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
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.
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.
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).
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 
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 
(ok) is infixed into T588, MacLeod (1989) advanced a tentative translation
of the verb as e¢-ok, “plant the feet.” Note also T1022 
,
which is quite similar to T588, lacking only the “sprout” affix. She
notes, however, that T588 is frequently suffixed by T178 
(la) and 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: 
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 
is ch’o and T110 
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) 
T739, and a “Smoking Squirrel”/God K 
T1030 glyph (E7). Yasugi and Saito (1991) suggest that T1030 with the T122
prefix 
,
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 
,
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.
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).
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:
| 819-Day Cycle | Days Elapsed | Days Elapsed, Mayan | Tzolk’in Position (Δtz) | Tzolk’in | Color | Direction |
|---|---|---|---|---|---|---|
| 0 | 0 | [0, 0] | 156 | 1 Kaban   |
Chak (Red)  |
Likin (East)  |
| 1 | 819 | [2, 4, 19] | 195 | 1 K’ib  |
Kan (Yellow)  |
Nohol (South)  |
| 2 | 1638 | [4, 9, 18] | 234 | 1 Men  |
Ek (Black)  |
Chikin (West)  |
| 3 | 2457 | [6, 14, 17] | 13 | 1 ’Ix  |
Sak (White)  |
Xaman (North)  |
| 4 | 3276 | [9, 1, 16] | 52 | 1 Ben  |
Chak (Red) |
Likin (East) |
| 5 | 4095 | [11, 6, 15] | 91 | 1 ’Eb  |
Kan (Yellow) |
Nohol (South) |
| 6 | 4914 | [13, 11, 14] | 130 | 1 Chuwen  |
Ek (Black) |
Chikin (West) |
| 7 | 5733 | [15, 16, 13] | 169 | 1 Ok  |
Sak (White) |
Xaman (North) |
| 8 | 6552 | [18, 3, 12] | 208 | 1 Muluk  |
Chak (Red) |
Likin (East) |
| 9 | 7371 | [1, 0, 8, 11] | 247 | 1 Lamat  |
Kan (Yellow) |
Nohol (South) |
| 10 | 8190 | [1, 2, 13, 10] | 26 | 1 Manik’  |
Ek (Black) |
Chikin (West) |
| 11 | 9009 | [1, 5, 0, 9] | 65 | 1 Kimi  |
Sak (White) |
Xaman (North) |
| 12 | 9828 | [1, 7, 5, 8] | 104 | 1 Chik’chan  |
Chak (Red) |
Likin (East) |
| 13 | 10647 | [1, 9, 10, 7] | 143 | 1 K’an  |
Kan (Yellow) |
Nohol (South) |
| 14 | 11466 | [1, 11, 15, 6] | 182 | 1 Ak’bal  |
Ek (Black) |
Chikin (West) |
| 15 | 12285 | [1, 14, 2, 5] | 221 | 1 ’Ik’ |
Sak (White) |
Xaman (North) |
| 16 | 13104 | [1, 16, 7, 4] | 0 | 1 ’Imix  |
Chak (Red) |
Likin (East) |
| 17 | 13923 | [1, 18, 12, 3] | 39 | 1 ’Ahaw  |
Kan (Yellow) |
Nohol (South) |
| 18 | 14742 | [2, 0, 17, 2] | 78 | 1 Kawak  |
Ek (Black) |
Chikin (West) |
| 19 | 15561 | [2, 3, 4, 1] | 117 | 1 ’Etz’nab  |
Sak (White) |
Xaman (North) |
| 20 | 16380 | [2, 5, 9, 0] | 156 | 1 Kaban |
Chak (Red) |
Likin (East) |
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).
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.
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.
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.
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).
| Symbol | Meaning |
|---|---|
| G | The nine-day cycle of the Lords of the Night, 1-9 |
| Y | The seven-day cycle of the terrestrial gods, 1-7 |
| T | The thirteen-day cycle of the celestial gods, the trecena 1-13 |
| v | The twenty named days, the veintena ’Imix-’Ahaw |
| c | One of the four colors red, yellow, black and white |
| d | One of the four directions east, south, west, north |
| tz | The tzolk’in, 260 days, T · v |
| Δtz | Distance between two positions in the tzolk’in |
| h | The haab, 365 days |
| Δh | Distance between two positions in the haab |
| hd | The haab day, 0-19 |
| hm | The named haab month, Pohp-Wayeb |
| CR | The Calendar Round, 18,980 days, 260 · 73 or 365 · 52 |
| ΔCR | Distance between two positions in the Calendar Round |
| e | The 819-day count, G · Y · T |
| Δe | Distance between two positions in the 819-day count |
| E | the 1460 · 819-day count cycle, 1,195,740 days |
| ΔE | Distance between two positions in cycle E |
| m | The decimal equivalent of a Long Count date |
| j | The Julian period day corresponding to m |
| m8 | An 819-day station |
| k | A k’in Long Count coeffecient |
| w | A winal Long Count coeffecient |
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
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.)
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:
| Index | Color | Direction | Veintena Days | ||||
|---|---|---|---|---|---|---|---|
| 0 | Chak (Red)
|
Likin (East)
|
’Imix 1 |
Chik’chan 5 |
Muluk 9 |
Ben 13 |
Kaban 17 |
| 1 | Sak (White)
|
Xaman (North)
|
Ik 2 |
Kimi 6 |
Ok 10 |
’Ix 14 |
’Etz’nab 18 |
| 2 | Ek (Black)
|
Chikin (West)
|
Ak’bal 3 |
Manik’ 7 |
Chuwen 11 |
Men 15 |
Kawak 19 |
| 3 | Kan (Yellow)
|
Nohol (South)
|
K’an 4 |
Lamat 8 |
’Eb 12 |
K’ib 16 |
’Ahaw 0 |
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).
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.
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:
| 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.
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):
| Long Count Position | Shift 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.
There’s a saying attributed to Eichler that there are five fundamental operations of arithmetic: addition, subtraction, multiplication, division, and modular forms.
—Andrew Wiles6
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.
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).
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.
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.
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.
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. 7 | 1 | Manik’ | 10 | Pohp | G6 | 0 | Palenque, N. Tab. Temple XVIII |
| B | 9.13.16.10.13 | 1 | Ben | 1 | Ch’en | G6 | 0 | Yaxchilan L.29, L.30 |
| C | 9.15.19.14.14 | 1 | ’Ix | 7 | Wo | G6 | 0 | Yaxchilan St. 11 |
| D | 9.16. 8.16.10 | 1 | Ok | 18 | Pohp | G6 | 0 | Yaxchilan St. 1 |
| E | 9.18.14. 7.10 | 1 | Ok | 18 | K’ayab | G6 | 0 | Quirigua St. K |
| F | -0. 0. 6.15. 0 | 1 | ’Ahaw | 18 | Sots | G6 | 0 | Palenque Temple of the Cross [as 12.19.13.3.0] |
| G | 9.17. 2.10. 4 | 1 | K’an | 7 | Yax | G6 | 0 | Copan T11, East Door South Panel |
| H | 9.10.10.11. 2 | 1 | ’Ik’ | 15 | Yaxk’in | G6 | 0 | Palenque Palace Tablet |
| I | 9.18. 7.10.13 | 1 | Ben | 11 | Sots | G6 | 0 | Palenque IS Vase |
| J | 9.12.18. 7. 1 | 1 | ’Imix | 19 | Ch’en | G6 | 0 | Palenque 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. 3 | 1 | Ak’bal | 16 | K’ank’in | G6 | 0 | Yaxchilan St. 4 |
| N | 9.11.15.11.11 | 1 | Chuwen | 19 | Pohp | G6 | 0 | Palenque House A Pier A |
| O | 10. 1.13.10. 4 | 1 | K’an | 17 | Sek | G6 | 0 | Walter Randel Stela |
| 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) |
||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0.0 | -0.3 |   1 Kaban |   5 Kumk’u | 345 | 1 | 819 | 2.4.19 | 2.4.16 |  1 K’ib |   9 Sots | 69 |
| 2 | 1638 | 4.9.18 | 4.9.15 |  1 Men |   18 Mol | 158 | 3 | 2457 | 6.14.17 | 6.14.14 |  1 ’Ix |   7 Mak | 247 |
| 4 | 3276 | 9.1.16 | 9.1.13 |  1 Ben |   16 K’ayab | 336 | 5 | 4095 | 11.6.15 | 11.6.12 |  1 ’Eb |  0 Sots | 60 |
| 6 | 4914 | 13.11.14 | 13.11.11 |  1 Chuwen | 9 Mol | 149 | 7 | 5733 | 15.16.13 | 15.16.10 |  1 Ok |  18 Keh | 238 |
| 8 | 6552 | 18.3.12 | 18.3.9 |  1 Muluk | 7 K’ayab | 327 | 9 | 7371 | 1.0.8.11 | 1.0.8.8 |  1 Lamat |   11 Sip | 51 |
| 10 | 8190 | 1.2.13.10 | 1.2.13.7 |  1 Manik’ | 0 Mol | 140 | 11 | 9009 | 1.5.0.9 | 1.5.0.6 |  1 Kimi | 9 Keh | 229 |
| 12 | 9828 | 1.7.5.8 | 1.7.5.5 |  1 Chik’chan |  18 Pax | 318 | 13 | 10647 | 1.9.10.7 | 1.9.10.4 |  1 K’an |  2 Sip | 42 |
| 14 | 11466 | 1.11.15.6 | 1.11.15.3 |  1 Ak’bal |  11 Yaxk’in | 131 | 15 | 12285 | 1.14.2.5 | 1.14.2.2 |  1 ’Ik’ | 0 Keh | 220 |
| 16 | 13104 | 1.16.7.4 | 1.16.7.1 |  1 ’Imix | 9 Pax | 309 | 17 | 13923 | 1.18.12.3 | 1.18.12.0 |  1 ’Ahaw |   13 Wo | 33 |
| 18 | 14742 | 2.0.17.2 | 2.0.16.19 |  1 Kawak | 2 Yaxk’in | 122 | 19 | 15561 | 2.3.4.1 | 2.3.3.18 |  1 ’Etz’nab |  11 Sak | 211 |
| 20 | 16380 | 2.5.9.0 | 2.5.8.17 | 1 Kaban | 0 Pax | 300 | 21 | 17199 | 2.7.13.19 | 2.7.13.16 | 1 K’ib |  4 Wo | 24 |
| 22 | 18018 | 2.10.0.18 | 2.10.0.15 | 1 Men |  13 Xul | 113 | 23 | 18837 | 2.12.5.17 | 2.12.5.14 | 1 ’Ix | 2 Sak | 202 |
| 24 | 19656 | 2.14.10.16 | 2.14.10.13 | 1 Ben |  11 Muwan | 291 | 25 | 20475 | 2.16.15.15 | 2.16.15.12 | 1 ’Eb |   15 Pohp | 15 |
| 26 | 21294 | 2.19.2.14 | 2.19.2.11 | 1 Chuwen | 4 Xul | 104 | 27 | 22113 | 3.1.7.13 | 3.1.7.10 | 1 Ok |  13 Yax | 193 |
| 28 | 22932 | 3.3.12.12 | 3.3.12.9 | 1 Muluk | 2 Muwan | 282 | 29 | 23751 | 3.5.17.11 | 3.5.17.8 | 1 Lamat |  6 Pohp | 6 |
| 30 | 24570 | 3.8.4.10 | 3.8.4.7 | 1 Manik’ |  15 Sek | 95 | 31 | 25389 | 3.10.9.9 | 3.10.9.6 | 1 Kimi | 4 Yax | 184 |
| 32 | 26208 | 3.12.14.8 | 3.12.14.5 | 1 Chik’chan |  13 K’ank’in | 273 | 33 | 27027 | 3.15.1.7 | 3.15.1.4 | 1 K’an |  2 Wayeb | 362 |
| 34 | 27846 | 3.17.6.6 | 3.17.6.3 | 1 Ak’bal | 6 Sek | 86 | 35 | 28665 | 3.19.11.5 | 3.19.11.2 | 1 ’Ik’ |  15 Ch’en | 175 |
| 36 | 29484 | 4.1.16.4 | 4.1.16.1 | 1 ’Imix | 4 K’ank’in | 264 | 37 | 30303 | 4.4.3.3 | 4.4.3.0 | 1 ’Ahaw | 13 Kumk’u | 353 |
| 38 | 31122 | 4.6.8.2 | 4.6.7.19 | 1 Kawak |  17 Sots | 77 | 39 | 31941 | 4.8.13.1 | 4.8.12.18 | 1 ’Etz’nab | 6 Ch’en | 166 |
| 40 | 32760 | 4.11.0.0 | 4.10.17.17 | 1 Kaban | 15 Mak | 255 | 41 | 33579 | 4.13.4.19 | 4.13.4.16 | 1 K’ib | 4 Kumk’u | 344 |
| 42 | 34398 | 4.15.9.18 | 4.15.9.15 | 1 Men |  8 Sots | 68 | 43 | 35217 | 4.17.14.17 | 4.17.14.14 | 1 ’Ix | 17 Mol | 157 |
| 44 | 36036 | 5.0.1.16 | 5.0.1.13 | 1 Ben | 6 Mak | 246 | 45 | 36855 | 5.2.6.15 | 5.2.6.12 | 1 ’Eb | 15 K’ayab | 335 |
| 46 | 37674 | 5.4.11.14 | 5.4.11.11 | 1 Chuwen |  19 Sip | 59 | 47 | 38493 | 5.6.16.13 | 5.6.16.10 | 1 Ok | 8 Mol | 148 |
| 48 | 39312 | 5.9.3.12 | 5.9.3.9 | 1 Muluk | 17 Keh | 237 | 49 | 40131 | 5.11.8.11 | 5.11.8.8 | 1 Lamat | 6 K’ayab | 326 |
| 50 | 40950 | 5.13.13.10 | 5.13.13.7 | 1 Manik’ |  10 Sip | 50 | 51 | 41769 | 5.16.0.9 | 5.16.0.6 | 1 Kimi | 19 Yaxk’in | 139 |
| 52 | 42588 | 5.18.5.8 | 5.18.5.5 | 1 Chik’chan | 8 Keh | 228 | 53 | 43407 | 6.0.10.7 | 6.0.10.4 | 1 K’an | 17 Pax | 317 |
| 54 | 44226 | 6.2.15.6 | 6.2.15.3 | 1 Ak’bal | 1 Sip | 41 | 55 | 45045 | 6.5.2.5 | 6.5.2.2 | 1 ’Ik’ | 10 Yaxk’in | 130 |
| 56 | 45864 | 6.7.7.4 | 6.7.7.1 | 1 ’Imix | 19 Sak | 219 | 57 | 46683 | 6.9.12.3 | 6.9.12.0 | 1 ’Ahaw | 8 Pax | 308 |
| 58 | 47502 | 6.11.17.2 | 6.11.16.19 | 1 Kawak |  12 Wo | 32 | 59 | 48321 | 6.14.4.1 | 6.14.3.18 | 1 ’Etz’nab | 1 Yaxk’in | 121 |
| 60 | 49140 | 6.16.9.0 | 6.16.8.17 | 1 Kaban | 10 Sak | 210 | 61 | 49959 | 6.18.13.19 | 6.18.13.16 | 1 K’ib | 19 Muwan | 299 |
| 62 | 50778 | 7.1.0.18 | 7.1.0.15 | 1 Men |  3 Wo | 23 | 63 | 51597 | 7.3.5.17 | 7.3.5.14 | 1 ’Ix | 12 Xul | 112 |
| 64 | 52416 | 7.5.10.16 | 7.5.10.13 | 1 Ben | 1 Sak | 201 | 65 | 53235 | 7.7.15.15 | 7.7.15.12 | 1 ’Eb | 10 Muwan | 290 |
| 66 | 54054 | 7.10.2.14 | 7.10.2.11 | 1 Chuwen |  14 Pohp | 14 | 67 | 54873 | 7.12.7.13 | 7.12.7.10 | 1 Ok | 3 Xul | 103 |
| 68 | 55692 | 7.14.12.12 | 7.14.12.9 | 1 Muluk | 12 Yax | 192 | 69 | 56511 | 7.16.17.11 | 7.16.17.8 | 1 Lamat | 1 Muwan | 281 |
| 70 | 57330 | 7.19.4.10 | 7.19.4.7 | 1 Manik’ | 5 Pohp | 5 | 71 | 58149 | |||||