Blinking snake

Somewhere in Time:
New Mathematical Methods for the 819 Day Count

Constellation band



We’re just going to some high place so we can see clearly, and at that high place you may be able to see the top, you may be able to see other new particles. But really, the idea is you’re trying to get up somewhere where you think you can see more clearly than you are right now. So I’d love to discover the top quark, but that’s not why I’m doing this. I just want to get to that place, and look around.
—Melissa Franklin1

Abstract

The 819-day count of the Mayan calendar comprises the three factors 7, 9 and 13, taken from smaller recurring cycles; the zero day (or starting point) of each 819-day sequence, known as an 819-day station, moves through the tzolk’in and haab in a pattern that is mathematically determined, with each station changing color and direction. Since the pattern’s behaviour may be described using just a few rigorous mathematical rules, the 819-day count may be viewed as a “finite automaton,” a term taken from computer science. Because tzolk’in and haab positions combine to form a mathematical coordinate in the Calendar Round, it is possible to use the Calendar Round position and the 819-day position to retrieve Long Count dates. This is due to the fact that the Calendar Round and 819-day count form a 1,195,740-day cycle, or 3,276 haabs. In this paper, I show computational methods for determining Long Counts from Calendar Round plus 819-day counts and methods for determining 819-day stations from Long Count dates, without resorting to full conversion to base 10 as do other methods. I describe new methods of computation based on modular mathematics, an area with useful applications to Mayan calendrics, using the Python programming language, which has a many features making research into Mayan mathematics particularly convenient. In the conclusion, I suggest some directions for future research.

1.  Introduction

Beauty comes always from the singularityof things.
—David Pye2

Recently, I tried to win an argument with my wife, regarding the date of the upcoming election day, by saying, “Now, who knows more about calendrical matters, you or me?” She responded without hesitation, “I do, of course. After all, I’m married to a calendar expert, and you certainly can’t make that claim, can you?” I retreated.

The 819-day count of the Mayan calendar is a ritual cycle that repeats endlessly; every 819 days, the cycle reaches what is known in the literature as a station, or, sometimes, resting place. The station can be defined as the moment when three other cycles are at specific points; these three smaller cycles are the thirteen numbered days, the Nine Lords of the Underworld and the Seven Lords of the Earth. The smallest possible commensurating cycle of these three factors (7, 9 and 13) is 819, because the three factors have no common divisor other than 1. As represented on Classic monuments, the count is represented as a distance number from the last previous station, the glyphs indicating the coordinates in the Calendar Round of the station so reached, and (often) a color/direction pair. A measure of the importance of the 819-day count can be realized by observing the amount of visual real estated occupied by the glyph set on Classic monuments. One can assume that the energy required to lay out and carve stone monuments would not be wasted on trivial ritual markers, but only upon matters of secular or religious significance. At the same time, it is notable that there are only 15 monuments known with 819-Day inscriptions upon them, indicating that many factors, including political and power dynamics, contributed to the choice of whether or not to commit the 819-Day phrase to stone.

1.1.  Factors

The three factors represent cycles of differing ritual importance for Mayans, judging from their relative presence on Classic monuments. Thirteen is, of course, the trecena (in the same sense that score stands for a unit of twenty, trecena means a unit of thirteen). The trecena and the Mayan phrase Oxlahun ti ku (“thirteen holies”) undoubtedly stand for the gods of the first thirteen numbers, or the Lords of the Day (Thompson, 1971). Since the trecena is present on virtually every single Classic monument, we can assume that it is of major importance.

Nine represents the nine-day cycle of the Lords of the Night, also known by the Mayan phrase Bolon ti ku (“nine holies”). In Mayanist circles, it is known as “Glyph G of the Lunar Series” (Thompson, 1929), even though the cycle has nothing to do with the Lunar glyphs. Most monuments contain glyphs referring to this cycle; for the most part, the names of these nine Lords of the Night are unknown (but see Frumker, 1993). Relative importance can be assumed to be generally equivalent to that of the trecena.

Seven, as Yasugi and Saito (1991) have shown, appears to indicate a cycle of seven “Lords of the Earth,” indicated in the inscriptions by “Glyph Y” of the Lunar (or Supplementary) Series. Although there is no evidence in the Books of Chilam Balam for this, so far as I know, a Mayan phrase similar to the previous two would be Wuk ti ku, or “seven holies.” Yasugi and Saito list only 22 monuments displaying glyph Y, not too many more than show the 819-day count, and thus the importance of this cycle may be judged to be markedly less than the other two.

It is widely known among Mayanists that Glyph G may be used to help reconstruct Long Count dates from Calendar Round dates; given a Calendar Round and a Lord of the Night glyph, Long Counts may be located accurately within Classic Mayan time. Nine is not a factor of the Calendar Round, so we can determine the minimum cycle of the G glyphs and the Calendar Round by simply multiplying 9 · 18,980, which is 170,820 days, or 468 haabs; a bak’tun is only 144,000 days. Those who have used some of the widely available computer programs that help determine imprecise Mayan dates will remember that all occurrences of any given Calendar Round within a bak’tun have differing glyph G positions, and that each additional Calendar Round gives a reduction by one in the glyph G position, due to the fact that division of 18,980 by 9 yields a remainder of 8. Because most Mayan dates occupy the 9th bak’tun, with very few falling outside that range, it is ordinarily not difficult to convert vague dates into accurate ones.

Yasugi and Saito’s research into the Lords of the Earth, or Glyph Y, cycle shows that Y glyphs could also be used in the same way, although not over so long a time scale. Seven is also not a factor of the Calendar Round, so the Calendar Round and the Y glyphs together make a 132,860-day cycle, as 7 · 18,980 = 132,860, or 364 haabs. Because division of 18,980 by 7 gives 3 as a remainder, the addition of a Calendar Round to any date makes the glyph Y position increase by three (or decrease by four, depending upon mathematical convenience). It is possible to have two dates in a bak’tun with co-incident Calendar Rounds and Y glyph positions. While the number of Long Count dates having associated G glyphs far exceeds those with Y glyphs, the possibilty of increasing temporal specificity using the seven-day Y glyph cycle is nonetheless a distinct one.

Since the Calendar Round and the 819-day count share thirteen as a factor, this eliminates garnering additional information from the trecena over and above what one already has from the Calendar Round. In fact, if one has an 819-day position, the positions in the subsidiary seven, nine and thirteen-day cycles may easily be extracted from it; no additional date information may be found from the three sub-cycles and the 819-day count. Therefore, the seven, nine and thirteen-day cycles are (mathematically) interesting insofar as they permit reconstruction of either Long Count dates or 819-day positions. Inscriptions bearing 819-day counts usually offer an additional coordinate, however: that of color/direction, which permits resolution to 3,276 days, as there are four colors and four directions. The color and direction co-vary (are locked together), and, since four is not a factor of 819, the pattern results in a 3,276-day cycle (Kelley, 1976).

Even though the 819-day count combines the three ritual cycles of seven, nine and thirteen days into one larger cycle, day 0 of the 819-day count does not begin on the zero (or first) points of the three sub-cycles. While the first point in the glyph G cycle is occupied by G1, day 0 of the 819-day count is set to G6; it is also position 1 in the trecena, when you might think that it would be set to 10 (to correspond to the G6 position). Day 0 is also set to seven in the glyph Y cycle (Yasugi and Saito explain that this position in the Y cycle could be viewed as either 0 or 7, i.e., the cycle could run either 0 to 6 or 1 to 7. Since they have not found a Y glyph with a numerical coeffecient for this position, they have arbitrarily chosen to refer to it as 7, mimicking the pattern of the trecena). An 819-day station of 0 therefore corresponds mathematically to (Y7, G6, T1), or to use modular notation (discussed later), to (0,6,1).

1.2.  The 819-Day Count in the Inscriptions

The inscriptional phrase, found in varying locations in date clauses on Classic monuments, opens with a distance number counting back to a station, followed by some form of the T588 819-day verb verb. Thompson (1971), who first noted its existence in the inscriptions (1943), has described the 819-day count as a “Soulless Mechanism,” while Barbara MacLeod (1989) demurs, calling it “A Soulful Mechanism.” Since it is a mechanism governed by specific, mathematical rules, I prefer to refer to the cycle as a “finite automaton.” This term, taken from computer science, simply indicates that the mechanism’s state, past, present and future, can always be calculated, given any other state; In fact, the whole automaton may be described solely in terms of such rules, as I intend to show in what follows.

The elementary mathematics of the 819-day count have been known since Thompson’s (1943) work, with Kelley’s (1976) work adding much to it, but only recently have there been any advancements in understanding the meaning of the cycle. The only indication of the significance of the count in Mayan spiritual life has been the amount of graphical real estate lavished upon it in monuments, and this has not been an informative indication. Approaching our own, Gregorian, calendar from the perspective of future archaeology, we might view our own ignorance of the importance of the 819-day count as similar to knowing that Easter is important in the Gregorian calendar, but being unable to determine its significance. In recent Western times, Christmas has occupied more “visual real estate,” so to speak, than has Easter; future epigraphers might be justified in missing the facts that the date of Easter determines the entire ritual calendar of the Catholic church for the coming year, that the accurate calculation of Easter is therefore tremendously important, and that Christmas is hardly a blip on the ecclesiastical radar screen. Our knowledge of the meaning of the 819-day count might conceivably be on a par with knowing only that Easter had something to do with something or someone “getting up.”

The 819-day count has not been found in the codices, although the verb is there; because T520 T520 (ok) is infixed into T588, MacLeod (1989) advanced a tentative translation of the verb as e¢-ok, “plant the feet.” Note also T1022 T1022, which is quite similar to T588, lacking only the “sprout” affix. She notes, however, that T588 is frequently suffixed by T178 T178 (la) and T181 T181 (halah); these suffixes have led to Linda Schele’s reading of this verb as walah, “to place” or “to seat” (Schele and Grube, 1997). Schele’s paraphrase of the glyphs is “on such-and-such a day, or so many days since that day, God K seated (or stood up) something.” Although this translation leaves much to be desired, my aim in this paper is not to offer any additional illumination on the meaning of the phrase but to attempt to reveal the full complexity of the cycle’s mathematics.

The phrase will sometimes close with a “One Rodent-Bone” glyph, or T758/T757:T110: Hun/one ch’ok/sprout MacLeod’s tentative translation of the “one-rodent-bone” glyph was hun ch’ok, “one offspring” or “one sprout.” It is one of the alternate forms of Glyph B, where T758 Ch’ok/sprout is ch’o and T110 ko is ko.

The tzolk’in and haab glyphs for the 819-day station come either after the distance number (cf. Yaxchilan L30) or, as in some examples from Palenque, in the position otherwise occupied by T757/T758. The most complete and complex examples do seem to come from Yaxchilan; a specific example is Lintel 30, E3-F6. Here, in addition to the distance number (E3-E4), T588 (E5), the direction (F5), the color (E6) and the “Rodent-Bone” (F7) glyphs, we have two additional glyphs: the “beetle,” or Glyph Y (F6) Glyph Y T739, and a “Smoking Squirrel”/God K Smoking squirrel T1030 glyph (E7). Yasugi and Saito (1991) suggest that T1030 with the T122 prefix T122, represents the thirteen celestial deities (oxlahun ti ku), that T739 may represent the seven terrestrial deities (wuk ti ku?) and that another glyph, which they describe as “a head glyph of a deity or a geometric glyph with X sign infixed,” might possibly be read as bolon ti ku, the nine infernal deities. I have not, however, been able to determine exactly to which glyph this last reference may be.

In at least three examples, again all from Yaxchilan, there is an additional glyph with a numerical coeffecient of 6 Proposed G6, a form of T540/T541 (Yaxchilan L29, C4; Yax. 1, C8; Yax. 11, D15(?)), associated with the Calendar Round date; since G6 is the Lord of the Night for all 819 stations, Thompson (1943, 1971) speculated that this might be a form of the G6 glyph, which would be fortunate, as we have only one other example of G6 in the corpus. Yasugi and Saito (1991), however, cast doubt on Thompson’s choice for G6, and suggest a form of the Y glyph in its place.

A listing of inscriptions containing 819-day stations is given in Appendix I.

1.3.  Colors and Directions in the 819-Day Count

Berlin and Kelley (1961; Kelley, 1976) have shown that each 819 day station is referenced to (or under the control of) a different direction/color; since there are four directions and corresponding colors, the 819-Day Count is usually said to have 3,276 days, adding the ritually important number 4. Each trip through the 819-day cycle decrements color and direction by one, and the color and direction may be determined solely on the basis of the veintena day. The mathematics are detailed below, but the pattern for colors (at each successive station) is: red (chak), yellow (k’an), black (ek) and white (sak). The directions are locked to the colors, following the pattern: east (likin), south (nohol), west (chikin) and north (xaman).

1.4.  The Tzolk’in and the 819-Day Count

Mathematically, we usually consider the 819 day count, or cycle, as beginning on day -3 of linear time, 1 Kaban 5 Kumk’u. As Lounsbury (1978) points out, this is merely a convenience, not a necessity, as we do not know which possible starting point the Mayans may have used (I will have more to say on this topic later in the paper). Because 819 is evenly divisible by thirteen, the trecena day for each 819 day station is a constant and is always one. And since 819 is one less than an even multiple of 20, the veintena decrements by one on each increment of the cycle; so too do the color and direction, as noted above. Since there are twenty veintena days, that means that it takes twenty 819 day cycles, or 16,380 (Mayan 2.5.9.0) days, for the same veintena day to recur. The position in the tzolk’in (Δtz) shifts by 39 days each station, in the following pattern:

Table 1: 819-Day Tzolk’in Shifts
819-Day Cycle Days Elapsed Days Elapsed, Mayan Tzolk’in Position (Δtz) Tzolk’in Color Direction
0 0 [0, 0] 156 1 Kaban 1Kaban Chak (Red) Chak, red Likin (East) East, Likin
1 819 [2, 4, 19] 195 1 K’ib 1Kib Kan (Yellow) Kan, yellow Nohol (South) South, Nohol
2 1638 [4, 9, 18] 234 1 Men 1Men Ek (Black) Ek, black Chikin (West) West, Chikin
3 2457 [6, 14, 17] 13 1 ’Ix 1Ix Sak (White) Sak, white Xaman (North) Xaman, North
4 3276 [9, 1, 16] 52 1 Ben 1Ben Chak (Red) Chak, red Likin (East) East, Likin
5 4095 [11, 6, 15] 91 1 ’Eb 1Eb Kan (Yellow) Kan, yellow Nohol (South) South, Nohol
6 4914 [13, 11, 14] 130 1 Chuwen 1Chuwen Ek (Black) Ek, black Chikin (West) West, Chikin
7 5733 [15, 16, 13] 169 1 Ok 1Ok Sak (White) Sak, white Xaman (North) Xaman, North
8 6552 [18, 3, 12] 208 1 Muluk 1Muluk Chak (Red) Chak, red Likin (East) East, Likin
9 7371 [1, 0, 8, 11] 247 1 Lamat 1Lamat Kan (Yellow) Kan, yellow Nohol (South) South, Nohol
10 8190 [1, 2, 13, 10] 26 1 Manik’ 1Manik Ek (Black) Ek, black Chikin (West) West, Chikin
11 9009 [1, 5, 0, 9] 65 1 Kimi 1Kimi Sak (White) Sak, white Xaman (North) Xaman, North
12 9828 [1, 7, 5, 8] 104 1 Chik’chan 1Chikchan Chak (Red) Chak, red Likin (East) East, Likin
13 10647 [1, 9, 10, 7] 143 1 K’an 1Kan Kan (Yellow) Kan, yellow Nohol (South) South, Nohol
14 11466 [1, 11, 15, 6] 182 1 Ak’bal 1Akbal Ek (Black) Ek, black Chikin (West) West, Chikin
15 12285 [1, 14, 2, 5] 221 1 ’Ik’ 1Ix Sak (White) Sak, white Xaman (North) Xaman, North
16 13104 [1, 16, 7, 4] 0 1 ’Imix 1Imix Chak (Red) Chak, red Likin (East) East, Likin
17 13923 [1, 18, 12, 3] 39 1 ’Ahaw 1Ahaw Kan (Yellow) Kan, yellow Nohol (South) South, Nohol
18 14742 [2, 0, 17, 2] 78 1 Kawak 1Kawak Ek (Black) Ek, black Chikin (West) West, Chikin
19 15561 [2, 3, 4, 1] 117 1 ’Etz’nab 1Etznab Sak (White) Sak, white Xaman (North) Xaman, North
20 16380 [2, 5, 9, 0] 156 1 Kaban 1Kaban Chak (Red) Chak, red Likin (East) East, Likin

We can immediately see, then, that the 819-Day count must be at minimum 16,380 days long, not merely 3,276 days. From examples of contrived numbers, we have good evidence that the Mayans knew about this specific cycle. The distance number that links Pakal’s birth on 9.8.9.13.0 8 ’Ahaw 13 Pohp with the initial date on the Tablet of the Cross at Palenque is 9.8.16.9.0, which is 83 of these 16,380 day cycles (1,660 of the 819-day counts) (Lounsbury, 1978).

1.5.  The Haab and the 819-Day Count

The haab and the 819-day count also form a repeating cycle, due to the fact that the haab position (Δh) shifts by 89 days on each increment of the count. Since 819 and 365 do not share a common divisor other than one, it necessarily takes 365 trips through the count for the haab day 5 Kumk’u to recur. The extreme length of the cycle, 298,935 days, is simply the product of 365 · 819; every position in the haab is visited on some 819-day station. A table showing all 365 positions in the haab is provided as Appendix II.

The minimum possible period for the 819-Day Count must then be 298,935 days, or Mayan 2.1.10.6.15. However, one trip through this 365-haab cycle will still not return us to day 1 Kaban of the tzolk’in, since 298,935 is not evenly divisible by 16,380 (there is a remainder of 4095). This suggests that there is a larger, recurring cycle which may be determined as discussed next.

1.6.  The Haab, the Tzolk’in and the 819-Day Count

The tzolk’in and the 819-day count combine to form a cyle of 16,380 days, which is 20 · 819, or 63 · 260. The haab and the 819-day count form a larger cycle of 298,935 days, which is 819 · 365. To reconcile the two cycles requires a larger cycle, which may have as factors 260, 365 and 819. 260 · 365 · 819 is 77,723,100 days, but we can reduce this by noting that, as is well-known, 260 and 365 have a greatest common divisor of 5; these are factors of the Calendar round. The Calendar Round is only 18,980 days long instead of 94,900 due to the fact that we can divide either of these factors by 5 (our greatest common divisor) and multiply the result times the other factor to determine the maximal length. I.e., 260 / 5 = 52, 52 · 365 = 18,980 and 365 / 5 = 73, 73 · 260 = 18,980. We thus have every reason to believe that, rather than dealing directly with 260, 365 and 819, we can work with the 18,980 days of the Calendar Round (determining the greatest common divisor of three numbers is rather more work than we need to undertake), and it therefore seems likely that the Calendar Round and the 819-Day count would have common factors. Indeed, this turns out to be the case as, not surprisingly, 18,980 and 819 share a greatest common divisor of 13. 18980 / 13 is 1460, and 1460 · 819 is 1,195,740 days, or Mayan 8.6.1.9.0. This is the same as 73 of the 20 · 819-Day cycles; remember that for each 20 · 819-day cycle, the haab position shifts by 5 days. It therefore takes 73 of the 5-day shifts (73 · 16,380) for the same tzolk’in and haab combination to recur.

Finally, the greatest common divisor of 1460 and 819 is one, showing that the number we have arrived at is the smallest possible cycle commensurating the Calendar Round and the 819-day cycle. To test this hypothesis, we can add our Mayan number, 8.6.1.9.0, to any reasonable Long Count date and determine the position in the 819-day cycle (Δe) and the position in the Calendar Round (ΔCR):

  0.0.0.0.0 4 ’Ahaw 8 Kumk’u, ΔCR = 7283, Δe = 3
+ 8.6.1.9.0
  8.6.1.9.0 4 ’Ahaw 8 Kumk’u, ΔCR = 7283, Δe = 3

Similarly, the standard (convenient) starting point for the 819-day count, -3 (three days before 0) or 1 Kaban 5 Kumk’u, with the addition of 8.6.1.9.0, becomes 8.6.1.8.17 1 Kaban 5 Kumk’u, with G6 as Lord of the Night; it is itself an 819-day station.

1.7.  Utility of the Larger Cycle

We can suppose that using the two coordinates Δe and ΔCR, it would be possible to determine positions within such a larger cycle of 1460 · 819 = 1,195,740 days. Clearly, this would be advantageous, as it would mean that given an 819-day count position or station and a Calendar Round date, we would be able to determine a Mayan date within 3,276 haabs. Given one additional piece of information, say the bak’tun, the date could be pinpointed within a much larger timeframe. Not having a better name at hand, I will refer to the 3,276 haab cycle as E.

In the next section, I will cover conventional means for performing 819-Day calculations, using adaptations of standard techniques as explained in Lounsbury (1978) and again in Lounsbury (n.d.), and following that, the calculation of 819-Day stations using only Long Count coefficients without the necessity of converting to decimal numbers. Next, I will describe new methods using modular arithmetic as applied to the same calculations; finally the reconstruction of Long Counts from Calendar Round and 819-Day stations is described. Many of my calculations were done directly in Mayan arithmetic (or a close simulation), using a package that I wrote in the Python programming language, which is available for Microsoft, Macintosh and Unix Operating Systems. Python functions are given for many operations in this paper; all of them are contained in a downloadable package.Appendix V: Python Resources includes some notes for obtaining, installing and using the Python Language. There are also instructions for obtaining and installing the mayalib package, which is the code for all the functions described in this paper.

2.  Conventional Mathematics

It sounds like just normal stuff, and yet it’s something about the universe that’s incredibly hard to measure, hardly anyone does it, and it’s totally cool.
—Melissa Franklin3

Most of the methods in this section are based on or adapted from Floyd Lounsbury’s, as described in “Formulae for Maya Calendrical Computations” (n.d.) and in “Maya Numeration, Computation, and Calendrical Astronomy” (1978). It will be useful to review here the symbols used in the previous pages (and introduce some more).

Table 2: Symbols Defined
SymbolMeaning
GThe nine-day cycle of the Lords of the Night, 1-9
YThe seven-day cycle of the terrestrial gods, 1-7
TThe thirteen-day cycle of the celestial gods, the trecena 1-13
vThe twenty named days, the veintena ’Imix-’Ahaw
cOne of the four colors red, yellow, black and white
dOne of the four directions east, south, west, north
tzThe tzolk’in, 260 days, T · v
ΔtzDistance between two positions in the tzolk’in
hThe haab, 365 days
ΔhDistance between two positions in the haab
hdThe haab day, 0-19
hmThe named haab month, Pohp-Wayeb
CRThe Calendar Round, 18,980 days, 260 · 73 or 365 · 52
ΔCRDistance between two positions in the Calendar Round
eThe 819-day count, G · Y · T
ΔeDistance between two positions in the 819-day count
Ethe 1460 · 819-day count cycle, 1,195,740 days
ΔEDistance between two positions in cycle E
mThe decimal equivalent of a Long Count date
jThe Julian period day corresponding to m
m8An 819-day station
kA k’in Long Count coeffecient
wA winal Long Count coeffecient

2.1.  Finding the 819-Day Station

Ordinary 819 day calculations are not particularly complicated, once you have the “Mayan Day” (m), i.e., the decimal equivalent of a Long Count date. Methods for the conversion of Long Counts to decimal numbers are quite common and will not be covered here. As an example, let us take the Long Count 12.19.4.12.0 9 ’Ahaw 18 Sak, which can be converted to decimal 1866480. Once this value is known, simply add 3 to it (1866483) and modulo4 it by 819:

Formula Python
Δe = (m + 3) % 819 
def st8(m):
  st = (m + 3) % 819
  return st

for our example this is:

Δe = (1866480 + 3) % 819
Δe = 801

meaning that it is 801 days past the last 819 day station. Or, using Distance Number format, 1 K’in, 4 Winals, 2 Tuns (1.4.2). The actual day of an 819 day station, then, would be marked as 0 K’in, 0 Winal, 0 Tun, or Mayan 0.0.0, and the largest Distance Number you could possibly see would be 818, or Mayan 18.4.2 (2.4.18 in normal form).

To find the actual day of the 819 day station (m8), subtract the 819 day position (801 in our example above) from the date of the monument, m:

Formula Python
m8 = m - Δe
m8 = 1866480 - 801
m8 = 1865679
 
def fm8(m,de):
  return m - de

The trecena for 819 day stations is always one; since 819 is evenly divisible by 13, and since we’re using three days before “zero” (4 ’Ahaw 8 Kumk’u) as a convenient base day, then 4 - 3 is 1.

The veintena can be found directly, by taking the 819 day station (m8) modulo 20:

Formula   Python
v = m8 % 20
v = 1865679 % 20
v = 19 (Kawak)
 
def fv(m8):
  return m8 % 20

In the case of our base date, “three days before zero,” we can determine the trecena, veintena and haab with very little difficulty:

 4 ’Ahaw (day 0) 8 Kumk’u
                      - 3
1 Kaban (day 17) 5 Kumk’u

Determining the Lord of the Night for the base date is equally simple; for m 0, the Lord of the Night is G9, and three days before that would necessarily be G6. Thus, all 819 day stations are under the influence or protection of G6 (819 is evenly divisible by 9, of course). Using m = 1866480 from our example, the Lord of the Night (G) is most easily determined by taking m modulo 9 and substituting G9 for 0; Long Count positions are not required with this method.

Formula   Python
G = m % 9
G = 1866480 % 9
G = 6
 
def fG(m):
  return m % 9

Another method for finding the Lord of the Night that depends on the winal and k’in positions of the Long Count is as follows:

Formula   Python
G = ((w · 2) + k) % 9  
def flcG(w,k):
  return ((w * 2) + k) % 9

The next station, at m 816, would be:

1 Kib (day 16) 9 Sots G6

2.2.  Finding the Tzolk’in

Once m for the 819 day station (m8) is known, then we can find the tzolk’in position using this formula:

Formula   Python
Δtz = (m8 + 159) % tz  
def ftz(m8):
  return (m8 + 159) % 260

We add 159 because the tzolk’in does not start on day 0 (4 ’Ahaw 8 Kumk’u) of the Long Count, but on 1 ’Imix, and 4 ’Ahaw is day 159. So, for our example,

Δtz = (1865679 + 159) % 260
Δtz = 1865838 % 260
Δtz = 78

And then, with the tzolk’in position, we can easily find both the trecena and veintena:

Formula   Python
T = (Δtz + 1) % 13  
def fT(dtz):
  T = (dtz + 1) % 13
  if T == 0 :
    T = 13
  return T

(We add 1 because day 0 in the tzolk’in is 1 ’Imix; our modulo function in this case does not return 0, but for other positions in the tzolk’in, a return value of 0 should be replaced with 13.)

T = (78 + 1) % 13
T = 1

And 1 is exactly what we expect and require.

For the tzolk’in position, we may use the following formula.

Formula   Python
v = (Δtz + 1) % 20  
def fV(dtz):
  return (dtz +1) % 20

v = (78 + 1) % 20
v = 19
v = Kawak
 

(We add 1 here also, because ’Imix is day 1 and ’Ahaw is day 0.)

2.3.  Finding Colors and Directions

Once we know the position in the tzolk’in, we can derive the color and direction very easily:

Formula   Python
c = Δtz % 4
c = 78 % 4
c = 2 (Black)
 
def fCD(dtz):
  return dtz % 4

d = Δtz % 4
d = 78 % 4
d = 2 (Chikin [West])
 

The color and direction may be found by looking up the value found by applying our formula in the following table:

Table 3: 819-Day Color and Direction Indices
IndexColorDirectionVeintena Days
0 Chak (Red) Chak, red Chak, red Likin (East) East, Likin ’Imix
Imix
1
Chik’chan
Chikchan
5
Muluk
Muluk
9
Ben
Ben
13
Kaban
Kaban
17
1 Sak (White) Sak, white Sak, white Xaman (North) Xaman, North Ik
Ik
2
Kimi
Kimi
6
Ok
Ok
10
’Ix
Ix
14
’Etz’nab
Etznab
18
2 Ek (Black) Ek, black Ek, black Chikin (West) West, Chikin Ak’bal
Akbal
3
Manik’
Manik
7
Chuwen
Chuwen
11
Men
Men
15
Kawak
Kawak
19
3 Kan (Yellow) Kan, yellow Kan, yellow Nohol (South) South, Nohol K’an
Kan
4
Lamat
Lamat
8
’Eb
Eb
12
K’ib
Kib
16
’Ahaw
Ahaw
0

2.4.  Finding the Haab

Again, once we know the Mayan day number of the 819 day station (m8), we can figure out the haab position using:

Formula   Python
Δh = (m8 + 348 ) % haab  
def fH(m8):
  return (m8 + 348) % 365

We add 348 to m8 because the haab does not start on day 0 (4 ’Ahaw 8 Kumk’u) of the Long Count, but at 0 Pohp; 8 Kumk’u is day 348. For our example, the position in the haab would be:

Δh = (m8 + 348) % 365
Δh = 1866027 % 365
Δh = 147

Once this position in the haab is known, it’s almost trivial to find the haab month (hm) and day (hd):

Formula   Python
hm = (int) Δh / 20
hm = 147 / 20
hm = 7 Mol (Pohp is month 0)
 
def fHMD(dh):
  hm = dh / 20
  hd = dh % 20
  return (hm, hd)

hd = Δh % 20
hd = 147 % 20
hd = 7
 

Therefore, our haab is 7 Mol, giving a complete Calendar Round date of 1 Kawak 7 Mol. Note that the Python function defined above returns what is called a tuple (a pair or more of values). When using such a function in a program, an easy way to do so is:

hm, hd = fHMD(dh)

Which automatically “unpacks” the tuple, putting the answers into the correct variables (hm and hd).

2.5.  Finding the Position in the Calendar Round

The Calendar Round (CR) starts on a day 1 Kaban 0 Pohp and runs for 18,980 days to end on 13 ’Ahaw 4 Kumk’u. Day 0 (4 ’Ahaw 8 Kumk’u) of the Long Count, in this system, has a CR position of 7283. Lounsbury (n.d.) gives several formulae for determining the CR position, but the most direct way to calculate it is by taking m for the 819 day cycle modulo 18,980 (the length of the Calendar Round):

Formula   Python
ΔCR = m8 % CR
ΔCR = 1865679 % 18980
ΔCR = 5639
 
def fCR(m8):
  return m8 % 18980

What we’re calculating here is not the absolute position but the distance from day 0. The absolute CR can be found by simply adding 7283 (remember from the above that day 0 of the Long Count, 4 ’Ahaw 8 Kumk’u, has a Calendar Round position of 7283):

Formula   Python
ΔCR = 5639 + 7283
ΔCR = 12922
 
def fCRa(m8):
  return (7283 + m8) % 18980

For a fuller explanation of why this works, and to see Lounsbury’s more detailed formulae, refer to the Calendar Round Page. Once you have found the relative position in the Calendar Round, you can easily find the tzolk’in and haab positions:

Formula   Python
Δtz = (ΔCR + 159) % tzolk’in
Δtz = (5639 + 159) % 260
Δtz = 5798 % 260
Δtz = 78
 
def fCRtzh(dCR):
  tz = (dCR + 159) % 260
  h = (dCR + 348) % 365
  return (tz, h)

Δh = (ΔCR + 348) % haab
Δh = (5639 + 348) % 365
Δh = 5987 % 365
Δh = 147
 

The Python function again returns a tuple, the pair being (Δtz, Δh). These positions can, of course, be used above for finding the haab day and month, and for finding the trecena and veintena, as described above.

2.6.  Finding the Long Count

Using m8 found above, we can simply apply the standard techniques explained on the Long Count Page, that is, performing a sequence of modulo operations followed by divisions on the number of days to be converted until the number is zero.

For our example, we have m8 set to 1865679; in this case, our operations are:

Table 4: Conversion of m8 to Long Count
Mayan Position Formula Value Result
K’in LC[0] m8 % 20
m8 = m8 / 20
LC[0] = 1865679 % 20 = 19
m8 = 1865679 / 20 = 93283
19
Winal LC[1] m8 % 18
m8 = m8 / 18
LC[1] = 93283 % 18 = 7
m8 = 93283 / 18 = 5182
7
Tun LC[2] m8 % 20
m8 = m8 / 20
LC[2] = 5182 % 20 = 2
m8 = 5182 / 20 = 259
2
K’atun LC[3] m8 % 20
m8 = m8 / 20
LC[3] = 259 % 20 = 19
m8 = 259 / 20 = 12
19
Bak’tun LC[4] m8 % 20
m8 = m8 / 20
LC[4] = 12 % 20 = 12
m8 = 12 / 20 = 0
12
LC = 12.19.2.7.19

A Python function to perform the conversion is shown here:

m8 = 1865679
def fMLC(m8):
  lc=[]          # Make an empty list
  i = 0
  bs = 20        # Set the base of the current position
  while(m8 > 0): # While we’ve got work to do
    if i == 1:   # If the position is the winal...
      bs = 18    # set the base to 18;
    else:
      bs = 20    # otherwise, set it to 20
    t = m8 % bs  # Calculate the Long Count coeffecient for this place
    lc.append(t) # Append the coeffecient to the list
    m8 = m8 / bs # Integer truncation; get rid of coeffecient
    i = i + 1    # Increment the position (place) index
  lc.reverse()   # Put the list in the correct order, since we
                 # started figuring it out from the smallest first
  return lc

This function performs several tasks, and is valid for any number; that is, it will correctly convert any decimal number, no matter how large, into a Long Count of however many places are required. The comments (after the #) should give a good idea of what’s going on, and for the given example, the value returned is [12, 19, 2, 7, 19]; the [ and ] are Python notation for a list of values, which is a (very) convenient notation for Long Counts.

3.  819-Day Count Positions from Long Count Dates

We find no vestige of a beginning, no prospect of an end.
—James Hutton5

It is often useful to work with Long Counts directly without going through an intermediate conversion to a decimal value as we did above. It is not difficult, once a Long Count date is known, to calculate the positions in the haab and tzolk’in, and in fact the method for finding 819-day positions described here is modelled on Lounsbury’s method for calculating the trecena position from Long Counts (Lounsbury, 1978, and Lounsbury, n.d.). He observed that the trecena coeffecient moved, forward or backward, directly according to the days passed. Each day that passed incremented or decremented the trecena by one, each winal incremented or decremented by 20, and so on. However, these shifts can be reduced by taking them modulo 13; therefore, the real shift for the winal is either 7 or -6, depending on which you prefer to use (the distance between the negative shift and the positive one must sum to 13), being equivalent. For tuns, the shift is either 9 or -4; for k’atuns, 11 or -2. This can be further simplified so that a table of all shifts, both positive and negative, need not be kept for all positions in the Long Count (bak’tuns, piktuns, and so on), but may instead be rather quickly calculated. He observed that, except for the tun position, each shift is 7 times the shift of the position to its right. In the tun position, the multiplier is 5 instead of 7.

A Python function to print out as many places of these shifts as are required is given here:

def print_trcoefs(n):
  i = 0
  c = 7
  q = 1
  l = []
  while(i < n):
    l.append(q)
    if(math.fabs(q) < math.fabs(q - 13)):
      print “l[%d] = %d [[%d]]  preferred: %d” % (i, q, q - 13, q)
    else:
      print “l[%d] = %d [[%d]]  preferred: %d” % (i, q, q - 13, q - 13)
    i = i + 1
    if i == 2:
      c = 5
    else:
      c = 7
    q = c * l[i - 1]
    q = q % 13

And here is sample output from the procedure when n is 20 (l stands for Long Count position, where l[0] is k’ins, l[1] is winals, and so on):

 l[0] =  1 [[-12]]  preferred:  1
 l[1] =  7 [[ -6]]  preferred: -6
 l[2] =  9 [[ -4]]  preferred: -4
 l[3] = 11 [[ -2]]  preferred: -2
 l[4] = 12 [[ -1]]  preferred: -1
 l[5] =  6 [[ -7]]  preferred:  6
 l[6] =  3 [[-10]]  preferred:  3
 l[7] =  8 [[ -5]]  preferred: -5
 l[8] =  4 [[ -9]]  preferred:  4
 l[9] =  2 [[-11]]  preferred:  2
l[10] =  1 [[-12]]  preferred:  1
l[11] =  7 [[ -6]]  preferred: -6
l[12] = 10 [[ -3]]  preferred: -3
l[13] =  5 [[ -8]]  preferred:  5
l[14] =  9 [[ -4]]  preferred: -4
l[15] = 11 [[ -2]]  preferred: -2
l[16] = 12 [[ -1]]  preferred: -1
l[17] =  6 [[ -7]]  preferred:  6
l[18] =  3 [[-10]]  preferred:  3
l[19] =  8 [[ -5]]  preferred: -5

The l[n] notation is the same as shown above in Table 3: Conversion of m8 to Long Count. That is, l[0] indicates the trecena shift for the k’in position, l[1] indicates the trecena shift for the winal position, and so on. The preferred column simply lists the smaller of the two possible numbers; typically, the number used would be the smaller of the two just because it’s easier to perform arithmetic on paper with smaller multipliers.

We can make a similar set of observations regarding the 819-Day Count. Each k’in that passes shifts the position by one day. Each winal by 20 days, each tun by 360 days. Each k’atun shifts by 7200, but this modulo 819 is 648; each bak’tun shifts by 675 days. Here is a table showing the shifts, but it only goes from l[0] (k’in) to l[13] (certainly more than will ordinarily be needed):

Table 5: Shift by Long Count Place for 819-Day Count
Long Count PositionShift in 819-Day Position
for Each Unit
l[0]1
l[1]20
l[2]360
l[3]648
l[4]675
l[5]396
l[6]549
l[7]333
l[8]108
l[9]522
l[10]612
l[11]774
l[12]738
l[13]18

Only l[0] - l[13] are listed, because the pattern starts to repeat: l[14] has the same shift as l[2], l[15] the same as l[3], and so on. If we treat the table of shifts as an array beginning with position l[2], we see that we can find the shift for any given l[n] by finding the shift value at index i, where

i = (n - 2) % 12

unless n = 0, when i = 1, or n = 1, when i = 20. For each Long Count place, we simply multiply the coeffecient times the appropriate i, and modulo the answer 819. We add or subtract this to or from our conventional starting point, i.e., three days before zero, or -3 1 Kaban 5 Kumk’u.

As an example, let us take the Long Count 9.16.9.0.0. We can see that l[0] and l[1] are both 0, so they have no effect on the position in the 819-Day Count. Therefore Δe, our position in the 819-day count, remains at 3. For all the Long Count positions listed, we have:

l[0]Δe = 3 (0 · 1 = 0)
l[1]Δe = 3 (0 · 20 = 0)
l[2]Δe = 3 + 783 ((9 · 360) % 819 = 783)
l[3]Δe = 3 + 783 + 540 ((16 · 648) % 819 = 540)
l[4]Δe = 3 + 783 + 540 + 342 ((9 · 675) % 819 = 342)
Δe = 1668 % 819
Δe = 30

Therefore, for 9.16.9.0.0 5 ’Ahaw 8 Sip, our 819-Day Count position is 30, meaning the last station was on 9.16.8.16.10 1 Ok 18 Pohp. To find the next station, simply add Mayan 2.4.19 to this date, which is 9.16.11.3.9 1 Muluk 7 Xul. The method also works for negative Long Count dates, since we only have to keep track of the distance from -3 1 Kaban 5 Kumk’u; to go back one station, we move 2.4.19, which is equivalent to zero, and we can see that our Long Count becomes -0.0.2.5.2 1 ’Etz’nab 16 Mak.

Following is a Python function to calculate the previous 819-Day position from any size Long Count date. The input to the function, l, is a simple list containing Long Count coeffecients. Lists, in Python, are expressed as shown before: [ 9, 16, 9, 0, 0 ]. To make calculation inside the function easier, the list is reversed (rewritten so that it appears in the order [ 0, 0, 9, 16, 9 ]) so that larger components appear last instead of first (just the same way we converted m8 into a Long Count, above).

#   Remember that we make special cases of k’in and winal.
# Index         0    1    2    3    4    5    6    7    8    9   10  11
l819coefs = [ 360, 648, 675, 396, 549, 333, 108, 522, 612, 774, 738, 18 ]

def f819st(l, sign): # List of LC coeffecients, sign is -1 or 1
  l.reverse()  # Put the k’in first.
  j = 0
  e = 3        # 819 Day station for 0.0.0.0.0
  for i in l:
    if j == 0:
      c = 1
    elif j == 1:
      c = 20
    else:
      c = l819coefs[(j - 2) % 12]
    if sign > 0:
      e = (e + (i * c)) % 819
    else:
      e = (e - (i * c)) % 819
    j = j + 1
  l.reverse()  # Put back original order
  return e

Suppose, however, that all we have is the position in the 819-Day Count and the position in the Calendar Round, such as 5 ’Ahaw 8 Sip? Can we recover the full Long Count from these two data? Yes, we can, and that will be covered in the next section, after a slight detour.

4.  Modular Mathematics

There’s a saying attributed to Eichler that there are five fundamental operations of arithmetic: addition, subtraction, multiplication, division, and modular forms.
—Andrew Wiles6

4.1.  Introduction

Anyone who has dealt with the Mayan calendar at all recognizes that modular arithmetic plays an important part in calendrical calculations. Math involving the tzolk’in, for example, uses modulo operations frequently, and calculations with the Calendar Round also require such math. Conversions to and from decimal integers require at least some facility in modular arithmetic. What may not be as widely known, however, is that such arithmetic has its own set of rules and that modular methods of representation find widespread application in the design of modern, parallel, computers, because multiplication of modular numbers can be far quicker than conventional methods. Addition, subtraction and multiplication of modular numbers is very simple, while division, comparison and positive or negative determinations are very difficult. Conversion of an integer number into its modular representation is fairly simple, but the reverse is not (Knuth, 1998).

A modular number is, basically, a list of r remainders that result from the division of a number, n, by rmoduli. Thus, the number 153 can be represented as (10, 13), where r1 = 13 and r2 = 20. The range of a modular number is just the product of the moduli, as long as such moduli have no common factors, and possible numbers in that range are 0 <= n < m (in other words, n can be any number from 0 through (r1 · r2) - 1). In our example, the range is 260, since 13 · 20 = 260, and n can be 0 through 259. Since this is the tzolk’in, though, we have to make some small adjustments. Because the tzolk’in begins on day 1 ’Imix, we need to set position 0 equal to 1 ’Imix, and we do that by adding 1 to our position, which is 153 in our example. Therefore, to find the trecena, we take (153 + 1) % 13, and (as usual), replace 0 with 13 when our modulo function returns it. And we do the same with the veintena: (153 + 1) % 20. For the example, then, we represent our n as (11, 14), or 11 ’Ix. This should be quite familiar, and the tuple representation of modular numbers should also be familiar.

What, however, do we do when we wish to convert a modular representation such as 11 ’Ix, (11, 14), back into an integer—in this case, a position in the tzolk’in? The usual method is to apply this formula (Lounsbury, n.d.):

Δtz = 40[(tr2 - tr1) - (v2 - v1)] + (v2 - v1) % 260

But there is another method that we can use; according to Knuth (1998), it is possible to convert any modular representation back into an integer representation. He gives two general proofs, one intuitive and one better suited to computer science. He then demonstrates that while the second method will work, it demands far more computation than is practical. H.L. Garner in 1958 (Knuth, 1998) suggested an even better proof which is quite practical for computers, and, since Mayan arithmetic ordinarily deals with re-entrant cycles comprised of only two factors (20 · 13, 52 · 365, etc.), can be adapted for hand calculation.

Garner’s generalized method, for only two relatively prime moduli, depends on a “magic number,” and is as follows:

v1 = u1 % m1
v2 = ((u2 - v1) · c12) % m2
vr = (v2 · m1) + v1

Where m1 is the modulus of the first element in the list of coordinates that is the modular representation of integer vr; m2 is the modulus of the second element. u1 is the first coordinate, u2 the second. c12 is a constant, or our “magic number,” v1 and v2 are intermediate values, and vr is our answer, the re-constituted integer. We can use the same method on numbers represented by more than two coordinates, but it becomes very much more complicated, and there’s not a lot of point to it in Mayan arithmetic, as we really don’t need to find numbers from three coordinates, and when we do, we can break them down into their constituent parts. However, in Appendix IV, I do provide formulae demonstrating that it is possible, using the three factors of the 819-day count, 7, 9 and 13; these formulae will allow you to determine 819-day stations directly from trecena, glyph G and glyph Y coordinates (T, G and Y).

The key problem here, though, is finding the constant, or magic number, c12. Again, we can find the answer in Knuth (1998), where he is describing Garner’s algorithm. Since our two moduli must be relatively prime, that is, share no common factors, that means that their greatest common divisor (gcd) must be 1. This can be computed, and in the process we can find our constant c12, using an algorithm developed by the Hindu mathematician Bháscara I in the sixth century CE. It is a modification of Euclid’s algorithm for finding the greatest common divisor, and is referred to as the “Extended Euclidean Algorithm,” or, as it was called by other Hindu mathematicians, kuṭṭaka, or, “The Pulverizer.” Knuth’s description of the Pulverizer can be found in Appendix III, followed by its Python implementation.

The Python function exgcd() (described in Appendix V) is an extremely useful one. Since its main job is to determine the greatest common divisor of a pair of numbers, it becomes a valuable aid to determining factors of cycles in the Mayan Calendar. Its secondary job is to find, as a byproduct, a magic number suitable for use in Garner’s algorithm described above. Running the function on any two numbers will return a tuple of three numbers: the greatest common divisor; the magic number we need; and a second magic number that we do not need.

At this point, we can apply these principles to finding the position in the tzolk’in.

4.2.  New Formula for the Tzolk’in

As stated above, we have a general method in the form of Garner’s algorithm for recovering a positional number from a modular number:

v1 = u1 % m1
v2 = ((u2 - v1) · c12) % m2
vr = (v2 · m1) + v1

If we plug our moduli for the tzolk’in(13, 20) into the Pulverizer, we get gcd = 1 (as expected) and c12 = -3. Modifying the formula above, then, we can establish our formula for position in the tzolk’in (Δtz) to be:

v1 = (T - 1) % 13
v2 = (((v - 1) - v1) · -3) % 20
Δtz = (v2 · 13) + v1

T - 1 and v - 1 are due to the initial position (0) of the tzolk’in being set at 1 ’Imix; and when v - 1 is less than 0 we need to make v = 19. Taking our previous example of 11 ’Ix(11, 14), we can substitute and work our formula as follows:

Formula Python
v1 = (11 - 1) % 13
v1 = (10) % 13
v1 = 10

v2 = (((v - 1) - v1) · -3) % 20
v2 = (((14 - 1) - 10) · -3) % 20
v2 = (((13) - 10) · -3) % 20
v2 = (((3) · -3) % 20
v2 = ((-9) % 20
v2 = -9

Δtz = (v2 · 13) + v1
Δtz = (-9 · 13) + 10
Δtz = (-117) + 10
Δtz = -107
Δtz = -107 + 260
Δtz = 153
 
def p260(T, v): # T 1-13, v 1-19, 0
  c12 = -3
  m1 = 13
  m2 = 20
  u1 = T - 1
  u2 = v - 1
  if u2 < 0:
    u2 = 19
  v1 = u1 % m1
  v2 = ((u2 - v1)*c12) % m2
  dtz = ((v2 * m1) + v1)
  return dtz

Which is precisely the answer we expect, and we can now apply these same principles to finding the position in the Calendar Round (ΔCR).

4.3.  New Formula for the Calendar Round

The same formula, with different magic numbers, can be used to recover any positional number from any set of two modular coordinates. The hardest part is determining our magic number, such as we require for the modular number that represents position in the Calendar Round (ΔCR). Our moduli are 260 and 73; we use 73 instead of 365 since 260 and 365 have 5 as the greatest common divisor, and 365/5 = 73, which then gives us the necessary greatest common divisor of 1. The Python function exgcd(260, 73) gives us the tuple (1, -16, 57), which is, as you recall, (gcd, magic # 1, magic # 2), and we are able to use -16 in our formula.

We can modify Garner’s algorithm to calculate ΔCR in this way:

Formula   Python
v1 = (Δtz - 156) % 260
v2 = ((Δh - v1) · -16) % 73
ΔCR = (v2 · 260) + v1
 
def pCR(tz, hb): # tz = 0-259, hb = 0-364
  c12 = -16
  m1 = 260
  m2 = 73
  u1 = tz - 156
  if u1 < 0:
    u1 = u1 + 260
  u2 = hb % m2
  v1 = u1 % m1
  v2 = ((u2 - v1) * c12) % m2
  dcr = ((v2 * m1) + v1)
  return dcr

Again, remember that the beginning of the Calendar Round is set at coordinates (156, 0), or 1 Kaban 0 Pohp, which is why we subtract 156 from Δtz, and add 260 if it’s less than 0

As an example, we shall take the CR date of 1 Kawak 7 Mol, substitute and work the formula (although not in as much detail as previously). Δtz = 78, and Δh = 147:

v1 = (Δtz - 156) % 260
v1 = (78 - 156) % 260
v1 = 182

v2 = ((Δh - v1) · -16) % 73
v2 = ((147 - 182) · -16) % 73
v2 = 49

ΔCR = (v2 · 260) + v1
ΔCR = (49 · 260) + 78
ΔCR = 12922

Which is what we expect. Finally, we are ready to tackle the full 819 · 1460 cycle (E) and, as a byproduct, establish a probable starting point for cycle E.

4.4.  Formula for the 819 · 1460 Day Cycle

It is hardly possible for me to recall to the reader who is not a practical geologist the facts leading the mind feebly to comprehend the lapse of time. He who can read Sir Charles Lyell’s grand work on the Principles of Geology, which the future historian will recognize as having produced a revolution in natural science, and yet does not admit how vast have been the past periods of time, may at once close this volume.
—Charles Darwin7

Our coordinates in this cycle are the position in the 819-Day Count, Δe, and the position in the Calendar Round, ΔCR. The position in the 1,195,740 (Mayan 8.6.1.9.0) day cycle is termed ΔE.

As before, we plug our two moduli, 819 and 1460, into the Pulverizer (exgcd()), and find that our greatest common divisor is, of course, 1, and that c12 = 23. Thus, our formula for the position in cycle E is:

Formula Python
v1 = (ΔCR) % 1460
v2 = ((Δe - v1) · 23) % 819
ΔE = ((v2 · 1460) + v1)
 
def pE(dcr, pe): # dcr = 0-18979, pe = 0-818
  c12 = 23
  m1 = 1460
  m2 = 819
  u1 = dcr % 18980
  u2 = pe % 819
  m = m1 * m2
  v1 = u1 % m1
  v2 = ((u2 - v1) * c12) % m2
  de = ((v2 * m1) + v1)
  return de

Cycle E begins when the Calendar Round position is 0, or 1 Kaban 0 Pohp, and the 819-Day Count is also 0. We need to determine where that zero point falls in relation to Long Count day 0, and the best way to do that is to simply plug in the correct values for ΔCR and Δe that serve as coordinates of Long Count day 0. We already know that Δe = 3, and by simply substituting 4 ’Ahaw 8 Kumk’u into our previous algorithm for position in the Calendar Round, we can easily find that ΔCR is 7283. We can now apply our above formula to the coordinates (7283, 3) as follows:

v1 = (ΔCR) % 1460
v1 = (7283 % 1460
v1 = 1443

v2 = ((Δe - v1) · 23) % 819
v2 = ((3 - 1443) · 23) % 819
v2 = 459

ΔE = ((v2 · 1460) + v1)
ΔE = ((459 · 1460) + 1443)
ΔE = 671583

If day 0 in the Long Count corresponds to day 671,583 in cycle E, we have now “calibrated” our algorithm. All we need to do to produce Long Count dates from it is to subtract 671,583 from whatever answer we get, and then take that result and convert into Long Count coeffecients by the usual methods.

By subtracting 671,583 from 0, we can also work out the beginning point of cycle E: -671,583 corresponds to the Long Count date -4.13.5.9.3 before the start of the current era (0.0.0.0.0 4 ’Ahaw 8 Kumk’u). The Calendar Round for this date is, of course, 1 Kaban 0 Pohp, which falls on Sunday, November 19, -4,952 Gregorian (using the 584285 correlation). This is, by the way, 87,297 days (239+ years) before the beginning of our own Julian Period8, which we use to correlate the Mayan and the Julian calendars, and from there the Gregorian.

As a final note, it may be instructive to determine the length of time required for cycle E, combined with some other well-known calendric cycles, to repeat. Since Y, G and T are all components of E, we really can’t extract any additional information from them. Colors and directions, of course, are also components, as is the Calendar Round. We need to look at cycles not incorporated directly into E, such as tuns, k’atuns and bak’tuns. At first glance, we might suppose that the tun, composed as it is of 18 winals, might repay examination. However, it turns out that 360 and E share a greatest common divisor of 180, and the smallest commensurating cycle is therefore:

cs = (E / 180) · 360
cs = 2391480
which is 2E.

What this means is that a date with winal 0, k’in 0, Calendar Round 7283 (4 ’Ahaw 8 Kumk’u) and 819-day position 3 recurs every 16.12.3.0.0 days (2,391,480 decimal, or 6,552 haabs). This is alignment on tun boundaries. If we up our standards a little bit, and require k’in, winal and tun to all be 0, then the repetition frequency goes up by a factor of twenty. Dates don’t recur for 16.12.3.0.0.0 days (47,829,600 decimal, 131,040 haabs). Increase the requirements to include k’atuns too, and it goes up by another factor of twenty: 16.12.3.0.0.0.0 (956,592,000 decimal, or 2,620,800 haabs). Increase yet again to include bak’tuns, another factor of twenty (we are now at a factor of 8,000) and we get: 16.12.3.0.0.0.0.0 days, 19,131,840,000 decimal, 52,416,000 haabs.

Let’s ask a different question. Remember the date at Coba? The one that reads 13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.0.0.0.0 4 ’Ahaw 8 Kumk’u? We know that this date is equivalent to 0.0.0.0.0 4 ’Ahaw 8 Kumk’u, and can therefore assume that the 819-day station is 3 (it’s not shown on the monument). The question is, how many 13s does it take before the lowest five Long Count coeffecients are 0 (0.0.0.0.0), the Calendar Round is 4 ’Ahaw 8 Kumk’u, and the 819-day count position is 3 again? This is a question with an answer on a very much larger scale than previously, where the largest cycle we found was about 52.4 · 106 haabs. Here, we are starting with something that is the equivalent of 28.3 · 1027 haabs!

Is there an answer? One way to find out is to start with day 0, and start inserting 13s at the large end of the number. Admittedly, this is extreme brute force; but it does work. We start with 0.0.0.0.0 and our next step is 13.0.0.0.0.0, at which point we calculate that date’s position in cycle E; we continue inserting 13s, until we find a date where the ΔE again equals that of day 0, or the computer explodes. Well, there is an answer.

It turns out that if we keep inserting 13s until we have 72 of them, we end up with

13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.0.0.0.0.0 4 ’Ahaw 8 Kumk’u

This is 9,305,547,427,296,816,673,725,170,526,315,789,473,684,210,526,315,789,473,684,210,526,315,789,473,684,210,526,315,789,473,682,240,000 days or 25,494,650,485,744,703,215,685,398,702,235,039,653,929,343,907,714,491,708,723,864,455,659,697,188,175,919,250,180,245,133,376,000 haabs, and in the vicinity of 25.4 · 1096 years. This is a very large number; current estimates of the number of atoms in the entire universe range from 1070 to 1090. It is not quite a googol, which is 101010, or 10100; that is, 10 followed by 100 zeroes.

I suspect, however, that larger cycles exist within the framework of the Mayan calendar, and I hope to throw some light upon this subject in the future.

5.  Conclusion

That was when I became very close to Taniyama. Taniyama was not a very careful person as a mathematician. He made a lot of mistakes, but he made mistakes in a good direction, and so eventually, he got right answers, and I tried to imitate him, but I found out that it is very difficult to make good mistakes.
—Goro Shimura9

While I’ve covered some important techniques that may help simplify the narrowing down of inscriptional dates, I have also shown some new methods that can be applied to

Equally important, I think, should be the realization that modular arithmetic techniques may be used in many ways in Mayan calendrics, not merely in calculations involving the 819-day count. The analytical methods used herein may also apply to other Mayan calendrical topics, thus removing the determination of some formulae from the realm of trial-and-error. When I first determined the existance of the 1460 · 819-day cycle, I did not think of researching Knuth’s books for assistance, instead spending a good bit of time trying to devise formulae based on Lounsbury’s for the minimum interval between Calendar Round dates. Only after failing miserably at this endeavor did I start looking at Knuth, and, once I found the topics and techniques I needed, it became clear that the hard part was over. I just needed to implement some functions in Python, plug in values, and out came answers suitable for direct use in other formulae, taken virtually unchanged from Knuth. I was lucky. I was able to make some good mistakes.

6.  Notes

  1. Nova: Race for the Top, Chedd-Angier Production Company and WGBH Educational Foundation, 23 January, 1990; Melissa Franklin, particle physicist, CDF Fermilab (now at Harvard University).
  2. Pye, David. The Nature and Aesthetics of Design, New York: Nostrand Reinhold, 1978.
  3. Nova: Race for the Top, Chedd-Angier Production Company and WGBH Educational Foundation, 23 January, 1990; Melissa Franklin, particle physicist, CDF Fermilab (now at Harvard University).
  4. Modulo: divide, throw away the answer and keep the remainder; in keeping with computer science notation, I use % to indicate this operation.
  5. James Hutton: (1726-1797) The father of modern geology, published The Theory of the Earth. with Proofs and Illustrations in 1785.
  6. Nova #2414: The Proof, BBC-TV Co-Production with WGBH Educational Foundation, 28 October, 1997; Martin Eichler, Twentieth Century number theorist; Andrew Wiles, prover of Fermat’s Last Theorem. “Modular forms,” however, have little to do with modular arithmetic.
  7. Charles Darwin: The Origin of Species, Abridged and Introduced by Richard Leakey, Hill and Wang, New York, 1979. Pp. 151-53.
  8. The Julian Period:, A period defined by Joseph Justus Scaliger (1548-1609) that begins at noon on January 1, 4713 BC, runs for 7,980 years, and ends at noon on January 1, 3268, all in the Julian calendar—not the Gregorian one that we use. For correlation with the Mayan calendar, the procedure is to find the decimal equivalent of a Mayan Long Count, add a constant (called “The Ahaw Equation”) to it, and convert the resulting Julian Period date into the Gregorian calendar. Many Mayanists prefer a correlation constant of 584285 (which is the one I use), or 584283. With the 584285 correlation, day 0 is equivalent to Wednesday, August 13, -3113 (3114 BCE). See The Julian Period.
  9. Nova #2414: The Proof, BBC-TV Co-Production with WGBH Educational Foundation, 28 October, 1997; Goro Shimura, co-author with Yutaka Taniyama, of the Taniyama-Shimura Conjecture (which is, due to Andrew Wiles’ efforts, no longer a conjecture).

7.  Appendix I: Known 819-Day Count Dates

The only listing of all known 819-Day dates is in (Kelley, 1976), but the distance numbers used to determine these stations are unfortunately not shown. Here is the list:

A 9.12. 4.13. 71Manik’10PohpG60Palenque, N. Tab. Temple XVIII
B 9.13.16.10.131Ben1Ch’enG60Yaxchilan L.29, L.30
C 9.15.19.14.141’Ix7WoG60Yaxchilan St. 11
D 9.16. 8.16.101Ok18PohpG60Yaxchilan St. 1
E 9.18.14. 7.101Ok18K’ayabG60Quirigua St. K
F-0. 0. 6.15. 01’Ahaw18SotsG60Palenque Temple of the Cross [as 12.19.13.3.0]
G 9.17. 2.10. 41K’an7YaxG60Copan T11, East Door South Panel
H 9.10.10.11. 21’Ik’15Yaxk’inG60Palenque Palace Tablet
I 9.18. 7.10.131Ben11SotsG60Palenque IS Vase
J 9.12.18. 7. 11’Imix19Ch’enG60Palenque TFC (South Jamb?)
K[no date]------Palenque Templo del Conde (N. Side, S. Pillar, Central Doorway)
L[no date]------Palenque Fallen Stucco Glyphs, Temple 4, N.)
M 9.17. 4.15. 31Ak’bal16K’ank’inG60Yaxchilan St. 4
N 9.11.15.11.111Chuwen19PohpG60Palenque House A Pier A
O10. 1.13.10. 41K’an17SekG60Walter Randel Stela

8.  Appendix II: Haab Shifts

Table 6: 819-Day Haab Shifts
819-Day
Cycle
Days Elapsed
Eng.     Mayan
Long Count Tzolk’in Haab Haab
Position
(Δh)
819-Day
Cycle
Days Elapsed
Eng.     Mayan
Long Count Tzolk’in Haab Haab
Position
(Δh)
000.0-0.31Kaban
1 Kaban
5Kumku
5 Kumk’u
345 18192.4.192.4.161Kib
1 K’ib
9Sots
9 Sots
69
216384.9.184.9.151Men
1 Men
18Mol
18 Mol
158 324576.14.176.14.141Ix
1 ’Ix
7Mak
7 Mak
247
432769.1.169.1.131Ben
1 Ben
16Kayab
16 K’ayab
336 5409511.6.1511.6.121Eb
1 ’Eb
seating or zeroSots
0 Sots
60
6491413.11.1413.11.111Chuwen
1 Chuwen
9Mol
9 Mol
149 7573315.16.1315.16.101Ok
1 Ok
18Keh
18 Keh
238
8655218.3.1218.3.91Muluk
1 Muluk
7Kayab
7 K’ayab
327 973711.0.8.111.0.8.81Lamat
1 Lamat
11Sip
11 Sip
51
1081901.2.13.101.2.13.71Manik
1 Manik’
seating or zeroMol
0 Mol
140 1190091.5.0.91.5.0.61Kimi
1 Kimi
9Keh
9 Keh
229
1298281.7.5.81.7.5.51Chikchan
1 Chik’chan
18Pax
18 Pax
318 13106471.9.10.71.9.10.41Kan
1 K’an
2Sip
2 Sip
42
14114661.11.15.61.11.15.31Akbal
1 Ak’bal
11Yaxkin
11 Yaxk’in
131 15122851.14.2.51.14.2.21Ik
1 ’Ik’
seating or zeroKeh
0 Keh
220
16131041.16.7.41.16.7.11Imix
1 ’Imix
9Pax
9 Pax
309 17139231.18.12.31.18.12.01Ahaw
1 ’Ahaw
13Wo
13 Wo
33
18147422.0.17.22.0.16.191Kawak
1 Kawak
2Yaxkin
2 Yaxk’in
122 19155612.3.4.12.3.3.181Etznab
1 ’Etz’nab
11Sak
11 Sak
211
20163802.5.9.02.5.8.171Kaban
1 Kaban
seating or zeroPax
0 Pax
300 21171992.7.13.192.7.13.161Kib
1 K’ib
4Wo
4 Wo
24
22180182.10.0.182.10.0.151Men
1 Men
13Xul
13 Xul
113 23188372.12.5.172.12.5.141Ix
1 ’Ix
2Sak
2 Sak
202
24196562.14.10.162.14.10.131Ben
1 Ben
11Muwan
11 Muwan
291 25204752.16.15.152.16.15.121Eb
1 ’Eb
15Pohp
15 Pohp
15
26212942.19.2.142.19.2.111Chuwen
1 Chuwen
4Xul
4 Xul
104 27221133.1.7.133.1.7.101Ok
1 Ok
13Yax
13 Yax
193
28229323.3.12.123.3.12.91Muluk
1 Muluk
2Muwan
2 Muwan
282 29237513.5.17.113.5.17.81Lamat
1 Lamat
6Pohp
6 Pohp
6
30245703.8.4.103.8.4.71Manik
1 Manik’
15Sek
15 Sek
95 31253893.10.9.93.10.9.61Kimi
1 Kimi
4Yax
4 Yax
184
32262083.12.14.83.12.14.51Chikchan
1 Chik’chan
13Kankin
13 K’ank’in
273 33270273.15.1.73.15.1.41Kan
1 K’an
2Wayeb
2 Wayeb
362
34278463.17.6.63.17.6.31Akbal
1 Ak’bal
6Sek
6 Sek
86 35286653.19.11.53.19.11.21Ik
1 ’Ik’
15Chen
15 Ch’en
175
36294844.1.16.44.1.16.11Imix
1 ’Imix
4Kankin
4 K’ank’in
264 37303034.4.3.34.4.3.01Ahaw
1 ’Ahaw
13Kumku
13 Kumk’u
353
38311224.6.8.24.6.7.191Kawak
1 Kawak
17Sots
17 Sots
77 39319414.8.13.14.8.12.181Etznab
1 ’Etz’nab
6Chen
6 Ch’en
166
40327604.11.0.04.10.17.171Kaban
1 Kaban
15Mak
15 Mak
255 41335794.13.4.194.13.4.161Kib
1 K’ib
4Kumku
4 Kumk’u
344
42343984.15.9.184.15.9.151Men
1 Men
8Sots
8 Sots
68 43352174.17.14.174.17.14.141Ix
1 ’Ix
17Mol
17 Mol
157
44360365.0.1.165.0.1.131Ben
1 Ben
6Mak
6 Mak
246 45368555.2.6.155.2.6.121Eb
1 ’Eb
15Kayab
15 K’ayab
335
46376745.4.11.145.4.11.111Chuwen
1 Chuwen
19Sip
19 Sip
59 47384935.6.16.135.6.16.101Ok
1 Ok
8Mol
8 Mol
148
48393125.9.3.125.9.3.91Muluk
1 Muluk
17Keh
17 Keh
237 49401315.11.8.115.11.8.81Lamat
1 Lamat
6Kayab
6 K’ayab
326
50409505.13.13.105.13.13.71Manik
1 Manik’
10Sip
10 Sip
50 51417695.16.0.95.16.0.61Kimi
1 Kimi
19Yaxkin
19 Yaxk’in
139
52425885.18.5.85.18.5.51Chikchan
1 Chik’chan
8Keh
8 Keh
228 53434076.0.10.76.0.10.41Kan
1 K’an
17Pax
17 Pax
317
54442266.2.15.66.2.15.31Akbal
1 Ak’bal
1Sip
1 Sip
41 55450456.5.2.56.5.2.21Ik
1 ’Ik’
10Yaxkin
10 Yaxk’in
130
56458646.7.7.46.7.7.11Imix
1 ’Imix
19Sak
19 Sak
219 57466836.9.12.36.9.12.01Ahaw
1 ’Ahaw
8Pax
8 Pax
308
58475026.11.17.26.11.16.191Kawak
1 Kawak
12Wo
12 Wo
32 59483216.14.4.16.14.3.181Etznab
1 ’Etz’nab
1Yaxkin
1 Yaxk’in
121
60491406.16.9.06.16.8.171Kaban
1 Kaban
10Sak
10 Sak
210 61499596.18.13.196.18.13.161Kib
1 K’ib
19Muwan
19 Muwan
299
62507787.1.0.187.1.0.151Men
1 Men
3Wo
3 Wo
23 63515977.3.5.177.3.5.141Ix
1 ’Ix
12Xul
12 Xul
112
64524167.5.10.167.5.10.131Ben
1 Ben
1Sak
1 Sak
201 65532357.7.15.157.7.15.121Eb
1 ’Eb
10Muwan
10 Muwan
290
66540547.10.2.147.10.2.111Chuwen
1 Chuwen
14Pohp
14 Pohp
14 67548737.12.7.137.12.7.101Ok
1 Ok
3Xul
3 Xul
103
68556927.14.12.127.14.12.91Muluk
1 Muluk
12Yax
12 Yax
192 69565117.16.17.117.16.17.81Lamat
1 Lamat
1Muwan
1 Muwan
281
70573307.19.4.107.19.4.71Manik
1 Manik’
5Pohp
5 Pohp
5 71581498.1.9.98.1.9.61Kimi
1 Kimi
14Sek
14 Sek
94
72589688.3.14.88.3.14.51Chikchan
1 Chik’chan
3Yax
3 Yax
183 73597878.6.1.78.6.1.41Kan
1 K’an
12Kankin
12 K’ank’in
272
74606068.8.6.68.8.6.31Akbal
1 Ak’bal
1Wayeb
1 Wayeb
361 75614258.10.11.58.10.11.21Ik
1 ’Ik’
5Sek
5 Sek
85
76622448.12.16.48.12.16.11Imix
1 ’Imix
14Chen
14 Ch’en
174 77630638.15.3.38.15.3.01Ahaw
1 ’Ahaw
3Kankin
3 K’ank’in
263
78638828.17.8.28.17.7.191Kawak
1 Kawak
12Kumku
12 Kumk’u
352 79647018.19.13.18.19.12.181Etznab
1 ’Etz’nab
16Sots
16 Sots
76
80655209.2.0.09.1.17.171Kaban
1 Kaban
5Chen
5 Ch’en
165 81663399.4.4.199.4.4.161Kib
1 K’ib
14Mak
14 Mak
254
82671589.6.9.189.6.9.151Men
1 Men
3Kumku
3 Kumk’u
343 83679779.8.14.179.8.14.141Ix
1 ’Ix
7Sots
7 Sots
67
84687969.11.1.169.11.1.131Ben
1 Ben
16Mol
16 Mol
156 85696159.13.6.159.13.6.121Eb
1 ’Eb
5Mak
5 Mak
245
86704349.15.11.149.15.11.111Chuwen
1 Chuwen
14Kayab
14 K’ayab
334 87712539.17.16.139.17.16.101Ok
1 Ok
18Sip
18 Sip
58
887207210.0.3.1210.0.3.91Muluk
1 Muluk
7Mol
7 Mol
147 897289110.2.8.1110.2.8.81Lamat
1 Lamat
16Keh
16 Keh
236
907371010.4.13.1010.4.13.71Manik
1 Manik’
5Kayab
5 K’ayab
325 917452910.7.0.910.7.0.61Kimi
1 Kimi
9Sip
9 Sip
49
927534810.9.5.810.9.5.51Chikchan
1 Chik’chan
18Yaxkin
18 Yaxk’in
138 937616710.11.10.710.11.10.41Kan
1 K’an
7Keh
7 Keh
227
947698610.13.15.610.13.15.31Akbal
1 Ak’bal
16Pax
16 Pax
316 957780510.16.2.510.16.2.21Ik
1 ’Ik’
seating or zeroSip
0 Sip
40
967862410.18.7.410.18.7.11Imix
1 ’Imix
9Yaxkin
9 Yaxk’in
129 977944311.0.12.311.0.12.01Ahaw
1 ’Ahaw
18Sak
18 Sak
218
988026211.2.17.211.2.16.191Kawak
1 Kawak
7Pax
7 Pax
307 998108111.5.4.111.5.3.181Etznab
1 ’Etz’nab
11Wo
11 Wo
31
1008190011.7.9.011.7.8.171Kaban
1 Kaban
seating or zeroYaxkin
0 Yaxk’in
120 1018271911.9.13.1911.9.13.161Kib
1 K’ib
9Sak
9 Sak
209
1028353811.12.0.1811.12.0.151Men
1 Men
18Muwan
18 Muwan
298 1038435711.14.5.1711.14.5.141Ix
1 ’Ix
2Wo
2 Wo
22
1048517611.16.10.1611.16.10.131Ben
1 Ben
11Xul
11 Xul
111 1058599511.18.15.1511.18.15.121Eb
1 ’Eb
seating or zeroSak
0 Sak
200
1068681412.1.2.1412.1.2.111Chuwen
1 Chuwen
9Muwan
9 Muwan
289 1078763312.3.7.1312.3.7.101Ok
1 Ok
13Pohp
13 Pohp
13
1088845212.5.12.1212.5.12.91Muluk
1 Muluk
2Xul
2 Xul
102 1098927112.7.17.1112.7.17.81Lamat
1 Lamat
11Yax
11 Yax
191
1109009012.10.4.1012.10.4.71Manik
1 Manik’
seating or zeroMuwan
0 Muwan
280 1119090912.12.9.912.12.9.61Kimi
1 Kimi
4Pohp
4 Pohp
4
1129172812.14.14.812.14.14.51Chikchan
1 Chik’chan
13Sek
13 Sek
93 1139254712.17.1.712.17.1.41Kan
1 K’an
2Yax
2 Yax
182
1149336612.19.6.612.19.6.31Akbal
1 Ak’bal
11Kankin
11 K’ank’in
271 1159418513.1.11.513.1.11.21Ik
1 ’Ik’
seating or zeroWayeb
0 Wayeb
360
1169500413.3.16.413.3.16.11Imix
1 ’Imix
4Sek
4 Sek
84 1179582313.6.3.313.6.3.01Ahaw
1 ’Ahaw
13Chen
13 Ch’en
173
1189664213.8.8.213.8.7.191Kawak
1 Kawak
2Kankin
2 K’ank’in
262 1199746113.10.13.113.10.12.181Etznab
1 ’Etz’nab
11Kumku
11 Kumk’u
351
1209828013.13.0.013.12.17.171Kaban
1 Kaban
15Sots
15 Sots
75 1219909913.15.4.1913.15.4.161Kib
1 K’ib
4Chen
4 Ch’en
164
1229991813.17.9.1813.17.9.151Men
1 Men
13Mak
13 Mak
253 12310073713.19.14.1713.19.14.141Ix
1 ’Ix
2Kumku
2 Kumk’u
342
12410155614.2.1.1614.2.1.131Ben
1 Ben
6Sots
6 Sots
66 12510237514.4.6.1514.4.6.121Eb
1 ’Eb
15Mol
15 Mol
155
12610319414.6.11.1414.6.11.111Chuwen
1 Chuwen
4Mak
4 Mak
244 12710401314.8.16.1314.8.16.101Ok
1 Ok
13Kayab
13 K’ayab
333
12810483214.11.3.1214.11.3.91Muluk
1 Muluk
17Sip
17 Sip
57 12910565114.13.8.1114.13.8.81Lamat
1 Lamat
6Mol
6 Mol
146
13010647014.15.13.1014.15.13.71Manik
1 Manik’
15Keh
15 Keh
235 13110728914.18.0.914.18.0.61Kimi
1 Kimi
4Kayab
4 K’ayab
324
13210810815.0.5.815.0.5.51Chikchan
1 Chik’chan
8Sip
8 Sip
48 13310892715.2.10.715.2.10.41Kan
1 K’an
17Yaxkin
17 Yaxk’in
137
13410974615.4.15.615.4.15.31Akbal
1 Ak’bal
6Keh
6 Keh
226 13511056515.7.2.515.7.2.21Ik
1 ’Ik’
15Pax
15 Pax
315
13611138415.9.7.415.9.7.11Imix
1 ’Imix
19Wo
19 Wo
39 13711220315.11.12.315.11.12.01Ahaw
1 ’Ahaw
8Yaxkin
8 Yaxk’in
128
13811302215.13.17.215.13.16.191Kawak
1 Kawak
17Sak
17 Sak
217 13911384115.16.4.115.16.3.181Etznab
1 ’Etz’nab
6Pax
6 Pax
306
14011466015.18.9.015.18.8.171Kaban
1 Kaban
10Wo
10 Wo
30 14111547916.0.13.1916.0.13.161Kib
1 K’ib
19Xul
19 Xul
119
14211629816.3.0.1816.3.0.151Men
1 Men
8Sak
8 Sak
208 14311711716.5.5.1716.5.5.141Ix
1 ’Ix
17Muwan
17 Muwan
297
14411793616.7.10.1616.7.10.131Ben
1 Ben
1Wo
1 Wo
21 14511875516.9.15.1516.9.15.121Eb
1 ’Eb
10Xul
10 Xul
110
14611957416.12.2.1416.12.2.111Chuwen
1 Chuwen
19Yax
19 Yax
199 14712039316.14.7.1316.14.7.101Ok
1 Ok
8Muwan
8 Muwan
288
14812121216.16.12.1216.16.12.91Muluk
1 Muluk
12Pohp
12 Pohp
12 14912203116.18.17.1116.18.17.81Lamat
1 Lamat
1Xul
1 Xul
101
15012285017.1.4.1017.1.4.71Manik
1 Manik’
10Yax
10 Yax
190 15112366917.3.9.917.3.9.61Kimi
1 Kimi
19Kankin
19 K’ank’in
279
15212448817.5.14.817.5.14.51Chikchan
1 Chik’chan
3Pohp
3 Pohp
3 15312530717.8.1.717.8.1.41Kan
1 K’an
12Sek
12 Sek
92
15412612617.10.6.617.10.6.31Akbal
1 Ak’bal
1Yax
1 Yax
181 15512694517.12.11.517.12.11.21Ik
1 ’Ik’
10Kankin
10 K’ank’in
270
15612776417.14.16.417.14.16.11Imix
1 ’Imix
19Kumku
19 Kumk’u
359 15712858317.17.3.317.17.3.01Ahaw
1 ’Ahaw
3Sek
3 Sek
83
15812940217.19.8.217.19.7.191Kawak
1 Kawak
12Chen
12 Ch’en
172 15913022118.1.13.118.1.12.181Etznab
1 ’Etz’nab
1Kankin
1 K’ank’in
261
16013104018.4.0.018.3.17.171Kaban
1 Kaban
10Kumku
10 Kumk’u
350 16113185918.6.4.1918.6.4.161Kib
1 K’ib
14Sots
14 Sots
74
16213267818.8.9.1818.8.9.151Men
1 Men
3Chen
3 Ch’en
163 16313349718.10.14.1718.10.14.141Ix
1 ’Ix
12Mak
12 Mak
252
16413431618.13.1.1618.13.1.131Ben
1 Ben
1Kumku
1 Kumk’u
341 16513513518.15.6.1518.15.6.121Eb
1 ’Eb
5Sots
5 Sots
65
16613595418.17.11.1418.17.11.111Chuwen
1 Chuwen
14Mol
14 Mol
154 16713677318.19.16.1318.19.16.101Ok
1 Ok
3Mak
3 Mak
243
16813759219.2.3.1219.2.3.91Muluk
1 Muluk
12Kayab
12 K’ayab
332 16913841119.4.8.1119.4.8.81Lamat
1 Lamat
16Sip
16 Sip
56
17013923019.6.13.1019.6.13.71Manik
1 Manik’
5Mol
5 Mol
145 17114004919.9.0.919.9.0.61Kimi
1 Kimi
14Keh
14 Keh
234
17214086819.11.5.819.11.5.51Chikchan
1 Chik’chan
3Kayab
3 K’ayab
323 17314168719.13.10.719.13.10.41Kan
1 K’an
7Sip
7 Sip
47
17414250619.15.15.619.15.15.31Akbal
1 Ak’bal
16Yaxkin
16 Yaxk’in
136 17514332519.18.2.519.18.2.21Ik
1 ’Ik’
5Keh
5 Keh
225
1761441441.0.0.7.41.0.0.7.11Imix
1 ’Imix
14Pax
14 Pax
314 1771449631.0.2.12.31.0.2.12.01Ahaw
1 ’Ahaw
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18 Wo
38
1781457821.0.4.17.21.0.4.16.191Kawak
1 Kawak
7Yaxkin
7 Yaxk’in
127 1791466011.0.7.4.11.0.7.3.181Etznab
1 ’Etz’nab
16Sak
16 Sak
216
1801474201.0.9.9.01.0.9.8.171Kaban
1 Kaban
5Pax
5 Pax
305 1811482391.0.11.13.191.0.11.13.161Kib
1 K’ib
9Wo
9 Wo
29
1821490581.0.14.0.181.0.14.0.151Men
1 Men
18Xul
18 Xul
118 1831498771.0.16.5.171.0.16.5.141Ix
1 ’Ix
7Sak
7 Sak
207
1841506961.0.18.10.161.0.18.10.131Ben
1 Ben
16Muwan
16 Muwan
296 1851515151.1.0.15.151.1.0.15.121Eb
1 ’Eb
seating or zeroWo
0 Wo
20
1861523341.1.3.2.141.1.3.2.111Chuwen
1 Chuwen
9Xul
9 Xul
109 1871531531.1.5.7.131.1.5.7.101Ok
1 Ok
18Yax
18 Yax
198
1881539721.1.7.12.121.1.7.12.91Muluk
1 Muluk
7Muwan
7 Muwan
287 1891547911.1.9.17.111.1.9.17.81Lamat
1 Lamat
11Pohp
11 Pohp
11
1901556101.1.12.4.101.1.12.4.71Manik
1 Manik’
seating or zeroXul
0 Xul
100 1911564291.1.14.9.91.1.14.9.61Kimi
1 Kimi
9Yax
9 Yax
189
1921572481.1.16.14.81.1.16.14.51Chikchan
1 Chik’chan
18Kankin
18 K’ank’in
278 1931580671.1.19.1.71.1.19.1.41Kan
1 K’an
2Pohp
2 Pohp
2
1941588861.2.1.6.61.2.1.6.31Akbal
1 Ak’bal
11Sek
11 Sek
91 1951597051.2.3.11.51.2.3.11.21Ik
1 ’Ik’
seating or zeroYax
0 Yax
180
1961605241.2.5.16.41.2.5.16.11Imix
1 ’Imix
9Kankin
9 K’ank’in
269 1971613431.2.8.3.31.2.8.3.01Ahaw
1 ’Ahaw
18Kumku
18 Kumk’u
358
1981621621.2.10.8.21.2.10.7.191Kawak
1 Kawak
2Sek
2 Sek
82 1991629811.2.12.13.11.2.12.12.181Etznab
1 ’Etz’nab
11Chen
11 Ch’en
171
2001638001.2.15.0.01.2.14.17.171Kaban
1 Kaban
seating or zeroKankin
0 K’ank’in
260 2011646191.2.17.4.191.2.17.4.161Kib
1 K’ib
9Kumku
9 Kumk’u
349
2021654381.2.19.9.181.2.19.9.151Men
1 Men
13Sots
13 Sots
73 2031662571.3.1.14.171.3.1.14.141Ix
1 ’Ix
2Chen
2 Ch’en
162
2041670761.3.4.1.161.3.4.1.131Ben
1 Ben
11Mak
11 Mak
251 2051678951.3.6.6.151.3.6.6.121Eb
1 ’Eb
seating or zeroKumku
0 Kumk’u
340
2061687141.3.8.11.141.3.8.11.111Chuwen
1 Chuwen
4Sots
4 Sots
64 2071695331.3.10.16.131.3.10.16.101Ok
1 Ok
13Mol
13 Mol
153
2081703521.3.13.3.121.3.13.3.91Muluk
1 Muluk
2Mak
2 Mak
242 2091711711.3.15.8.111.3.15.8.81Lamat
1 Lamat
11Kayab
11 K’ayab
331
2101719901.3.17.13.101.3.17.13.71Manik
1 Manik’
15Sip
15 Sip
55 2111728091.4.0.0.91.4.0.0.61Kimi
1 Kimi
4Mol
4 Mol
144
2121736281.4.2.5.81.4.2.5.51Chikchan
1 Chik’chan
13Keh
13 Keh
233 2131744471.4.4.10.71.4.4.10.41Kan
1 K’an
2Kayab
2 K’ayab
322
2141752661.4.6.15.61.4.6.15.31Akbal
1 Ak’bal
6Sip
6 Sip
46 2151760851.4.9.2.51.4.9.2.21Ik
1 ’Ik’
15Yaxkin
15 Yaxk’in
135
2161769041.4.11.7.41.4.11.7.11Imix
1 ’Imix
4Keh
4 Keh
224 2171777231.4.13.12.31.4.13.12.01Ahaw
1 ’Ahaw
13Pax
13 Pax
313
2181785421.4.15.17.21.4.15.16.191Kawak
1 Kawak
17Wo
17 Wo
37 2191793611.4.18.4.11.4.18.3.181Etznab
1 ’Etz’nab
6Yaxkin
6 Yaxk’in
126
2201801801.5.0.9.01.5.0.8.171Kaban
1 Kaban
15Sak
15 Sak
215 2211809991.5.2.13.191.5.2.13.161Kib
1 K’ib
4Pax
4 Pax
304
2221818181.5.5.0.181.5.5.0.151Men
1 Men
8Wo
8 Wo
28 2231826371.5.7.5.171.5.7.5.141Ix
1 ’Ix
17Xul
17 Xul
117
2241834561.5.9.10.161.5.9.10.131Ben
1 Ben
6Sak
6 Sak
206 2251842751.5.11.15.151.5.11.15.121Eb
1 ’Eb
15Muwan
15 Muwan
295
2261850941.5.14.2.141.5.14.2.111Chuwen
1 Chuwen
19Pohp
19 Pohp
19 2271859131.5.16.7.131.5.16.7.101Ok
1 Ok
8Xul
8 Xul
108
2281867321.5.18.12.121.5.18.12.91Muluk
1 Muluk
17Yax
17 Yax
197 2291875511.6.0.17.111.6.0.17.81Lamat
1 Lamat
6Muwan
6 Muwan
286
2301883701.6.3.4.101.6.3.4.71Manik
1 Manik’
10Pohp
10 Pohp
10 2311891891.6.5.9.91.6.5.9.61Kimi
1 Kimi
19Sek
19 Sek
99
2321900081.6.7.14.81.6.7.14.51Chikchan
1 Chik’chan
8Yax
8 Yax
188 2331908271.6.10.1.71.6.10.1.41Kan
1 K’an
17Kankin
17 K’ank’in
277
2341916461.6.12.6.61.6.12.6.31Akbal
1 Ak’bal
1Pohp
1 Pohp
1 2351924651.6.14.11.51.6.14.11.21Ik
1 ’Ik’
10Sek
10 Sek
90
2361932841.6.16.16.41.6.16.16.11Imix
1 ’Imix
19Chen
19 Ch’en
179 2371941031.6.19.3.31.6.19.3.01Ahaw
1 ’Ahaw
8Kankin
8 K’ank’in
268
2381949221.7.1.8.21.7.1.7.191Kawak
1 Kawak
17Kumku
17 Kumk’u
357 2391957411.7.3.13.11.7.3.12.181Etznab
1 ’Etz’nab
1Sek
1 Sek
81
2401965601.7.6.0.01.7.5.17.171Kaban
1 Kaban
10Chen
10 Ch’en
170 2411973791.7.8.4.191.7.8.4.161Kib
1 K’ib
19Mak
19 Mak
259
2421981981.7.10.9.181.7.10.9.151Men
1 Men
8Kumku
8 Kumk’u
348 2431990171.7.12.14.171.7.12.14.141Ix
1 ’Ix
12Sots
12 Sots
72
2441998361.7.15.1.161.7.15.1.131Ben
1 Ben
1Chen
1 Ch’en
161 2452006551.7.17.6.151.7.17.6.121Eb
1 ’Eb
10Mak
10 Mak
250
2462014741.7.19.11.141.7.19.11.111Chuwen
1 Chuwen
19Kayab
19 K’ayab
339 2472022931.8.1.16.131.8.1.16.101Ok
1 Ok
3Sots
3 Sots
63
2482031121.8.4.3.121.8.4.3.91Muluk
1 Muluk
12Mol
12 Mol
152 2492039311.8.6.8.111.8.6.8.81Lamat
1 Lamat
1Mak
1 Mak
241
2502047501.8.8.13.101.8.8.13.71Manik
1 Manik’
10Kayab
10 K’ayab
330 2512055691.8.11.0.91.8.11.0.61Kimi
1 Kimi
14Sip
14 Sip
54
2522063881.8.13.5.81.8.13.5.51Chikchan
1 Chik’chan
3Mol
3 Mol
143 2532072071.8.15.10.71.8.15.10.41Kan
1 K’an
12Keh
12 Keh
232
2542080261.8.17.15.61.8.17.15.31Akbal
1 Ak’bal
1Kayab
1 K’ayab
321 2552088451.9.0.2.51.9.0.2.21Ik
1 ’Ik’
5Sip
5 Sip
45
2562096641.9.2.7.41.9.2.7.11Imix
1 ’Imix
14Yaxkin
14 Yaxk’in
134 2572104831.9.4.12.31.9.4.12.01Ahaw
1 ’Ahaw
3Keh
3 Keh
223
2582113021.9.6.17.21.9.6.16.191Kawak
1 Kawak
12Pax
12 Pax
312 2592121211.9.9.4.11.9.9.3.181Etznab
1 ’Etz’nab
16Wo
16 Wo
36
2602129401.9.11.9.01.9.11.8.171Kaban
1 Kaban
5Yaxkin
5 Yaxk’in
125 2612137591.9.13.13.191.9.13.13.161Kib
1 K’ib
14Sak
14 Sak
214
2622145781.9.16.0.181.9.16.0.151Men
1 Men
3Pax
3 Pax
303 2632153971.9.18.5.171.9.18.5.141Ix
1 ’Ix
7Wo
7 Wo
27
2642162161.10.0.10.161.10.0.10.131Ben
1 Ben
16Xul
16 Xul
116 2652170351.10.2.15.151.10.2.15.121Eb
1 ’Eb
5Sak
5 Sak
205
2662178541.10.5.2.141.10.5.2.111Chuwen
1 Chuwen
14Muwan
14 Muwan
294 2672186731.10.7.7.131.10.7.7.101Ok
1 Ok
18Pohp
18 Pohp
18
2682194921.10.9.12.121.10.9.12.91Muluk
1 Muluk
7Xul
7 Xul
107 2692203111.10.11.17.111.10.11.17.81Lamat
1 Lamat
16Yax
16 Yax
196
2702211301.10.14.4.101.10.14.4.71Manik
1 Manik’
5Muwan
5 Muwan
285 2712219491.10.16.9.91.10.16.9.61Kimi
1 Kimi
9Pohp
9 Pohp
9
2722227681.10.18.14.81.10.18.14.51Chikchan
1 Chik’chan
18Sek
18 Sek
98 2732235871.11.1.1.71.11.1.1.41Kan
1 K’an
7Yax
7 Yax
187
2742244061.11.3.6.61.11.3.6.31Akbal
1 Ak’bal
16Kankin
16 K’ank’in
276 2752252251.11.5.11.51.11.5.11.21Ik
1 ’Ik’
seating or zeroPohp
0 Pohp
0
2762260441.11.7.16.41.11.7.16.11Imix
1 ’Imix
9Sek
9 Sek
89 2772268631.11.10.3.31.11.10.3.01Ahaw
1 ’Ahaw
18Chen
18 Ch’en
178
2782276821.11.12.8.21.11.12.7.191Kawak
1 Kawak
7Kankin
7 K’ank’in
267 2792285011.11.14.13.11.11.14.12.181Etznab
1 ’Etz’nab
16Kumku
16 Kumk’u
356
2802293201.11.17.0.01.11.16.17.171Kaban
1 Kaban
seating or zeroSek
0 Sek
80 2812301391.11.19.4.191.11.19.4.161Kib
1 K’ib
9Chen
9 Ch’en
169
2822309581.12.1.9.181.12.1.9.151Men
1 Men
18Mak
18 Mak
258 2832317771.12.3.14.171.12.3.14.141Ix
1 ’Ix
7Kumku
7 Kumk’u
347
2842325961.12.6.1.161.12.6.1.131Ben
1 Ben
11Sots
11 Sots
71 2852334151.12.8.6.151.12.8.6.121Eb
1 ’Eb
seating or zeroChen
0 Ch’en
160
2862342341.12.10.11.141.12.10.11.111Chuwen
1 Chuwen
9Mak
9 Mak
249 2872350531.12.12.16.131.12.12.16.101Ok
1 Ok
18Kayab
18 K’ayab
338
2882358721.12.15.3.121.12.15.3.91Muluk
1 Muluk
2Sots
2 Sots
62 2892366911.12.17.8.111.12.17.8.81Lamat
1 Lamat
11Mol
11 Mol
151
2902375101.12.19.13.101.12.19.13.71Manik
1 Manik’
seating or zeroMak
0 Mak
240 2912383291.13.2.0.91.13.2.0.61Kimi
1 Kimi
9Kayab
9 K’ayab
329
2922391481.13.4.5.81.13.4.5.51Chikchan
1 Chik’chan
13Sip
13 Sip
53 2932399671.13.6.10.71.13.6.10.41Kan
1 K’an
2Mol
2 Mol
142
2942407861.13.8.15.61.13.8.15.31Akbal
1 Ak’bal
11Keh
11 Keh
231 2952416051.13.11.2.51.13.11.2.21Ik
1 ’Ik’
seating or zeroKayab
0 K’ayab
320
2962424241.13.13.7.41.13.13.7.11Imix
1 ’Imix
4Sip
4 Sip
44 2972432431.13.15.12.31.13.15.12.01Ahaw
1 ’Ahaw
13Yaxkin
13 Yaxk’in
133
2982440621.13.17.17.21.13.17.16.191Kawak
1 Kawak
2Keh
2 Keh
222 2992448811.14.0.4.11.14.0.3.181Etznab
1 ’Etz’nab
11Pax
11 Pax
311
3002457001.14.2.9.01.14.2.8.171Kaban
1 Kaban
15Wo
15 Wo
35 3012465191.14.4.13.191.14.4.13.161Kib
1 K’ib
4Yaxkin
4 Yaxk’in
124
3022473381.14.7.0.181.14.7.0.151Men
1 Men
13Sak
13 Sak
213 3032481571.14.9.5.171.14.9.5.141Ix
1 ’Ix
2Pax
2 Pax
302
3042489761.14.11.10.161.14.11.10.131Ben
1 Ben
6Wo
6 Wo
26 3052497951.14.13.15.151.14.13.15.121Eb
1 ’Eb
15Xul
15 Xul
115
3062506141.14.16.2.141.14.16.2.111Chuwen
1 Chuwen
4Sak
4 Sak
204 3072514331.14.18.7.131.14.18.7.101Ok
1 Ok
13Muwan
13 Muwan
293
3082522521.15.0.12.121.15.0.12.91Muluk
1 Muluk
17Pohp
17 Pohp
17 3092530711.15.2.17.111.15.2.17.81Lamat
1 Lamat
6Xul
6 Xul
106
3102538901.15.5.4.101.15.5.4.71Manik
1 Manik’
15Yax
15 Yax
195 3112547091.15.7.9.91.15.7.9.61Kimi
1 Kimi
4Muwan
4 Muwan
284
3122555281.15.9.14.81.15.9.14.51Chikchan
1 Chik’chan
8Pohp
8 Pohp
8 3132563471.15.12.1.71.15.12.1.41Kan
1 K’an
17Sek
17 Sek
97
3142571661.15.14.6.61.15.14.6.31Akbal
1 Ak’bal
6Yax
6 Yax
186 3152579851.15.16.11.51.15.16.11.21Ik
1 ’Ik’
15Kankin
15 K’ank’in
275
3162588041.15.18.16.41.15.18.16.11Imix
1 ’Imix
4Wayeb
4 Wayeb
364 3172596231.16.1.3.31.16.1.3.01Ahaw
1 ’Ahaw
8Sek
8 Sek
88
3182604421.16.3.8.21.16.3.7.191Kawak
1 Kawak
17Chen
17 Ch’en
177 3192612611.16.5.13.11.16.5.12.181Etznab
1 ’Etz’nab
6Kankin
6 K’ank’in
266
3202620801.16.8.0.01.16.7.17.171Kaban
1 Kaban
15Kumku
15 Kumk’u
355 3212628991.16.10.4.191.16.10.4.161Kib
1 K’ib
19Sots
19 Sots
79
3222637181.16.12.9.181.16.12.9.151Men
1 Men
8Chen
8 Ch’en
168 3232645371.16.14.14.171.16.14.14.141Ix
1 ’Ix
17Mak
17 Mak
257
3242653561.16.17.1.161.16.17.1.131Ben
1 Ben
6Kumku
6 Kumk’u
346 3252661751.16.19.6.151.16.19.6.121Eb
1 ’Eb
10Sots
10 Sots
70
3262669941.17.1.11.141.17.1.11.111Chuwen
1 Chuwen
19Mol
19 Mol
159 3272678131.17.3.16.131.17.3.16.101Ok
1 Ok
8Mak
8 Mak
248
3282686321.17.6.3.121.17.6.3.91Muluk
1 Muluk
17Kayab
17 K’ayab
337 3292694511.17.8.8.111.17.8.8.81Lamat
1 Lamat
1Sots
1 Sots
61
3302702701.17.10.13.101.17.10.13.71Manik
1 Manik’
10Mol
10 Mol
150 3312710891.17.13.0.91.17.13.0.61Kimi
1 Kimi
19Keh
19 Keh
239
3322719081.17.15.5.81.17.15.5.51Chikchan
1 Chik’chan
8Kayab
8 K’ayab
328 3332727271.17.17.10.71.17.17.10.41Kan
1 K’an
12Sip
12 Sip
52
3342735461.17.19.15.61.17.19.15.31Akbal
1 Ak’bal
1Mol
1 Mol
141 3352743651.18.2.2.51.18.2.2.21Ik
1 ’Ik’
10Keh
10 Keh
230
3362751841.18.4.7.41.18.4.7.11Imix
1 ’Imix
19Pax
19 Pax
319 3372760031.18.6.12.31.18.6.12.01Ahaw
1 ’Ahaw
3Sip
3 Sip
43
3382768221.18.8.17.21.18.8.16.191Kawak
1 Kawak
12Yaxkin
12 Yaxk’in
132 3392776411.18.11.4.11.18.11.3.181Etznab
1 ’Etz’nab
1Keh
1 Keh
221
3402784601.18.13.9.01.18.13.8.171Kaban
1 Kaban
10Pax
10 Pax
310 3412792791.18.15.13.191.18.15.13.161Kib
1 K’ib
14Wo
14 Wo
34
3422800981.18.18.0.181.18.18.0.151Men
1 Men
3Yaxkin
3 Yaxk’in
123 3432809171.19.0.5.171.19.0.5.141Ix
1 ’Ix
12Sak
12 Sak
212
3442817361.19.2.10.161.19.2.10.131Ben
1 Ben
1Pax
1 Pax
301 3452825551.19.4.15.151.19.4.15.121Eb
1 ’Eb
5Wo
5 Wo
25
3462833741.19.7.2.141.19.7.2.111Chuwen
1 Chuwen
14Xul
14 Xul
114 3472841931.19.9.7.131.19.9.7.101Ok
1 Ok
3Sak
3 Sak
203
3482850121.19.11.12.121.19.11.12.91Muluk
1 Muluk
12Muwan
12 Muwan
292 3492858311.19.13.17.111.19.13.17.81Lamat
1 Lamat
16Pohp
16 Pohp
16
3502866501.19.16.4.101.19.16.4.71Manik
1 Manik’
5Xul
5 Xul
105 3512874691.19.18.9.91.19.18.9.61Kimi
1 Kimi
14Yax
14 Yax
194
3522882882.0.0.14.82.0.0.14.51Chikchan
1 Chik’chan
3Muwan
3 Muwan
283 3532891072.0.3.1.72.0.3.1.41Kan
1 K’an
7Pohp
7 Pohp
7
3542899262.0.5.6.62.0.5.6.31Akbal
1 Ak’bal
16Sek
16 Sek
96 3552907452.0.7.11.52.0.7.11.21Ik
1 ’Ik’
5Yax
5 Yax
185
3562915642.0.9.16.42.0.9.16.11Imix
1 ’Imix
14Kankin
14 K’ank’in
274 3572923832.0.12.3.32.0.12.3.01Ahaw
1 ’Ahaw
3Wayeb
3 Wayeb
363
3582932022.0.14.8.22.0.14.7.191Kawak
1 Kawak
7Sek
7 Sek
87 3592940212.0.16.13.12.0.16.12.181Etznab
1 ’Etz’nab
16Chen
16 Ch’en
176
3602948402.0.19.0.02.0.18.17.171Kaban
1 Kaban
5Kankin
5 K’ank’in
265 3612956592.1.1.4.192.1.1.4.161Kib
1 K’ib
14Kumku
14 Kumk’u
354
3622964782.1.3.9.182.1.3.9.151Men
1 Men
18Sots
18 Sots
78 3632972972.1.5.14.172.1.5.14.141Ix
1 ’Ix
7Chen
7 Ch’en
167
3642981162.1.8.1.162.1.8.1.131Ben
1 Ben
16Mak
16 Mak
256 3652989352.1.10.6.152.1.10.6.121Eb
1 ’Eb
5Kumku
5 Kumk’u
345
819-Day
Cycle
Days Elapsed
Eng.     Mayan
Long Count Tzolk’in Haab Haab
Position
(Δh)
819-Day
Cycle
Days Elapsed
Eng.     Mayan
Long Count Tzolk’in Haab Haab
Position
(Δh)

9.  Appendix III: The Pulverizer

This is Knuth’s description of the Pulverizer:

This extension of Euclid’s algorithm can be described conveniently in vector notation:

Algorithm X (Extended Euclid’s algorithm). Given nonnegative integers u and v, this algorithm determines a vector (u1, u2, u3) such that uu1 + uu2 = u3 = gcd(u, v). The computation makes use of auxiliary vectors (v1, v2, v3), (t1, t2, t3); all vectors are manipulated in such a way that the relations

ut1 + vt2 = t3; uu1 + vu2 = u3; uv1 + vv2 = v3;

hold throughout the calculation.

X1. [Initialize.] Set (u1, u2, u3) ← (1, 0, u),    (v1, v2, v3) ← (0, 1, v).
X2. [Is v3 = 0?] If v3 = 0, the algorithm terminates.
X3. [Divide, subtract.] Set q ← ⌊u3 / v3, and then set

(t1, t2, t3) ← (u1, u2, u3) - (v1, v2, v3)q,

(u1, u2, u3) ← (v1, v2, v3), (v1, v2, v3) ← (t1, t2, t3).

Return to step X2.

For example, let u = 40902, v = 24140. At step X2 we have
qu1u2u3v1v2v3
010409020124140
101241401-116762
11-116762-127378
2-1273783-52006
33-52006-10171360
1-1017136013-22646
213-22646-366168
9-366168337-57134
2337-57134-71012030

The solution is therefore 337 · 40902-571 · 24140 = 34 = gcd(40902, 24140).

This is the Python implementation of the Pulverizer:

def exgcd(u, v, pr = 0): 
  u1, u2, u3 = 1, 0, u  # X1
  v1, v2, v3 = 0, 1, v  # X1

  q = 0                 # X1
  while v3 > 0:         # X1
    q = u3/v3           # X2
    # Modification of Knuth’s algorithm from Tim Peters on the Python list;
    # this gets rid of the temporary variables and speeds it up.
    u1, u2, u3, v1, v2, v3 = v1, v2, v3, u1 - v1 * q, u2 - v2 * q, u3 - v3 * q
  return u3, u1, u2     # X3: gcd, magic number 1, magic number 2

For the example above, the greatest common divisor is 34, and u1 is 337, or c1, while u2 is -571, or c2. In this case, we might expect c12 to be -192427 (the product of 337 · -571), but, again according to Knuth, “... we may take cij = a.” (Knuth, v2, 3.4.2) In other words, just use c1 instead of carrying out the multiplication that would give us c12 (what Knuth is calling cij).

10.  Appendix IV: Formulae for Recovering 819-Day Count Positions from 7, 9 and 13 Coordinates

Since it may prove desirable to determine the 819-day count position (Δe) from the three constituent coordinates Y (Y glyphs), G (G glyphs) and T (trecena), without recourse to the Long Count coeffecients, I’ll include here the methods for determining the value.

Garner’s method (Knuth, 1998), expanded to include three coordinates that are relatively prime instead of two as we have been using it, will do the job quite nicely.

v1 = u1 % m1
v2 = (u2 - v1) · c12
v2 = v2 % m2
v3 = (((u3 - v1) · c13) - v2) · c23
v3 = v3 % m3
u = v1 + (v2 · m1) + (v3 · m2 · m1)

Where m1 = 7, m2 = 9, m3 = 13, and Δe = u.

Finding our constants, or “magic numbers,” is very simple. We need to find the values c12, c13 and c23, and we can do this by feeding the combinations required to the Pulverizer (Python function exgcd()); these combinations are:

Table 7: Exgcd() Combinations
To determine: Use: Result (magic number):
c12 exgcd(7, 9) 4
c13 exgcd(7, 13) 2
c23 exgcd(9, 13) 3

At this point, it might seem that we have enough information to recover 819-day positions from Y, G and T, but this is not the case; the hardest part is yet to come. As given so far, the algorithm demands input values in one set of ranges, but the values that are used on monumental inscriptions do not match the required set. For example, while the algorithm wants G in the range 0-8, values on the inscriptions range from G6-G5. Therefore, we need to remap the input value set onto another set, and we do that by following these rules and their formulae:

Table 8: Input Transformations
Variable Actual Input Values
and Order
Mathematical Equivalents Expected Input Values
and Order
Formula
Y 7, 1, 2, 3, 4, 5, 6 0, 1, 2, 3, 4, 5, 6 0, 1, 2, 3, 4, 5, 6
u1 = Y
if u1 == 7:
  u1 = 0
G 6, 7, 8, 9, 1, 2, 3, 4, 5 6, 7, 8, 0, 1, 2, 3, 4, 5 0, 1, 2, 3, 4, 5, 6, 7, 8
u2 = (Y - 6) % 9
T 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
u3 = (T - 1) % 13

What these rules mean is that our minimum input value set (7, 6, 1) [Y7, G6, T1] is transformed into the mathematically more useful set (0, 0, 0), and a similar transformation of the maximum set (6, 5, 13) [Y6, G5, T13] results in (6, 8, 12).

Incorporating Garner’s algorithm modified for 3 coordinates, the special constants found by the use of exgcd() and the formulae for the transformation of input values into a single Python function is straightforward, and results in:

def pe(Y, G, T):
    m1 = 7      # Radix for Y
    m2 = 9      # Radix for G
    m3 = 13     # Radix for T
    c12 = 4     # Magic number for 7 and 9
    c13 = 2     # Magic number for 7 and 13
    c23 = 3     # Magic number for 9 and 13
    u1 = Y      # Adjust input range for Y glyphs
    if u1 == m1:
        u1 = 0

    u2 = (G - 6) % m2  # Adjust input range for G glyphs

    u3 = (T - 1) % m3  # Adjust input range for trecena

        v1 = u1 % m1

    v2 = (u2 - v1) * c12
    v2 = v2 % m2

    v3 = (((u3 - v1) * c13) - v2) * c23
    v3 = v3 % m3

    pe = v1 + (v2 * m1) + (v3 * m2 * m1)
    return pe  # The resultant position in the 819-day count

The converse function, to return Y, G and T coordinates from any given position in the 819-day count (Δe) is almost trivial to implement:

def dpe(p):  # p = 0-818
    Y = p % 7
    if Y == 0:
        Y = 7
    G = p % 9
    G = (G + 6) % 9
    if G == 0:
        G = 9
    T = p % 13
    T = (T + 1) % 13
    if T == 0:
        T = 13
    return(Y, G, T)

11.  Appendix V: Python Resources

The Python programming language is a very sophisticated but easy to use and learn language that has widespread application in many fields, not just in computer science. The first step in using Python to aid research in the Mayan calendar is to obtain and install it on your computer; since it is freely available for virtually all kinds of computers and operating systems, this is ordinarily not difficult. For users of Win95/98/NT, probably the majority of computer-knowledgable Mayanists, this is an extremely simple step. As of this writing, the latest version available to the general public was version 2.4.2, and it can be downloaded from the Python web site. Follow the link and obey the instructions, which will install all you need (and more) to run everything described in this paper.

Once you’ve installed Python and verified that it is working, you should download the Mayan mathematics package I’ve written: mayalib.zip, which contains mayalib.py and hello.py, a tiny test program to make sure that both Python and mayalib have been installed correctly. You can use WinZip to unpack the zip file, or, on Unix, gzip. Either way, once you’ve unzipped mayalib.zip, find out where your Python distribution has been installed. Unless you overrode the destination during the install process, this will be in “c:\Program Files\Python”. Assuming that this directory is correct, put a copy of mayalib.py (note: the name may change; please see the README.TXT file that comes with the distribution to see the latest installation instructions) into “c:\Program Files\Python\lib”. Ensure that “c:\Program Files\Python” is in your path, cd to the directory where you have the copy of hello.py and type, in a DOS box: python hello.py. The hello.py program should tell you the date, in Mayan, if everything has been installed correctly.

Once installed, you can write Python programs for the Mayan calendar very easily. On Windows, use your favorite text editor (Notepad will do in a pinch), and type in program lines. At the Python website, there is a good tutorial available, written by Guido van Rossum, the creator of the language. Although it is aimed primarily at users who know some other programming language, it is nonetheless clear and concise. O’Reilly publishes Learning Python by Mark Lutz and David Ascher, which fills the niche for an introduction to Python for those with less programming experience. If you can find a copy, my own book Teach Yourself Python in 24 Hours, is adequate.

Programs that use the mayalib package should include the line from mayalib import * at or near the top of the file. As an example, here is a Python program that lists the first 20 819-day stations, beginning with the usual starting day on -3, 1 Kaban 5 Kumk’u; run it by typing python 20.py in a DOS box:

from mayalib import *

b = mayanum()  # Create a Mayan date for day 0
b.calculate()  # Calculate trecena, veintena, Calendar Round, etc.
b = b - 3      # Start at three days before 0
               #  Another way to get three days before 0 would be:
               #  b = mayanum ( [ -3 ] )
               #  b.calculate ( )
b.calculate()  # You have to recalculate subsidiary information after operations,
               # because -, +, /, *, etc. only work on LC positions.
print "Starting day:", b, b.gregorian()
for i in range(21):
    print "%s:  819-day position %d (%d,%d,%d), %s" % ( b, b.st819, b.Y, b.glord, b.trecena, colordirection ( b ) )
    b = b + 819 # You could also do this as b = b + "2.4.19"
    b.calculate()

print "Ending day:", b.gregorian()
#
# Set the correlation (the "'Ahaw Equation") to something else;
# default is 584285.
SetCorrelation ( 584283 )
#
print "Ending day:", b.gregorian()

Included in mayalib.zip is the above program, 20.py, the test program hello.py and full documentation in HTML format; you will need a web browser in order to use it (instructions can be found in the README.TXT file). The mayalib documentation assumes you know more about the Mayan calendar than programming, however, and comes with a full set of examples.

12.  References


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