Mayan-Aztec Calendar: The Century Cycle & Earth-Venus Conjunctions
Keith M. Hunter
An Extended Use of the Mayan 260-Day Tzolkin
Continuing on from the previous essay, it is clear then just how the 260-day Tzolkin cycle of the Mayan people can be used to track and refine successive Earth-Venus conjunctions based upon a ‘lowest common multiple’ (LCM) relationship, where both orbital bodies return to an almost exact starting point after a fixed period of time. In the short term a Venus-Earth ratio of 13 to 8 was identified (13 / 8 = 1.625), with also a far more refined ratio of 3251 to 2000 (3251 / 2000 = 1.6255), denoting a more long range cycle.
In addition to this however, it is also to be acknowledged that the 260-day Mayan calendar cycle may be employed to most accurately track to extreme accuracy ‘bulk batches’ of periodic Earth-Venus Synodic type conjunctions. And indeed, evidence that the Mayan people did just this, including also the Aztecs of the later period, is very well established. An examination of their method is thus called for.
Earth-Venus Synodic Conjunctions
To begin, it is only proper that one identify just what a synodic conjunction is. Essentially, starting out with an initial alignment of the Sun, Venus, and the Earth, due to the fact that Venus is nearer to the sun and completes one orbit sooner than the Earth, over the course of time Venus will outpace the Earth such that it will eventually catch up to the planet and ‘lap’ it. The point at which this happens will be a further conjunction event. However, both planets will not in fact be at their original starting positions when this occurs, as they would under a LCM type conjunction.
In determining the time interval of a synodic conjunction between any two planets, a very precise formula is used:
(a x b) / (a – b) = Synodic conjunction time
Where: a = orbital period of 1st planet & b = orbital period of 2nd planet
By substituting in the known orbital periods for the Earth and Venus (designated respectively ‘a’ and ‘b’) into the above formula, the time interval for a successive synodic conjunction between the two planets is determined as follows:
Earth Orbital Period = 365.24218408 days
(365.24218408 x 224.69526222) / (365.24218408 – 224.69526222) =
To further understand the positions of the two planets one may simply divide this calculated value by the orbital periods of the Earth and Venus:
583.92021140 / 365.24218408 = 1.598720621
As one can see, the greater speed of Venus demonstrates that in catching up to the Earth it will have completed 2 full orbits and be 0.598720621 into its third, whereas the Earth itself will have only completed 1 full orbit and be 0.598720621 into its second. This shows quite conclusively then that in re-aligning in conjunction in this way (i.e. synodic) the planets are not positioned at their original starting points. That being said however, there is a further point to mention, and that is that there is a distinct association between the noted Earth-Venus synodic time interval, and the 13 / 8 LCM conjunction described previously. Mathematically, there are almost exactly 5 synodic conjunctions to 8 earth years:
583.92021140 x 5 = 2919.601057 days
Due to the lack of perfection in the 8 / 13 ratio one can of course see that this is not an exact match.
Mayan-Aztec Calendar cycles: The Sheaf & the Century
As one can see then from the above, the key time cycle of interest, the Earth-Venus Synodic interval, is equal to about 583.92021140 days. That the Mayan people and also the Aztecs were interested in accurately tracking the successive passage of such conjunctions is beyond doubt. Their method, as will be examined, is indeed somewhat ingenious.
Right up until the arrival of the Spanish in the New world in the late 15th century, there was a time cycle actively tracked by the Indian people of Central America, from very ancient times. Indeed, the time cycle in question is so old that the exact moment of its inception as an astronomical cycle, to be first recorded by the people of this region, is unknown. The actual name given to it is the Sheaf.
The Sheaf, as a time interval, is generated by carefully combining a calendar year of 365 days, with the Tzolkin ‘forward advance of Venus measure’ of 260 days. When each of these measures are evaluated, it is so determined that the lowest common multiple (LCM) associated with them i.e. the lowest number that they will each divide into without a remainder, is 18980 days. Thus, after exactly 1 Sheaf of 18980 days, both time intervals would re-synchronise, just as for example, a LCM planetary conjunction cycle would. This fact is evident in the following relations:
365 / 260 = 73 / 52
The fraction so generated here of 73 / 52 is the smallest fraction that the Year & Tzolkin can be reduced to. It is also the case that they are the numbers via which one reveals the LCM between the two cycles:
365 x 52 = 18980 days
As one can see then, after exactly 52 calendar years, 73 completed periods of the forward advance of Venus will have occurred, which will be the starting point for a new (Sheaf) round.
18980 days, Refined
In considering the time period of 18980 days, on the face of it, it does not appear to be of too much significance. However, a very careful examination of this time measure reveals a most exacting association to the Earth-Venus synodic period. This is revealed when one is taken to correct it by subtracting from it a time interval equal to precisely 1/100th of a single 260-day Tzolkin cycle i.e. 2.6 days:
18980 – 2.6 = 18977.4 days
With this newly derived figure in hand, one may then proceed to divide it by the measure of the noted Earth-Venus synodic period:
18977.4 / 583.92021140 = 32.49998823
Immediately upon viewing one cannot help but see that the returned value is almost exactly 32.5. And indeed were the measure of the synodic period to be 583.92 days precisely then this would in fact be the answer. Moreover, in considering this further one may also recall the basic 13 / 8 ratio, extended to a 2000 year cycle, as was noted in the previous essay: 3250 / 2000. Here then one may see a ready harmonic of the same sequence of numbers, involved in the ‘Venus side’ of the ratio.
It is of course very obvious though that the value of 32.5 does not denote a whole number of Earth-Venus Synodic conjunctions. However, by doubling up the refined Sheaf value: 2 x 18977.4, one obtains 37954.8 days, which does indeed contain a whole number of conjunctions: almost precisely 65 to be exact.
37954.8 / 583.92021140 = 64.99997647
Indeed, the full measure of 2 Sheafs (uncorrected) of 37960 days was actively recognised by both the Maya and Aztecs as a distinct cycle, known as a Century, being 104 calendar years of 365 days in duration.
The Importance of the Mayan-Aztec Sheaf & Century
In the stone carvings (stele) of the Maya and the Aztecs, the Sheaf time cycle of 52 calendar years is well evidenced amongst the ruins of their once populous cities. But more than this however, far from being an incidental astronomical cycle to be appreciated by just their learned men; throughout the Mayan and Aztec settlements of the entire region, right up until the arrival of the Spanish, the successive completion of each Sheaf period was actively celebrated by performing elaborate ceremonies that involved everyone living within their cities. Culturally therefore, the Sheaf itself as a time cycle was of immense importance of the people of this region.
Proceed to Part 3:
Venus Conjunctions, the Pleiades & the Sheaf