Keith M. Hunter

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.

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

Venus Orbital Period = 224.69526222 days

(365.24218408 x 224.69526222) / (365.24218408 – 224.69526222) =

583.92021140 days

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

583.92021140 / 224.69526222 = 2.598720621

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

365.24218408 x 8 = 2921.937473 days

Due to the lack of perfection in the 8 / 13 ratio one can of course see that this is not an exact match.

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

260 x 73 = 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.

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.

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