Mayan cycles

• Calendar
• Month
• Period 65 days
• Tzol'Kin
• Period 819 days
• Period 3120 days (12 tzolkin)
• 13 tun
• Katun
• Period of eclipses
• Period V-E-Ln-R
• Calendar round
• Aztech century
• Cycles 130, 260 and 520 years
• Baktun
• Period 676 years
• Period 3742 years
• Period 5125 years

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) also as bases of other cycles.

Month

Mayans record integer duration, 29 or 30 days. In fact period of lunations (E,L) vary: 29.26-29.80 days. (Main periods of change are c. 413 days and c. 8*413 i.e. 3300 days.)

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).

Tzol'Kin

Sacred cycle of Mayans and Aztechs, 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.

• 259.52061 days: (V,E)*4/9 = 583.92138 days *4/9
• 259.55578 days: 9.5*L = 9.5*27.321661 days
• 259.95536 days: 1.5*J/25 = 17.7929747 years/25 = 0.711719 years
• 259.965 days: Ey*3/4 = (E,Ln)*3/4 = 346.620 days *3/4
• 259.9787 days: (E,R)*1/3 = 779.9361 days *1/3

Period 819 days

Synodical period (R,J) is c. 816.43 days (2.2353 years).
Conjunctions M-E-R repeats with period c. 821.8 days (2.250 years), alignments E-M-R have the same period.
Similar configurations occur with period 2*J = 25*Ey (c. 23.725 years = 10.5*2.2595 years).

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.

13 tun

Period 13*18*20 days = 4680 days (12.81 years).
Double period of conjunctions V-E-R :
8*(V,E) = 12.789516 years = 2* 6.394758 years,
6*(E,R) = 12.812088 years = 2* 6.406044 years.

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.

Period of eclipses

Period 46 tzolkin, 46*z= 11960 days (32.7447 years). It corresponds to 405 lunations (405*29.5306 days).
It reminds Douglass period of annual rings (32.8 years).
Period of beats (R/3,V) = 32.82 years.
Three Jovian periods modulated by Babylonian period make:
[3*J, B]= [35.586,427.1]= 32.85 years.

Examples of Solar eclipses in the Middle and South America:

Date 1	Date 2	Interval
7.10.804	6. 7.837	837.541-804.794 = 32.747 years
26. 9.805	25. 6.838	838.511-805.766 = 32.745 years
10. 2.826	9.11.858	858.886-826.141 = 32.745 years
25. 5.830	21. 2.863	863.170-830.426 = 32.744 years

Period 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).

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.

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,...).

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 megalit 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).

Aztech century

Two calendar rounds:
13*2920 days = 8*4745 days = 2*18980 = 37960 days (104 years),
46*260 days = 65*584 days = 104*365 days = 37960 days (104 years).

Conjunctions M-V-E occured 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.

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.

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 of 65 conjunctions Venus-Earth:
65*(V,E) = 65*583.9214 = 37954.9 days (103.91482 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.

Aztech 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).

13 Mayan katuns is 256.3 years. Triple conjunctions J-S (observed from the Earth) returns after 257-258 years.

Baktun

Period 20*18*20*20 = 144000 days, i.e.394.25 years.
Baktun do not coincide with multiples of period Y (12*11960 = 46*3120 = 143520 days) nor with multiples of ecliptic year (416*(E,Ln) = 416* 346.6 days = 144186 days).

Period c. 144240 days is common multiple of conjunctions of inner planets:
144240 days = 185*779.68 =210*686.86 =247*583.97 =642*224.67 days.
Such duration has e.g. interval 9.19.5.7.8 - 10.19.6.1.8 (Bohm&Bohm). Baktuns step through seasons (4*394.25=1577 years).

Period 676 years

Cycle of 13 calendar rounds (13*52 years), i.e. 57*J, 23*S, 34* (J,S),
see Multiple conjunctions.

Period 3742 years

Period V-E-Ln-R and calendar round differ by one tzolkin. Common multiple:
1366560 days= 72 calendar rounds = 73 periods V-E-Ln-R;
1366560 days= 2^5* 3^2* 5* 13* 73 days = 5256 tzolkin= 3744 haab (3741.44 years);
1366560 days= 117* 32 haab = 2340*584 days = 1752*780 days.

Period 5125 years

Period 13 baktun, 12 Babylonian cycles.
1872000 days = 7200 tzolkin = 600 *Y = 260 katun.

Particular baktuns were numbered 1-13:
-----------------------------------------------------------
                                             13.   4 Ahau
 1.   3 Ahau     2.   2 Ahau    3.  1 Ahau    4.   0 Ahau
 5.  12 Ahau     6.  11 Ahau    7. 10 Ahau    8.   9 Ahau
 9.   8 Ahau    10.   7 Ahau   11.  6 Ahau   12.   5 Ahau
-----------------------------------------------------------
13.  4 Ahau

Origin is assumed to be at the beginning of 13 baktun (4 Ahau).

Correlation of Mayan and Julian calendar