Mayan cycles

Calendar

Ancient Egyptians realized that 365.25/365 = 1461/1460. Period 1460 years was called Sothis (Sirius).
Similarly Mayans has found that 365.2422/365 = 1508/1507. So, 1507 tropical years last integer number of days:
1507*365.2422=1508*365=550420 days.

Number 550420 has these divisors:
550420 = 2^2 *5 *13 *29 *73.
Mayan selected 13 (13 god numbers) and 20 (20 god days). Their combination 13*20 = 260 days was called "tzolkin".

Conjunctions of Earth with Mars repeat with period 780 days: 780 = 2^2 *3 *5 *13 (3*tzolkin).
Integer period of Earth, 365 days: 365 = 5 *73.
Mayan used the prime divisors (2,5,13,29,73) completed by other small numbers (3,7,9,...) for deriving of cycles.

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Tzol'Kin

Sacred cycle of Mayans and Aztecs, z=260 days. It is probably certain "quantum" for synodical periods of inner planets and moon.
Tzolkin is equal to third of synodical period Earth-Mars and to 3/4 of ecliptic year.
50*z make 3 orbital periods of Jupiter, (so 600*z make Babylonian period 427 years).
Tzolkin is also equal to c. 4/9 of (V,E) and 19/2 of sidereal moon.

Stable resonance (-V/3, E, R/2) has period 129.94 days, i.e. z/2.

 

Calendar rounds

Calendar round

The smallest common multiple of tzolkin (260-days) and haab (365-days):
nsn(365,260) = 18980 days = 52 Haab (51.965 years).
Approximately 52-year cycle seems to be combination of 4-year period of Venus with 13 year period of Mercury.

Ø       Period 4 years:

Period of moon tides (Wood,4.001 years), 5/2*(V,E), 99/2 lunations:
99*(E,L) = 99*29.53059 days = 2923.53 days =8.00418 years,
5*(V,E) = 5*583.9214 = 2919.61 days = 7.99345 years.
(E.g. conjunctions V-E: 6.4.1977, 3.4.1985, 1.4.1993,...).

Ø       Period 13 years:

Mercury returns to the same (Earth) season with period c. 13 years (mean value 13.006 years, Smith,Bruce, 1972).
E.g. 1990 and 2003 begin with these conjunctions:
9.1.1990, 3.5.1990, 8.9.1990, 24.12.1990, 14.4.1991, ...
11.1.2003, 7.5.2003, 11.9.2003, 27.12.2003, 17.4.2004,...
During 41 conjunctions Mercury-Earth Mercury completes 54 (i.e. 41+13) orbits. Mayan megalith with 43 stones (2 reference points)?

Calendar round (18980 days) differs from 52 orbits of Earth (tropical 18992.6 days, sidereal 18993.3 days) by c. 12.6-13.3 days.
To keep exact length of year, c. 13 day corrections are necessary every 52 years, i.e. 1 day every 4 years.
Half period of 65 conjunctions Venus-Earth:
65*(V,E)/2 = 65*583.9214/2 = 18977.45 days (51.95741 years).

Aztecan century

Two calendar rounds:

Ø       13*2920 days = 8*4745 days = 2*18980 = 37960 days (104 years)

Ø       46*260 days = 65*584 days = 104*365 days = 37960 days (104 years).

Conjunctions M-V-E occurred in years (precision 40°):
-3631.23, -3527.18, -3423.41, -3319.35, -3215.28, -3111.56, -3007.46, -2903.78, -2799.70, -2695.97, -2591.91, -2488.14...
Data separation makes 103.65-104.15 years.

Selected configurations M-V-E: (conjunction & alignment, accuracy 40°, interval 0-2100) 

   5 Nov 12 AD (103.77276) #1723200  (   5.86721)    109 Dec  1 AD (104.05202) #1761205  ( 109.92145)
 213 Sep  6 AD (103.76454) #1799105  ( 213.68821)    317 Sep 25 AD (104.05202) #1837110  ( 317.74245)
 421 Oct 26 AD (104.08487) #1875127  ( 421.82955)    525 Jul  8 AD (103.69884) #1913003  ( 525.53060)
 629 Aug 17 AD (104.10951) #1951029  ( 629.64234)    733 Apr 17 AD (103.66598) #1988893  ( 733.31053)
 837 May 12 AD (104.06845) #2026904  ( 837.38120)    941 Feb  9 AD (103.74812) #2064798  ( 941.13153)
1045 Feb 28 AD (104.05202) #2102803  (1045.18578)   1148 Dec  4 AD (103.76454) #2140703  (1148.95254)
1252 Dec 26 AD (104.06023) #2178711  (1253.01499)   1356 Oct  1 AD (103.76454) #2216611  (1356.78175)
1460 Oct 20 AD (104.05202) #2254616  (1460.83599)   1564 Nov 17 AD (104.07666) #2292630  (1564.91488)
1668 Aug 15 AD (103.71526) #2330512  (1668.63235)   1772 Sep 22 AD (104.10130) #2368535  (1772.73588)
1876 May 27 AD (103.67420) #2406402  (1876.41229)   1980 Jun 22 AD (104.06845) #2444413  (1980.48296)
2084 Mar 16 AD (103.73169) #2482301  (2084.21686)

 

Inequality (E,Ln,-R/2)= (R/2,Ey/1)= (1.88089/2, 0.94899/1) = 104.4 years.
From (Ln/4,-E/5,R/9) = 0 follows (E,-R,Ln/8) = 26 years. So (Ln/4,2*(E,R)) = (4.65, 4.27) = 52 years.

Period of 65 conjunctions Venus-Earth:

Ø       65*(V,E) = 65*583.9214 = 37954.9 days (103.91482 years)

Also inequality (J/13, S/32)= 104.0 years, i.e.(5/J-4/S)/(4/J-5/S) = 5.97122/3.49995 = 1.706; (12/7=1.7143).

Period 676 years

Cycle of 13 calendar rounds (13*52 years), "oxalajuj tiku". I.e. 57*J, 23*S, 34* (J,S).

Natural month

Tun-Uc

Calendar with 28 days cycle of ‘natural months‘.

Sometimes it appears to be more favourable to divide year to 13 months. Orbital period of Venus is approximately 8/13 of our year. For observer on the Earth, every (V,E), i.e. c. 583.92 days, Venus passes Earth (in direction of Earth motion). Moon passes this place (against Venus direction) every c. 29.53 days. So, with regard to Venus, orbital period of Moon is little bit shorter.

It holds: ((E,L),(E,V)) = (29.53, -583.92) = 28.11 days.

Tzotz tun

Calculational year, bat cycle, 364 days = 4*91 days = 13*28 days.

Mayan add k*364 days (Dresden codex, F 45) to date 8.17.11.1.10, k=(2-5),10,15,40,60,80.
Multiple 20*364= 7280 days= 19.932 years is the period of symmetric configurations, alignment of axes J-S and U-N.

Other periods

Period 65 days

Greatest common divisor of periods (V,E) and (E,R) is c. 195 days (780/4 = 585/3 = 195), i.e. 3*65 days.
Mayans derived some data from multiples of 65 days:
65 = 5*13 (=780/12 = 585/9 = 260/4).  Also (V,E/2) = 975.3 days = 5* 195 days = 15*65 days;
11 synodical months (324.83649 days) is c. 325 days = 5*65 days (177+148 days).

Katun

Mayan katun, 20*18*20 days = 7200 days (19.7125 years) looks like modulated cycle of conjunctions (J,S):
[(J,S),(5J,2S)] = [19.86,2650] = 19.71 years.

Beats of inner planets R1 = (E/4,M) = 6.575 years, R2 = (R/3,V) = 32.82 years, where R2/R1 is c. 5/1.

It also holds approximately: R1 = [J,S/2] = 6.570 years. Three periods R1 make c. 19.71-19.72 years.

Three katuns, i.e.59.14 years, cover 37*(V,E).
E.g. dates 9.10.16.10.13, 9.11.16.10.13 (dates of "cosmic monster" from Yaxchilan) differ by katun.

 

Period 819 days

Synodical period (R,J) is c. 816.43 days (2.2353 years).
One of the periods of atlantic oscillation makes c. 27 months, i.e. 2.25 years.
Alignments E-M-R (conjunctions M-E-R) repeats with period c. 820 days (2.25 years);
7*(M,E) = 7*115.87 = 811.1 days, 8*(M,R) = 8*100.88 = 807.0 days.
Nodal (draconic) months: 30*27.1222=816.37 days.
Similar configurations of E-M-R occur with period 2*J = 25*Ey (c. 23.725 years = 10.5*2.2595 years).

819 days = 7*9*13 days; ( Number 9 control cycle of "lords of nigths".)
7*13 days = 91 days (364/4),
9*13 days = 117 days, approximation of (M,E)=115.87 days

E.g. see 9.15.19.1.1, 9.15.19.14.14, 9.16.1.0.0 (stela 11, Yaxchilan).

 

Period 2200 days

Period 2200 days (6.023 years) is a common multiple for M and (M,E) M/(M,E) = 19/25 (Bohm&Bohm).

Period 3120 days (12 tzolkin)

Ecliptic year and conjunctions Earth-Mars are synchronized with period Y = 4*(E,R) = 9*(E,Ln), i.e. 8.5413 years (c. 3120 days). Let us note, that 12*z = 12*260 days = 3120 days.

E.g. 10.11.5.14.5 - 9.17.8. 8.5 = 1521285 - 1421445 = 99840 days = 32*3120 days (first two "serpent numbers").

Perioda V-E-Ln-R

Earth(E), Mars(R), lunar line of nodes(Ln) and Venus(V) repeat the same configuration with period:
6*Y= 72 tzolkin= 72*260 d= 32*585 days= 18720 days (51.25 years).

 

Cycles 130, 260 and 520 years

Period Y is approximately 16/15 of 8-year period. Common multiple (15*8.54,16*8.00 years) equals to beats (8.0, 8.542), i.e. makes c. 125-130 years.

Aztecan century 37960 days (104 years) contains 65 conjunctions V-E. But conjunctions M-E in the meantime shift, by c. 0.587 (M,E).
Did Mayans try to correct this deviation?
189800 days =10*18980 days =1637.9366*115.8775 days (520 years).

Perioda 3742 let

Period V-E-Ln-R and calendar round differ by one tzolkin. Common multiple:

Ø       1366560 days = 72 calendar rounds = 73 period V-E-Ln-R;

Ø       1366560 days = 25* 32* 5* 13* 73 days = 5256 tzolkin= 3744 haab (3741.44 years)

Ø       1366560 days = 117* 32 haab = 2340*584 days = 1752*780 days.

Planetary interactions