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On the origin of the different Mayan Calendars Thomas Chanier

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Thomas Chanier. On the origin of the different Mayan Calendars. 2014. ￿hal-01018037v1￿

HAL Id: hal-01018037 https://hal.archives-ouvertes.fr/hal-01018037v1 Submitted on 3 Jul 2014 (v1), last revised 14 Jan 2015 (v3)

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the origin of the different Mayan Calendars

T. Chanier∗1 1 Department of Physics, University of Namur, rue de Bruxelles 61, B-5000 Namur, Belgium

The Maya were known for their astronomical proficiency. Whereas Mayan mathematics were based on a vigesimal system, they used a different base when dealing with long periods of time, the Long Count Calendar (LCC), composed of different Long Count Periods: the Tun of 360 days, the Katun of 7200 days and the of 144000 days. There were three other calendars used in addition to the LCC: a civil year Haab’ of 365 days, a religious year Tzolk’in of 260 days and a 3276- day cycle (combination of the 819-day Kawil cycle and 4 colors-directions). Based on astronomical arguments, we propose here an explanation of the origin of the LCC, the Tzolk’in and the 3276-day cycle. The study provides new evidences on the astronomical knowledge of the Maya.

The Maya had a very elaborate and accurate Long the Tzolk’in, which has been associated with various as- Count Calendar (LCC) that was used to pinpoint his- tronomical cycles. Three Tzolk’in correspond to Mars torical events from a selected beginning time. The small- synodic period, 16 Tzolk’in equal 11 of Saturn synodic est unit of time is the day (Kin); 20 Kin form a , periods (+2 days), and 23 Tzolk’in are equivalent to 15 18 Winal form a Tun (360 Kin), 20 Tun form a Katun Jupiter synodic periods (-5 days).2 It has been tentatively (7200 Kin), and 20 Katun form a Baktun (144000 Kin). connected to the eclipse half-year (173.31 days) because 2 The LCC represents a date D as a set of coefficients Tzolk’in are very close to 3 eclipse half-years.3 Finally, it (Ci,...,C3,C2,C1,C0) such that: D = C0 + C1 × 20 + has been noted that the Tzolk’in approximates the length n i−1 4 Pi=2 Ci × 18 × 20 with C0 = mod(D,20), C1 = of time Venus is visible as a morning or evening star. i int(mod(D,360)/20) and Ci = int(mod(D, 18 × 20 )/(18 However, these interpretations fail to link the Tzolk’in × 20i−1) for i> 1. The day count usually restarts when to the LCPs. The Kawil cycle has been attributed to 5,6 C4 reaches 13, such as the date is given as a set of 5 co- the observation of Jupiter and Saturn because 19 (6) efficients: D ≡ mod(D, 13 × 144000) = C4.C3.C2.C1.C0. Kawil correspond to 39 (13) Jupiter (Saturn) synodic pe- The Maya used three other independent calendars: a reli- riod. Four numbers of possible astronomical significance gious year (Tzolk’in), a civil year (Haab’) and a 3276-day have been discovered on the walls of a residential struc- cycle. One Tzolk’in of 260 days comprised 13 months ture in , and have been dated from (numerated from 1 to 13) containing 20 named days the early 9th century CE. The Xultun numbers are given (Imix, Ik, Akbal, Kan, Chicchan, Cimi, Manik, Lamat, in Table I. They are such that X1 = 365 × 3276 and Muluc, Oc, Chuen, Eb, Ben, Ix, Men, Cib, Caban, Et- X3 = X2 + 2X0. znab, Cauac, and Ahau). One Haab’ of 365 days com- prised 18 named months (Pop, Uo, Zip, Zotz, Tzec, Xul, Xi LCC D [day] Xi/56940 Yaxkin, Mol, Chen, Yax, Zac, Ceh, Mac, Kankin, Muan, X0 2.7.9.0.0 341640 6 X Pax, Kayab, and Cumku) with 20 days each (Winal) 1 8.6.1.9.0 1195740 21 X plus 1 extra month (Uayeb) with 5 nameless days. The 2 12.5.3.3.0 1765140 31 X Tzolk’in and the Haab’ coincide every 73 Tzolk’in or 52 3 17.0.1.3.0 2448420 43

Haab’ or a Calendar Round. Their least common multi- 7 TABLE I. Xultun numbers Xi. 56940 = LCM(365,780) is ple (LCM) is 73 × 260 = 52 × 365 = 18980 days. In the their largest common divisor. Calendar Round, a date is represented by αXβY with the religious month 1 ≤ α ≤ 13, X one of the 20 re- ligious days, the civil day 0 ≤ β ≤ 19, and Y one of the 18 civil months, 0 ≤ β ≤ 4 for the Uayeb. According This paper explains a possible origin of the LCPs, the to the Goodman-Martinez-Thompson (GMT) correlation religious year Tzolk’in and the relationship between the to the Gregorian calendar, which is based on historical Kawil cycle and the 4 colors-directions. This new in- facts, the Mayan Calendar began on 11 August 3114 BC terpretation, based on the periodic movements of the or 0(13).0.0.0.0, 4 Ahau 8 Cumku and ended on 21 De- planets in the night sky, gives rise to different day cy- cember 2012. This corresponds to a 13 Baktun cycle or cles that have been identified in the and a period of approximately 5125 years. The 3276-day cy- monuments. Attempts are made to provide a meaning cle is a combination of the Kawil, a 819-day cycle, and a to the Xultun numbers. The study shows that Mayan 4-day cycle corresponding to the 4 colors-directions. astronomers knew the synodic periods of all the planets of the Solar System visible to the naked eye: Mercury, The origin of the Long Count Periods (LCPs) is un- Venus, Mars, Jupiter and Saturn. known. A common assumption is the desire of the cal- The Maya were known for their astronomical skills as endar keeper to maintain the Tun in close agreement exemplified by the Codex, a bark-paper with the tropical/solar year of approximately 365.24 of the 11th or 12th century CE. On page 24 of this days.1 There is no consensus concerning the origin of Codex is written the so-called Long Round number noted 2

Planet P [day] Prime factorization structure. Mercury 116 22 × 29 3 Venus 584 2 × 73 Name i Ci [day] Pi Di Earth 365 5 × 73 N 6 - 018 /13/73/ P0 Ci 18 Mars 780 22 × 3 × 5 × 13 N 5 Tun 1360 /13/73/ P0 Ci 360 × × 4 Jupiter 399 3 7 19 N i 3 Katun 27200 /13/73/ P0 C 7215 Saturn 378 2 × 3 × 7 N 3 Baktun 3 144000 /13/73/ P0 Ci 144304 Lunar 177 3 × 59 N 2 Pictun 4 2880000 /13/73/ P0 Ci 2886428 senesters 178 2 × 89 1 Calabtun 5 57600000 N /13/73/ P Ci 57866020 Pentalunex 148 22 × 37 0 Kinchiltun 6 1152000000 N /13/73/C0 1215186420

8,9 TABLE II. Planet cycles and their prime factorizations. TABLE III. Divisibility of N = 20757814426440 days by a 6−i polynomial expression of the type Pi = 13 × 73 × (Pn 18 × n =0 20 ). Di = int(N /Pi). 9.9.16.0.0 in the LCC or 1366560 days, a whole multiple of the Tzolk’in, the Haab’, the Calendar Round, the Tun, Venus and Mars synodic periods: LR = 1366560 = 5256 Table IV gives the coincidence of the 338-day period × 260 = 3744 × 365 = 72 × 18980 = 3796 × 360 = with the LCPs, as well as the Tzolk’in and the 234-day 2340 × 584 = 1752 × 780. Only the moon, Mercury, period. As described earlier, the Tun and 338-day period Venus, Earth (solar year), Mars, Jupiter, and Saturn are coincide every 60840 days, or 234 Tzolk’in. The 338-day visible to the naked eye. Their respective mean synodic period also coincides with the Winal every 3380 days, periods are given in Table II. Evidences have been found corresponding to 13 Tzolk’in, a cycle that does not coin- in different Mayan Codices that Mayan astronomers ob- cide with the 234-day period. Rather, the 234-day period served the periodic movements in the night sky of Mer- coincides with the Winal every 2340 days or 9 Tzolk’in = cury, Venus, and Mars, but it is unclear whether they 9 × 260 = LCM(9,13,20). This 2340-day cycle is present tracked the movements of Jupiter and Saturn.10 The pe- in the on pages D30c-D33c and has been riods relevant for the prediction of solar/lunar eclipses are attributed to a Venus-Mercury almanac because 2340 = the pentalunex of 148 days (5 Moon synodic periods of 20 × 117 = 5 × 585 is an integer multiple of Mercury 29.53 days) and the lunar semesters of 177 or 178 days (6 and Venus mean synodic periods (+1 day).11 Another Moon synodic periods), which are the time intervals be- explanation may be of divination origin because 117 = 9 tween subsequent eclipse warning stations present in the × 13. In Mesoamerican mythology, there are a set of 9 Eclipse Table on pages 51 to 58 of the Dresden Codex.9 Gods called the Lords of the Night12–15 and a set of 13 The LCM of these numbers is N = 20757814426440 days Gods called the Lords of the Day.15 Each day is linked (including the planet synodic periods and the two lunar with 1 of the 13 Lords of the Day and 1 of the 9 Lords semesters) or N † = 768039133778280 days (also includ- of the Night in a repeating 117-day cycle. This cycle co- ing the pentalunex). N is such that: incides with the Tzolk’in every 2340 days. On Table IV figures a date D = LCM(338,7200) = 1216800 days or int(N /13/73/144000) = 338 + 360 + 7200 + 144000. (1) 8.9.0.0.0 in the LCC or 5 February 219 CE according to the GMT correlation. This date may have marked the N † gives the same result but divided by 13, 37 and 73. beginning of the reign of Yax Moch Xoc, the founder of That defines the Tun, Katun, and Baktun as a polyno- the dynasty, in 219 CE.16 mial expression of int(N /13/73) of the form 18 × 203 × 3 × n (C0 +Pn=1 18 20 ) with C0 = 338. The LCM(338,360) LCP D [day] L [day] M [day] N [day] L/260 L/234 = 169 × 360 = 234 × 260 = 60840; 338 and 365 are rel- Winal 20 3380 260 2340 13 14.44444 atively prime numbers: the LCM(338,365) = 338 × 365 Tun 360 60840 4680 4680 234 260 = 123370; 234 and 365 are also relatively prime num- Katun 7200 1216800 93600 93600 4680 5200 bers: the LCM(234,365) = 234 × 365 = 85410. The Baktun 144000 24336000 1872000 1872000 93600 104000 LCM(260,365) is 73 × 260 = 52 × 365 = 18980 days or a Calendar Round. That may define the Tzolk’in. N is TABLE IV. Coincidence of the 234-, 260- and 338-day periods such that the mod(N ,LR) = 341640 days or 2.7.9.0.0 in with the LCPs. L = LCM(D,338), M = LCM(D,260) and N = LCM(D,234). the LCC. This is the Xultun number X0 (Table I). It is to be noted that X0 is a whole multiple of the Tzolk’in, Haab’, Tun, Venus and Mars synodic periods: 341640 = 1314 × 260 = 936 × 365 = 949 × 360 = 585 × 584 = 438 We now examine the cycles that can be constructed × 780. X0 is also the largest common divisor of LR and from the combination of the Tzolk’in (260 = 13 × 20), N . Table III gives the divisibility of N by a polynomial the Haab’ (365 = 5 × 73) and the LCPs (D = 18 × 20n). n expression of the type Pi = 13 × 73 × (Pn 18 × 20 ). Because 13 is not present in the prime factorizations of The integer parts Di are such that Di/Di−1 ≈ 20 for the LCPs, we obtain LCM(260,D) = 13 ×D. There is, for 0

The Kawil, a 819-day cycle, is connected with a 4-day cycle (4 colors-directions), coinciding every 3276 days. The Haab’ and the Tzolk’in coincide with the 3276-day cycle every LCM(365,3276) = 365 × 3276 = 1195740 days or 8.6.1.9.0 in the LCC. This is the Xultun number X1 (Table I). X1 = LCM(9,4,819,260,365) is the time dis- tance between two days of the same Lord of the Night, color-direction, Kawil, Tzolk’in and Haab’ date. X0 and X1 are such that 341640 × 819 = 1195740 × 360 = 2391480 or Y = 16.12.3.0.0 in the LCC. Y represents the same as X1 at the end of a Tun (0 Winal, 0 Kin). FIG. 1. Pyramid of during an equinox. The pyra- The 3276-day cycle coincides with the Tun every 32760 mid is situated at , Yucatan, . days = LCM(360,819) = 7 × LCM(260,360). Table V shows the coincidence of the 32760-day cycle with the Some mesoamerican monuments may be symbolical planet synodic periods, giving rise to whole multiples of representations of the different Mayan calendars. This the Tzolk’in and the 2340-day cycle. N and N † are di- is the case of the pyramid of Kukulkan (Figure 1) built vidable by 32760. This provides further proofs of the as- sometime between the 9th and the 12th century CE at tronomical knowledge of the Maya. It shows that Mayan Chichen Itza. The pyramid shape may be linked to the astronomers calculated the synodic periods of at least Long Count Calendar (Table III). It is constituted of 9 Venus, Mars and Saturn: the LCM(584,365,780,378) = platforms with 4 stairways of 91 steps each leading to 7174440 = 219 × 32760. The use of the Tzolk’in and the platform temple corresponding to the 3276-day cy- the commemoration of the 13 Baktun cycle implies that cle: 3276 = LCM(4,9,91) = 22 × 32 × 7 × 13 = 4 × Mayan astronomers knew the synodic periods of Mercury 819, the coincidence of the 4 colors-directions with the and Jupiter as well. Kawil. The dimensions of the pyramid may be of signif- Various religious cycles can be constructed by taking icance: the width of its base is 55.30 m (37 zapal), its the LCM of 260 and the planet synodic periods (Ta- height up to the top of the platform temple is 30 m (20 ble VI). The most important one in Mayan religion is zapal) and the width of the top platform is 19.52 m (13 the Calendar Round, which is the LCM(260,365) = 73 zapal), taking into account the Mayan zapal length mea- × 260 = 52 × 365 = 18980 days. There is a coinci- surement such that 1 zapal ≈ 1.5 m.17 The width of the dence between Venus synodic period and the Tzolk’in: base and 3276 are such that 37 × 3276 = LCM(148,3276) the LCM(260,584) = 65 × 584 = 146 × 260 = 104 × 365 = 121212 (Table V). The pyramid height and the width of = 37960 days (2 Calendar Rounds), the length of the the top platform represents the Tzolk’in. The stairways 4 divide the 9 platforms of each side of the pyramid into 18 directions. Based on astronomical observations of the segments which, combined with the pyramid height, rep- Solar System, the LCC gives rise to three different peri- resents the 18 Winal of a Tun. The Haab’ is represented ods of 234 days, 260 days (Tzolk’in) and 338 days. The by the platform temple making the 365th step with the correlation between the 234-day period and the Tzolk’in, 4 × 91 = 364 steps of the 4 stairways. Each side of the a 2340-day cycle identified in the Dresden Codex, was pyramid contains 52 panels corresponding to the Calen- certainly used for divination purposes. The correlation dar Round: 52 × 365 = 73 × 260 = 18980. During an between the 338-day period and the Tzolk’in gives rise equinox, the Sun casts a shadow (7 triangles of light and to a 13-Tzolk’in cycle that has not yet been identified in shadow) on the northern stairway representing a serpent the Codices. The 13 Baktun cycle originates from the snaking down the pyramid (Figure 1). A 1820-day cycle correlation between the Tzolk’in and the Baktun. The or 7 Tzolk’in = 5 × 364 is present on pages 31-32a of the 3276-day cycles was chosen to commensurate with the Dresden Codex. This may also correspond to 32760 = planet synodic cycles, the Tzolk’in, the 2340-day div- LCM(360,819) = 7 × LCM(260,360). inatory cycle and the Tun. These results indicate that The study presented here describe a possible explana- Mayan astronomers calculated the synodic periods of all tion of the origin of the Long Count Calendar (LCC) the planets visible to the naked eye, as well as the basic and its connection with the Tzolk’in and Haab’ calen- cycles for solar/lunar eclipse prediction, the pentalunex dars. It provides a better understanding of the 3276- and the lunar semesters. day cycle, the coincidence of the Kawil and the 4 colors- ∗ e-mail: [email protected]

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