interval of 8660 days is not the only one which replicates the CR sequence found by successively adding intervals of 11 x 144,000 days. In fact, one finds that there are 83 possible sequences of these same CRs between that employing 8660 days (the smallest possible interval) and that employing eleven bak’tuns. However, there are specific reasons to assume that the smallest interval of 8660 days is the one intended, because (1) three times this can reasonably fall within an elite human lifetime, and (2) because that number is itself the smallest available count. Other intermediate counts, as well as counts based on different multiples of bak’tuns, were explored--and these are certainly possible mathematically-- but nothing productive resulted.


3-11 Pik Stations: A Hypothesis


The Tikal bone data suggests the possibility that the Maya noted intervals of 1 and 3 times 8660 days counted from 4 Ajaw 8 Kumk’u. These stations, based on intervals of 8660 days and 25,980 days, would be similar to those of the 819-Day Count. The Long Count positions of several of these suggested stations, which I am calling Single and Triple Stations, demonstrate meaningful relationships with the chronologies of the most important monuments on which the 3-11-Pik formula is found.

In his discussion of the Chichen Itza Caracol Stela text and  the Tikal bone , Nikolai Grube (2003) relates the 3-11 Pik phrase to the vaster cycle of three times eleven bak'tuns but without any consideration of a smaller set of intervals. He notes that the Caracol Stela has an example of the  phrase (following mention of the 4 Ajaw 8 Kumk'u 'changing of the hearth')  in which ’his 3-11 Pik’ is said to be "bound" (the verb is k’ahlaj ) to the 17th tun ( of the contemporary K'atun 1 Ajaw. No further denouement has been offered until now.



Long Count Positions of Relevant Single and Triple Stations


Following is a list of the proposed Triple Stations for the last part of Bak’tun 8, all of Bak’tun 9, and much of Bak’tun 10.


I am using 'Triple' for the station/interval 3 x 8660 days. This is 25,980
days or in Maya notation.

le' stations—cited as needed for specific monuments--separate intervals of 8660 days, or in Maya notation)      12 Ajaw 8 Mak

9. 0. 8. 6.0        5 Ajaw 13 Pax

9. 4. 0. 9.0       11 Ajaw 13 Pohp       4 Ajaw 18 Zotz'     (52 out from the base of       10 Ajaw 3 Mol       3 Ajaw 8 Zak

9.18. 9. 3.0       9 Ajaw 13 K'ank'in

10.2. 1. 6.0       2 Ajaw 18 K'ayab       8 Ajaw 18 Wo       1 Ajaw 3 Xul   7 Ajaw 8 Ch'en           13 Ajaw 13 Kej  (60 out from the base of



Chronology of K’ahk’-u-Pakal and The Caracol Stela at Chichen Itza:

Here are the re
levant 1-11-Pik (Single) and 3-11 Pik (Triple) stations which
lap with Kahk'upakal's life:

Presumed birth of K'ahk'upaka
l  takes place in this interval
le)                                  10. 0.17. 5. 0
Casa Co
lorada                      10. 2. 0. 1. 9       6 Muluk 12 Mak
(first mention of K'-u-P)
le Station                        10. 2. 1. 6. 0      2 Ajaw 18 K'ayab
l Stela                         10. 2 .17.0. 0      13 Ajaw 18 Yaxk'in
”binding” ( K’AL) event
linking his 3-11-Pik with the 17th tun (last mention of K'-u-P)
lose of K'atun 1 Ajaw           10. 3.  0. 0. 0      1 Ajaw 3 Yaxk'in
le)                                  10. 3.  5. 7. 0

The text
links (binds? records?) his 3-11-Pik and the 17th tun of K'atun 1 Ajaw (current
k'atun which ends on This "binding" is part of a verba
let, the other half of which is the Creation event or the changing of
the hearth (at the edge of the sky, etc.) on 4 Ajaw 8 Kumk'u. So
the narrative time
line is an important clue: it appears to have been
foreordained when the 13-Bak'tun cyc
le was set in place that K'ahk’upakal would
have this "binding" on

We have a Trip
le station on,
and within its first Sing
le unit we have two events: (1) the dedication of
the Caraco
l stela with the "binding" and (2) the close of K'atun 1 Ajaw on Since they both fa
ll within the first unit of a bundle of three,
the "binding" concept is even more reasonab
le. There are in fact three
things bound if we inc
lude the 3-11-Pik station/interval itself.


Chronological Timeline for Aj Wosal on Naranjo Altar 1:


9.4 .0. 9.0     1 Ajaw 13 Pohp      Triple 3-11-Pik station  (Aj Wosal not yet born)

9.5 4.10.0   13 Ajaw 18 K'ank'in  Single 11-Pik station    6 K'an 3 Zip             Accession of Aj Wosal at ~ age 12 or younger    2 Ajaw 18 Ch'en      Single 11-Pik station    7 Kaban 5 K'ayab    a house dedication event by Aj Wosal, with titles of 35th successor to the founder and successor to the Jaguar Paddler at Nah-Jo-Chan    7 Ak'bal 11 Zotz'      (reached via DN given, with 1-day correction) piling of skulls, pooling of blood (war event); the subjects are the One-Ajaw God and 6-Earth-Place Aj Wosal      9 Ajaw 3 Wayeb     Aj Wosal celebrates a k'atun ending      7 Ajaw 3 K'ank'in    Aj Wosal celebrates a k'atun ending   4 Ajaw 18 Zotz'      Triple 3-11-Pik station       5 Ajaw 3 Ch'en      Aj Wosal celebrates a k'atun ending; this is the dedicatory date; HE NOW HAS THE 3-11- PIK AJAW TITLE


Parents are named, then DN of 12 k'atuns counts forward to Bak'tun 10   6 Ajaw 3 Kumk'u    Single 11-Pik station                                 Single 11-Pik station   10 Ajaw 3 Mol        Triple 3-11-Pik station

(skipping over a lot of time here)

10.0.0. 0. 0   7 Ajaw 18 Zip         "will end 10 bak'tuns"

(9. 8.0. 0. 0)     5 Ajaw 3 Ch'en      "will happen his scattering of incense" 


A Sketch of Kahk’ Tiliw Chan K’awiil’s Modus Operandi on Quirigua Stela F:


This is a complex monument. Its iconographic panel of
K'ahk' Ti
liw Chan K'awiil in all his glory occupies the north side; the west
side opens with an Initia
l Series featuring his accession (at B6) on 12 Kaban 5 K'ayab. His names and tit
les continue through B10.

The 3-11 Pik tit
le does not appear on this side of the monument.

That Trip
le Station happens just under 4 tuns later on 3 Ajaw 8 Zak--the only tun
ending among the 3-11Trip
les in Bak'tun 9. It seems highly relevant that he
accedes just before this important station. I consider the re
lationship between this station and his
accession to be the rationa
le for all the calendric gymnastics on this
monument, in both current and deep time. It's the most ca
lendrically complex
la at Quirigua.

On other monuments he carries out Period Ending dedications for
and; he mentions some of these in passing on Ste
la F West after
jumping ahead to the main event on  6 Kimi 4 Tzek. This is the
capture of "Eighteen Rabbit"--Wax
lajun u-Baah K'awiil of Copan, his former
lord, recorded in the continuing text of Stela F, west. The rest of this
side has the CRs of and, and inc
ludes a reference to
the 'changing of the hearth' (at 4 Ajaw 8 Kumk'u--though that CR is
unstated).There's a restatement of his accession at the bottom of this side,
as if we were going to forget, and then he segues to the CR of,
as the ha
lf period of 13 Ajaw 18 Kumk'u. The lajuntun is featured
on the east side. He is definite
ly out to cover all the local bases, tying
them to his accession, before moving on.

The dedicatory PE of the monument is in the Initia
l Series of the east side.
It's a fu
ll IS with Lunar Series leading to his incense-scattering for the
"One Ajaw" monument itse
lf--whose name includes what looks like Pik-Witz
(-yi) Tun.Then fo
llows 'u tz'ap-aw, 'he plants it' and the thing being 'planted' is called
'u-Ajaw-Pik Witz.

This is a dramatic high point.

The 'planting' of his "royal pik mountain" takes place in the second Single unit following the Triple of


Here is an outline of the chronology:


West Side:  12 Kaban 5 K’ayab                   accession    3 Ajaw 8 Zak                            Triple station                                                       PE on other monument                                                       PE on other monument    6 Kimi 4 Tzek                           capture of “18 Rabbit” of Copan

( 4 Ajaw 8 Kumk’u)                         changing of the hearth

                                                                        DN counted from 12 Kaban 5 K’ayab

(                                                     Single station


(                                                    half period of 13 Ajaw 18 Kumk’u


East Side:  1 Ajaw 3 Zip                                IS and Dedicatory Date

                                                                        scattering of incense to/for the Pik Witzi(l) Tuunil

                                                                        (name of One Ajaw Stone)

                                                                        Planting of his Ajaw Pik Witz

                                                                        Full name/Titles of Kahk’ Tiliw, 3-11-Pik Ajaw                                                                                                                                         Single station


(DNs counting back into the primordial ooze follow…)


His "royal pik mountain" is planted on this spot. Now his name and titles
appear, c
losing with 3-11-Pik Ajaw.

And then we
launch into outrageous DNs; there's an excellent discussion of
them in Thompson's MHW: An Introduction on p. 315.


He is tying his 3-11-Pik Triple to the dedication of the monument.


A few more thoughts about Naranjo Altar 1 and Quirigua Stela F


These monuments share some important features, and may tell us how the 3-11-Pik formula operates when a king lays claim to it.


Both monuments have huge counts into the past. Not every monument that has a huge count has the formula, of course, but there really aren't that many huge DNs like this in the corpus.


In both texts, there is a war or sacrifice event. The 'piling of skulls, pooling of blood' seems to be a sacrifice of many people, and at Quirigua, the capture and sacrifice of "18 Rabbit" is K'ahk' Tiliw's centerpiece. On both monuments, the text jumps ahead of a chronological narrative to highlight the sacrifice, then goes back to a count of period endings.


Both texts feature a series of period endings celebrated by that king. This is not very common, all texts considered. It may be one of the privileges that accompanies the taking of the title.


On Naranjo Altar 1, the king takes the title a few years after the Triple, in association with his first chronological event following the Triple. On Quirigua F, the king takes the title at the period ending tuns after the Triple. Why does he wait so long? We know he celebrated the PEs in between; most are documented, including some jo'tuns. Stela F seems designed to feature--if I may speculate--all the privileges and associations of the title: his accession less than four years before the Triple, and the characteristics of sacrifice, PE tallies and counts into deep time. I can only assume that by, or even a k'atun later, he wasn't ready to take the title yet, and it has to mean that he hadn't logged his three Singles yet. That would not have happened until after, the first Single station after the Triple of That next unit would count as his third Single and is the second Single after the Triple--so he fixed his period ending, and all the hoopla of Stela F--into that unit.


This does suggest that it's not simply about having a Triple fall into one's reign, but also about straddling three Single units in one's lifetime and celebrating important Period Endings. Kahk' Tiliw's first unit would thus be the one prior to, within which we may assume he was born.



Xkalumk’in Miscellaneous Text 5


There are corrected drawings of this Xkalumk'in text. In this one, several pieces are out of place or, in the case of the sajal glyph, upside down. The second 'u-k'aba' is totally butchered. But none of this affects what I have discovered about it.


The first glyph in this text is 3-11-Pik, followed by Bolon Tz'akab (or perhaps Tz'ak-bu, depending on language). The third consists of ma-cha-la, and my best guess at this is that it is a participle machal, based on the root /mach/ 'take with the hands'. The form /machal/ 'ser asido, agarrado o tenido' appears in the Motul. It seems to be a subject-deleted focus construction, as in '(it is the) 3-11 pik and Bolon Tz'akbu ( or Tz'akab) (which are) taken'.


Now we have a nominal phrase, part of which is readable. 'U k'in Kokom kelem, ba wi? 'u-k'aba' would be 'the priest of the Kokom lineage, handsome youth, first ?? is his name...'. There's a part that's inscrutable, a part that's missing, then sajal 'secondary lord', and then bahiiy u ch'ab y-ak'il ti'al  'his body is the creation/darkness of, belonging to...' and what follows is his presumed father. Nothing else here sheds further light on the function of the 3-11-Pik, but here are two relevant items:


(1) Carl Callaway (personal communication), citing Grube, notes that 3 Ajaw 8 Zak--a Triple station in my schema--is actually recorded via a 'tun in k'atun X Ajaw' formula, and is the earliest date found at Xkalumk'in--on Lintel 2. It doesn't tell us anything about the formula either, but it's important to have it.


(2) The presence of Bolon Tz'akab as the other of the pair of things 'taken' (machal) made me wonder whether there was an 819-Day Count station in proximity, so I did the math and found the relevant 819 stations. But what constitutes proximity? It's terribly important to distinguish signal from background, and to try to know how the Maya would have made that distinction. The nearest 819-DC station is 24 days ahead of the Triple at 1 K'an 12 Kej (a south station). That seems more than coincidental, considering that Bolon Tz'akab--which has a number of 819-DC associations**-- is paired with the 3-11-Pik as one of two things 'taken'.


[** Dave Kelley demonstrated the similarities in structure between the Dresden New Year pages and the 819-DC; they both involve God K as a primary protagonist; God K is K'awiil in the Classic and Bolon Tz'akab in the Postclassic, etc.]


I am just about ready to suggest that Bolon Tz'akab (K'uh) was the name for the 819-DC at Xkalumk'in if not elsewhere. Mathematically, the tz'akab ' 'count' plus instrumental suffix' portion of 819 would be 91, or 7 x 13. Tz'ak also has the meanings 'count', 'follow in succession', 'generations in kinship', 'addition', 'step(s) of a stair', 'joints on a piece of cane'.


I take from this yet more evidence that an almanac of 3-11-Pik Triple stations was kept and that they (as expected) paid attention to its relationship to the 819-DC cycle and by extension, Jupiter and Saturn and the 364-day computing year (4 x 819 = 9 x 364).


One more thing about this text at Xkalumk'in: the two large basal register bird heads which flank the text are labeled both PIK (hand on jaw) and CH'E'EN (the bundle of reeds or whatever they are emerging from the nose--it's the HV of CH'E'EN). So this location is a pik-cave.


Early Classic Copan


Three monuments and a fourth fragment (jamb fragment CPN 519) have the 3-11 Pik formula. Of these, Altar M’ is not readable in the Morley photo. Stela 49 has recently been assigned to the Copan king Ruler 6 (Stuart, 2008); only the Jol (‘skull’) portion of his name can be read. While there are no dates on Stela 49, the date is firmly associated with his successor Bahlam Nehn (Ruler 7). This presents the possibility that the Triple Station motivated the reference to the 3-11Pik formula on Stela 49.


The Papagayo Step is assigned to the 4th successor of Yax K'uk' Mo; his name is K’al Tuun Hix. He was responsible for the construction of the Papagayo structure commemorating the Bak’tun ending 8 Ajaw 13 Kej (Stuart 2008). His name appears in the vertical text which is separate from the one containing the formula. This monument was intentionally burned and buried in ancient times, so there's not much left. The empty space suggests that the formula is at the top of the second column in a pair of columns, so what follows it is lost.


That said, both these monuments have some interesting data: the name Bolon Tz'ak K'uh, and a reference to 'earth-caves' or perhaps parcels of land (kab-ch'e'en, in one case written as 2-u-kab-ch'e'en). The speculative range (cf. Stuart above) for the reign of K’al Tuun Hix is -;  he is sandwiched between others (also with no firm dates) such that his reign could have been about ten years long. Given that he built the Papagayo following the great Bak’tun ending, one might assume his association with the Triple Station We lack sufficient data to understand how he manipulated it. Perhaps a recognition of the importance of this Triple station will help fix the span of his rulership.


The jamb fragment has only recently come to my attention. It is part of a much longer text, most of which is lost. It reads: 3-11 Pik la-ta, TZ’AK-ma, u-pa-ta-bu. The Tzak, as Marc Zender has pointed out, is composed of the pair ‘dark (ak’ab) and shiny (leem)’. A transliteration would be: 3-11 pik lat, tz’ahk-oom, ‘u –pat-bu  “3 times 11 piks accrued, will have been counted, (then) he fashions it…”. There’s no information with which to correlate activity surrounding a Triple station, but there is a strong suggestion of a count from a chronological datum. It appears to ba a late monument as Yax Pasaj is mentioned in one of the remaining frragments.



The 3-11 Pik Formula as a Precession Almanac


3 x 8660 days(25,980 days) is 71.18 years in a 365-day calendar or 71.13 years employing a tropical year of 365.2422.


A one-day correspondence in the earth’s precession cycle is 70.59 years by modern calculations, using a full cycle of 25,765 years (one of the modern estimates). The cycle of 25,765 years divided by 365 yields 70.589041 years.  A calculation employing the mean tropical year of 365.2422 days yields a one-day correspondence of 70.54 years. Employing this 3-11 Pik formula, this amounts to an error of .59 years, or 219 days, in the naked-eye observation of precession over 70-something years whether dividing by 365 or by 365.2422.


Alternatively, a cycle of 25,920 years has been proposed by astronomers.

66 bak'tuns equals 26,021.09 tropical years (256 TYs more than 25765 and 101 TYs less than 25920). Since we know they had an accurate understanding of the length of the tropical year (via the codical formula 365.2422 x 1507 = 365 x 1508) it seems highly likely that they'd have brought this knowledge into play in making precession calculations.


The full precession cycle approximates 66 bak'tuns: 144,000 x 66 = 9,504,000 days or 26,021.089 years using a tropical year of 365.2422 days. Half of 66 bak'tuns is 3 times 11 bak'tuns; this is exactly what the 3-11-bak’tun formula encodes. However, the smaller count would only represent one day’s movement of background stars in the ecliptic with respect to a point such as an equinox or solstice. If one takes a full circle of such intervals of 71.13 years into account, then the sky must have been divided into 365 “degrees”.


Furthermore, the relationship between the two counts which create the same CR sequence is not a direct correspondence of 365:1. If we consider 26,021.089 to be a best approximation of the 66-bak’tun round, dividing this by 365 we get 71.29 years as one degree of precession. We have seen that the result of dividing  (3 x 8660) by 365 is 71.17 and dividing it by 365.2422 we get 71.13.


Dividing 26021.089 by (2 x 182.90993) gives us the expected result of 71.13. Or, going about it in reverse, multiplying (3 x 8660) by half of 365—182.5—yields 4741350, which falls short of (3 x 11 x 144,000) by 10650 days, and the result would not reproduce the same sequence of CRs in both counts. So perhaps the Maya had an ideal system and a real one which more closely the actual precession round, whose value is still in dispute with a difference of 155 years.


Returning to the Long Count as we know it, and the 3-11-bak'tun formula as prescribed on the Burial 116 bone, we really should compare the product of 66 bak'tuns to that of 65 bak'tuns, which appears to motivate the thirteen-bak’tun “era cycle”.


66 x 144,000 is 9,504,000; dividing this by 365.2422, we get 26,021.082 years as an approximation of the full precession cycle. Rounding to 26,021 we have a difference of 256 years between this and 25,765 and a difference of 101 years between this and 25,920.


66 Bak’tuns vs. 65 Bak’tuns


If we use 65 bak'tuns instead (setting the 3-11-bak'tun formula aside), we have 65 x 144,000 which is 9,360,000. Dividing this by 365.2422, we get 25,626.83 years. Rounding to 25,627 and subtracting this from 25,765, we get a difference of 138 years. The difference between this and 25,920 is 293 years.  is So 66 bak'tuns is a closer approximation employing this calculation. To summarize:


65 bak'tuns equals 25,626.83 tropical years (138 TYs less than 25765 and 293 TYs less than 25920).

66 bak'tuns equals 26,021.09 tropical years (256 TYs more than 25765 and 101 TYs less than 25920).


66 bak'tuns = 144,000 x 66 = 9,504,000 days


9504000 divided by 365.2422 =26,021.089 tropical years


65 bak'tuns = 144,000 x 65 = 9,360,000 days


9360000 divided by 365.2422  = 25,626.83 tropical years


Here are other relationships between 65 and 66 bak'tuns, 8660 days and their factors and multiples:


In finding the triple stations nearest the completion of a 13-bak'tun cycle and a 65 baktun cycle:


8660 x 3 = 25,980 days or a "Triple" station interval counted forward from 4 Ajaw 8 Kumk'u.


144,000 x 13 = 1,872,000 days

1872000 divided by 25980 = 72.055427 OR... 72 with a remainder of 1440 days or 4 tuns!

Hence there will be a Triple station coming up on

This seems quite intentional. And look what we get when we take this out to 65 bak'tuns:


144,000 x 65 = 9,360,000


9360000 divided by 25980 = 360 .27713 OR... 360 with a remainder of 7200 days or 1 k'atun.

So there will be a Triple out there just one k'atun shy of 65 bak'tuns out from 4 Ajaw 8 Kumk’u. They had two ways of recording this number, comparable to the Long and Short Counts, if one considers, for example, the huge DNs at Quirigua vs. those at Coba.