The 3-11-Pik Formula
An 8660-day interval
(220.127.116.11 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 (18.104.22.168.0) 6 Ajaw 8 Mak; the second is (22.214.171.124.0)
8 Ajaw 8 Mol, and the third is (126.96.36.199.0)
10 Ajaw 8 Zip. Looper determined
that these CRs are separated by an interval of
only 8660 days, and the first is counted from
(188.8.131.52.0) 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 184.108.40.206.0 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
‘3-11-Pik Formula’, with the greater cycle counted
by eleven-bak’tun leaps
from the Creation date of 220.127.116.11.0 4 Ajaw 8 Kumk'u.
18.104.22.168.0 4 Ajaw
8 Kumk’u 22.214.171.124.0 4 Ajaw 8 Kumk’u
+ 126.96.36.199 +188.8.131.52 0
184.108.40.206.0 6 Ajaw
8 Mak 1.4 .0.0.0.0 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.
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
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
count. Other intermediate counts, as well
as counts based on different multiples
of bak’tuns, were explored--and these are
but nothing productive resulted.
3-11 Pik Stations: A
bone data suggests the possibility that the
Maya noted intervals of 1 and 3 times 8660
days counted from 220.127.116.11.0 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
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 (10.2.17.0.0) of the contemporary K'atun 1 Ajaw. No further
denouement has been offered until now.
Long Count Positions of Relevant
Single and Triple
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 18.104.22.168 in Maya notation.
('Single' stations—cited as needed for
specific monuments--separate intervals of
8660 days, or 22.214.171.124. in Maya notation)
126.96.36.199.0 12 Ajaw 8 Mak
0. 8. 6.0
5 Ajaw 13 Pax
9. 4. 0.
9.0 11 Ajaw 13 Pohp
188.8.131.52.0 4 Ajaw 18 Zotz' (52
out from the base of 184.108.40.206.0)
220.127.116.11.0 10 Ajaw 3 Mol
18.104.22.168.0 3 Ajaw 8 Zak
3.0 9 Ajaw 13
6.0 2 Ajaw
10.5.13.9.0 8 Ajaw 18 Wo
1 Ajaw 3 Xul
7 Ajaw 8 Ch'en
Ajaw 13 Kej (60 out from the base of 22.214.171.124.0)
Chronology of K’ahk’-u-Pakal and The Caracol Stela at Chichen Itza:
Here are the relevant 1-11-Pik (Single) and 3-11 Pik (Triple) stations which
overlap with Kahk'upakal's life:
Presumed birth of K'ahk'upakal takes place in this interval
(Single) 10. 0.17. 5. 0
10. 2. 0. 1. 9
6 Muluk 12 Mak
(first mention of K'-u-P)
Triple Station 10.
2. 1. 6. 0 2 Ajaw 18 K'ayab
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)
close of K'atun 1
Ajaw 10. 3. 0. 0.
0 1 Ajaw 3
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 10.3.0.0.0). This "binding" is part of a verbal
couplet, the other half of which is the Creation event or the changing of
the hearth (at the edge of the sky, etc.) on 126.96.36.199.0 4 Ajaw 8 Kumk'u. So
the narrative timeline is an important clue: it appears to have been
foreordained when the 13-Bak'tun cycle was set in place that
have this "binding" on 10.2.17.0.0.
We have a Triple station on 10.2.1.6.0,
and within its first Single unit we have two events: (1) the dedication of
the Caracol stela with the "binding" and (2) the close of K'atun 1 Ajaw on
10.3.0.0.0. Since they both fall within the first unit of a bundle of three,
the "binding" concept is even more reasonable. There are in fact three
things bound if we include the 3-11-Pik station/interval itself.
Timeline for Aj Wosal
on Naranjo Altar 1:
9.0 1 Ajaw 13 Pohp Triple 3-11-Pik station (Aj Wosal not yet born)
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
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
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
Ajaw 3 Mol
Triple 3-11-Pik station
over a lot of time here)
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
A Sketch of Kahk’ Tiliw
Chan K’awiil’s Modus Operandi on Quirigua Stela
This is a complex
monument. Its iconographic panel of
Chan K'awiil in all his glory
occupies the north side; the west
side opens with an Initial
Series featuring his accession (at B6) on
188.8.131.52.17 12 Kaban 5 K'ayab. His names and titles
continue through B10.
The 3-11 Pik title does not appear on this side of the monument.
That Triple Station
happens just under 4 tuns later on 184.108.40.206.0 3 Ajaw 8 Zak--the only tun
ending among the 3-11Triples in Bak'tun 9. It seems highly relevant that he
accedes just before this important station. I consider the relationship between this station and his
accession to be the rationale for all the calendric gymnastics on this
monument, in both current and deep time. It's the most calendrically complex
stela at Quirigua.
On other monuments he carries out Period Ending dedications for 220.127.116.11.0
and 18.104.22.168.0; he mentions some of these in passing on Stela F West after
jumping ahead to the main event on 22.214.171.124.6 6 Kimi 4 Tzek. This is the
capture of "Eighteen Rabbit"--Waxlajun u-Baah K'awiil of Copan, his former
overlord, recorded in the continuing text of Stela F, west. The rest of this
side has the CRs of 126.96.36.199.0 and 188.8.131.52.0, and includes 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 184.108.40.206.0,
as the half period of 220.127.116.11.0 13 Ajaw 18 Kumk'u. The lajuntun is featured
on the east side. He is definitely out to cover all the local bases, tying
them to his accession, before moving on.
The dedicatory PE of the monument is in the Initial Series of the east side.
It's a full IS with Lunar Series leading to his incense-scattering for the
"One Ajaw" monument itself--whose name includes what looks like Pik-Witz
(-yi) Tun.Then follows 'u tz'ap-aw, 'he plants it' and the thing being 'planted' is called
This is a dramatic high point.
The 'planting' of his "royal pik mountain" takes place in the second Single unit following the Triple of 18.104.22.168.0.
Here is an
outline of the chronology:
22.214.171.124.17 12 Kaban 5 K’ayab accession
3 Ajaw 8 Zak Triple station
126.96.36.199.0 PE on other monument
PE on other monument
188.8.131.52.6 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
half period of 184.108.40.206.0 13 Ajaw 18
220.127.116.11.0 1 Ajaw 3 Zip IS and
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
counting back into the primordial ooze follow…)
His "royal pik mountain" is planted on this spot. Now his name and titles
appear, closing 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.
more thoughts about Naranjo Altar 1 and Quirigua Stela F
monuments share some important features, and may tell us how the 3-11-Pik formula operates when a king lays claim to it.
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.
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.
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
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
18.104.22.168.0--33 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 22.214.171.124.0, 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 126.96.36.199.0, the first Single station after the Triple of 188.8.131.52.0. 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.
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
184.108.40.206.0, within which we may assume he was born.
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 220.127.116.11.0 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
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 18.104.22.168.4. 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).
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.
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 22.214.171.124.0 is firmly associated with his successor Bahlam Nehn (Ruler 7). This presents the possibility that the Triple Station 126.96.36.199.0 motivated the reference to the
3-11Pik formula on Stela 49.
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 188.8.131.52.0 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.
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 184.108.40.206.0 -
220.127.116.11.0; 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 18.104.22.168.0. 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.
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.
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
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:
equals 25,626.83 tropical years (138 TYs less than 25765 and 293 TYs less than 25920).
equals 26,021.09 tropical years (256 TYs more than 25765
and 101 TYs less than 25920).
= 144,000 x 66 = 9,504,000 days
divided by 365.2422 =26,021.089 tropical years
= 144,000 x 65 = 9,360,000 days
divided by 365.2422 = 25,626.83 tropical years
other relationships between 65 and 66
bak'tuns, 8660 days and their factors and multiples:
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
13 = 1,872,000 days
divided by 25980 = 72.055427 OR... 72 with a remainder of 1440 days or 4 tuns!
will be a Triple station coming up on 22.214.171.124.0.
seems quite intentional. And look what we get when we take this out to 65 bak'tuns:
65 = 9,360,000
divided by 25980 = 360 .27713 OR... 360 with a remainder of 7200 days or 1
So there will be a Triple out there just one k'atun shy of
65 bak'tuns out from 126.96.36.199.0 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.