Mayan astronomy - Cycles

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Calendar

Numbers

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:


Number 550420 has these divisors:

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 = 22 *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.

Tzol'Kin

Sacred cycle of Mayans and Aztecs, z=260 days. It is probably certain "quantum" for synodical periods of inner planets and moon.
Tzol'Kin - Correlation
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.


Tzol'Kin - Sequence
Some configurations (alignments) of the planets Venus-Earth-Mars follow each other
exactly with the period of Tzolkin:

9.12.2002 + TZOLKIN => 26.8.2003 + TZOLKIN => 12.5.2004
s20021209 s20030826 s20040512

13.5.2009 + TZOLKIN => 28.1.2010 + TZOLKIN => 15.10.2010
s20090513 s20100128 s20101015

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 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).

Period of rounds

Aztecan century

Two calendar rounds:

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...
Dataseparation makes 103.65-104.15 years.

Conjunctions M-V-E
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:
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).

Aztecan Period

Long count

Raster of dates

Mayan syntax of dates makes possible to observe accuracy of conjunctions of Earth (E) with Mars (R) and lunar line of nodes (Ln). These conjunctions E-Ln-R frame certain raster with period 12 tzolkin, i.e. 3120 days (8.542 years).
In the following schema the rows step with period of conjunctions Earth-Mars. Period 13 tun (18 tzolkin) is highlighted (bold).

    Beginning of Long count:
    -------------------------
    0. 0. 0. 0. 0 (      0)
    0. 0. 2. 3. 0 (    780)
    0. 0. 4. 6. 0 (   1560)
    0. 0. 6. 9. 0 (   2340)
    --------------------------
    0. 0. 8.12. 0 (   3120)
    0. 0.10.15. 0 (   3900)
    0. 0.13. 0. 0 (   4680)
    0. 0.15. 3. 0 (   5460)
    --------------------------
    0. 0.17. 6. 0 (   6240)
    0. 0.19. 9. 0 (   7020)
    0. 1. 1.12. 0 (   7800)
    0. 1. 3.15. 0 (   8580)
    --------------------------
    0. 1. 6. 0. 0 (   9360)
    .....
    End of Long count:
    .....
    -----------------------
    12.19. 2.12. 0 (1865760)
    12.19. 4.15. 0 (1866540)
    12.19. 7. 0. 0 (1867320)
    12.19. 9. 3. 0 (1868100)
    -----------------------
    12.19.11. 6. 0 (1868880)
    12.19.13. 9. 0 (1869660)
    12.19.15.12. 0 (1870440)
    12.19.17.15. 0 (1871220)
    -----------------------
    13. 0. 0. 0. 0 (1872000)

Period Y represents accurately value of 12 tzolkin in the Mayan calendar.

Y = 12z = 12∙260 days = 3120 days (8.5413 years)

Mayan syntax of dates allows us to monitor the accuracy of conjunctions Earth (E) with Mars (R) and node line of the Moon (Ln). Conjunction E-Ln-R creates a kind of grid with a period 12 tzolkin.

The following diagram lines proceed with periods of conjunctions Earth-Mars. Periods of 13 tons (18 tzolkin) is highlighted in bold.

Zero

Zero

Number zero may appear in mayan dates. E.g. 10.11.4.0.14 = 10.11.3.18.14.
Congruences
We can test connections of Mayan dates with help of congruences (remainder classes).
E.g. the following dates are congruent according to module 260 days (tzolkin):
Mayan date      D [days]   D/260   D mod 260]
 -------------------------------------------
 9.16. 0.13.17  1411477   5428.7577    197
 9.16. 4. 6.17  1412777   5433.7577    197
 9.16.12. 5.17  1415637   5444.7577    197
 9.17.19.13.16  1425516   5482.7538    196

Katun

Mayan katun, 20*18*20 days = 7200 days (19.7125 years) reminds cycle of conjunctions (J, S) = 19.86 years. Three katuns (ie. 59.14 years), cover approximately 37 conjunctions of V-E.

But it's - in all probability - entirely something else!? Foundation of the Mayan calendar "long count" is built of Katuns... (Also e.g. dates 9.10.16.10.13, 9.11.16.10.13, ie. dates of "cosmic monster" from Yaxchilan, differ by katun.).

When observing the conjunction of Moon and Jupiter perceive cycle 9 * (E, J) = 3590 days = 9.83 years. Average duration of two such cycles 2 * 9.83 = 19.66 years differ by Maya Katun only about 20 days (1 tun).

In the following outline approximate dates of conjunction of the Sun-Earth-Moon-Jupiter are listed. (A similar sequence - only about 14 days shifted - forms also the conjunctions of the Sun-Earth-Moon-Jupiter). We deliberately chose the sequence that ends at the time the strongest earthquake observed - May 22, 1960 (Valvidia in Chile):

     1891,60 1891 Aug  4 AD	1891,599	E: 313,7 J: 341,6 Moon: 134,7
   * 1901,46 1901 Jun 15 AD	1901,461	E: 265,5 J: 278,3 Moon:  75,5
     1911,25 1911 Mar 29 AD	1911,247	E: 189,4 J: 218,5 Moon:   0,4
     1921,11 1921 Feb  6 AD	1921,109	E: 138,7 J: 163,1 Moon: 305,7
     1930,97 1930 Dec 18 AD	1930,971	E:  87,3 J: 104,9 Moon: 252,9
     1940,84 1940 Oct 30 AD	1940,838	E:  38,1 J:  41,1 Moon: 216,2
     1950,62 1950 Aug 13 AD	1950,623	E: 321,2 J: 332,1 Moon: 142,3
   * 1960,40 1960 May 24 AD	1960,403	E: 244,2 J: 267,2 Moon:  56,1
          

Directions of conjunctions move in each step (with a duration of 9.83 years on average) about 60 dg, ie after 6 steps (3 Katun) returns to the same place.


:

Other aproximations:

Application of Long count

Earth(E), Mars(R), node line of Moon(Ln) and Venuše(V) repeat the same configuration with the period 6Y:

6∙Y = 72 tzolkin= 72∙260 d= 32∙585 days= 18720 days (51.25 years)

The difference of the first two "serpent numbers" 10.11.5.14.5, 9.17.8. 8.5 makes 32Y:

32∙Y= 192 tzolkin = 10.11.5.14.5-9.17.8. 8.5= 1521285-1421445= 99840 days (273.3 years)

Accuracy of Long count

It seems natural to consider as the basic points of the Long Count moments of conjunctions Earth with Mars.

Accuracy of long count
With regard to conjunctons E-R: conjunctions (E,R) do not last 3*260 days, but at average a less

(E,R) = 2.1353487 years, i.e. 779.9361 days

Difference 0.0639 days/780 days, i.e. 1 day/12207 days (33.42 years), c. 4 days/130 years.
Mayan tropical year 365.2422 days is more exact: today value is 365.24219879, difference 0.00000121 * 86400 s / year,
i.e.. 1 s/ 9.565 years (S.M.Brisbin 26.3 s/day?!).

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).

0-3
9.10.15.3.0 1 Ahau 13 Pax 1373460 1881852 440 Mar 25 AD 440,24 (E-Moon,Vlow)
9.10.17.7.19 1 Cauac 17 Uo 1374279 1882671 442 Jun 22 AD 442,48 Derived 819
9.10.19.12.18 1 Etznab 6 Yaxkin 1375098 1883490 444 Sep 18 AD 444,73 Derived 819
9.11.1.17.17 1 Caban 15 Zac 1375917 1884309 446 Dec 16 AD 446,97 Derived 819

m04400325 m04420622 m04440920 m04461218
4-7
9.11.4.4.16 1 Cib 4 Pax 1376736 1885128 449 Mar 14 AD 449,21 Derived 819
9.11.6.9.15 1 Men 8 Uo 1377555 1885947 451 Jun 11 AD 451,45 Derived 819
9.11.8.14.14 1 Ix 17 Xul 1378374 1886766 453 Sep 7 AD 453,69 Derived 819
9.11.11.1.13 1 Ben 6 Zac 1379193 1887585 455 Dec 5 AD 455,94 Derived 819


m04490316 m04510613 m04530909 m04551207
8-11
9.11.13.6.12 1 Eb 15 Muan 1380012 1888404 458 Mar 3 AD 458,18 Derived 819
9.11.15.11.11 1 Chuen 19 Pop 1380831 1889223 460 May 30 AD 460,42 Derived 819
9.11.17.16.10 1 Oc 8 Xul 1381650 1890042 462 Aug 27 AD 462,66 Derived 819
9.12.0.3.9 1 Muluc 17 Yax 1382469 1890861 464 Nov 23 AD 464,91 Derived 819

m04580303 m04600530 m04620827 m04641123 .....
46-49
9.15.19.14.14 1 Ix 7 Uo 1411134 1919526 543 May 18 AD 543,39 cycle 819 days (Yaxchilan)
9.16.2.1.13 1 Ben 16 Xul 1411953 1920345 545 Aug 14 AD 545,63 Derived 819
9.16.4.6.17 6 Caban 10 Zac 1412777 1921169 547 Nov 16 AD 547,89 Yaxchilan
9.16.6.11.16 6 Cib 19 Muan 1413596 1921988 550 Feb 12 AD 550,13 Derived 819
m05430518 m05450814 m05471116 m05500212

Natural month

Tun-Uc

c_csync 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).

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").

Period 18720 days (72 tzolkin)

Period of alignment 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).

Period 130 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).

Period 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).

Particular baktuns had numbers 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).

Baktuns step through seasons 4*394.25=1577 years ().

Period 3742 years

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

Period 5125 years

Mayan combined (orbital?) period c. 394 years and (Babylonian) period (of planetary configurations) into cycle 5125 years.

5125 years = 12*427.1 years = 13*394.23 years

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

Various recurrence of planetary positions are observable with the period of 13 Baktuns...
Baktuns (korelace 508392)

4 Ahau
0.0.0.0.0.0 4 Ahau 8 Cumku ( 0) 508392 3322 Nov 26 BC -3320,17 Start of dating
1.0.0.0.0.0 4 Ahau 3 Kankin (1872000) 2380392 1805 Mar 11 AD 1805,20 13.baktun
2.0.0.0.0.0 4 Ahau 18 Chen (3744000) 4252392 6930 Jul 21 AD 6930,56 26.baktun

baktun33211127 m18050311 m69300721



3 Ahau
0.1.0.0.0.0 3 Ahau 13 Chen ( 144000) 652392 2927 Feb 25 BC -2925,91 1.baktun
1.1.0.0.0.0 3 Ahau 8 Zec (2016000) 2524392 2199 Jun 13 AD 2199,46 14.baktun

baktun29270225 m21990613


2 Ahau
0.2.0.0.0.0 2 Ahau 3 Uayeb ( 288000) 796392 2533 May 27 BC -2531,65 2.baktun
1.2.0.0.0.0 2 Ahau 18 Kankin (2160000) 2668392 2593 Sep 15 AD 2593,72 15.baktun

baktun25330527 m25930915


3 Ahau
0.3.0.0.0.0 1 Ahau 8 Yax ( 432000) 940392 2139 Aug 27 BC -2137,39 3.baktun
1.3.0.0.0.0 1 Ahau 3 Xul (2304000) 2812392 2987 Dec 19 AD 2987,97 16.baktun

baktun21390827 m29871219


13 Ahau
0.4.0.0.0.0 13 Ahau 13 Pop ( 576000) 1084392 1745 Nov 26 BC -1743,13 4.baktun
1.4.0.0.0.0 13 Ahau 13 Muan (2448000) 2956392 3382 Mar 23 AD 3382,23 17.baktun

baktun17451126 m33820323


12 Ahau
0.5.0.0.0.0 12 Ahau 3 Zac ( 720000) 1228392 1350 Feb 26 BC -1348,87 5.baktun
1.5.0.0.0.0 12 Ahau 18 Xul (2592000) 3100392 3776 Jun 25 AD 3776,49 18.baktun

baktun13500226 m37760625



11 Ahau
0.6.0.0.0.0 11 Ahau 8 Uo ( 864000) 1372392 956 May 28 BC -954,61 6.baktun
1.6.0.0.0.0 11 Ahau 8 Pax (2736000) 3244392 4170 Sep 28 AD 4170,75 19.baktun

baktun09560528 m41700928




10 Ahau
0.7.0.0.0.0 10 Ahau 18 Zac (1008000) 1516392 562 Aug 28 BC -560,36 7.baktun
1.7.0.0.0.0 10 Ahau 13 Yaxkin (2880000) 3388392 4564 Dec 31 AD 4565,01 20.baktun

baktun05620828 m45641231



9 Ahau
0.8.0.0.0.0 9 Ahau 3 Zip (1152000) 1660392 168 Nov 27 BC -166,10 8.baktun
1.8.0.0.0.0 9 Ahau 3 Kayab (3024000) 3532392 4959 Apr 5 AD 4959,27 21.baktun

baktun01681127 m49590405



8 Ahau
0.9.0.0.0.0 8 Ahau 13 Ceh (1296000) 1804392 228 Feb 27 AD 228,16 9.baktun
1.9.0.0.0.0 8 Ahau 8 Mol (3168000) 3676392 5353 Jul 8 AD 5353,53 22.baktun


m02280227 m53530708



7 Ahau
0.10.0.0.0.0 7 Ahau 18 Zip (1440000) 1948392 622 May 29 AD 622,42 10.baktun
1.10.0.0.0.0 7 Ahau 18 Kayab (3312000) 3820392 5747 Oct 11 AD 5747,79 23.baktun

m06220529 m57471011



6 Ahau
0.11.0.0.0.0 6 Ahau 8 Mac (1584000) 2092392 1016 Aug 28 AD 1016,68 11.baktun
1.11.0.0.0.0 6 Ahau 3 Chen (3456000) 3964392 6142 Jan 13 AD 6142,05 24.baktun

m10160828 m61420113



5 Ahau
0.12.0.0.0.0 5 Ahau 13 Zotz (1728000) 2236392 1410 Nov 28 AD 1410,94 12.baktun
1.12.0.0.0.0 5 Ahau 13 Cumku (3600000) 4108392 6536 Apr 17 AD 6536,31 25.baktun

m14101128 m65360417