ASTRONOMY AND MATHEMATICS
all rights reserved, 2000
Although systems of mathematics
have been around since man learned to count on his fingers and toes, the Mayans took the science two steps further. First, they understood and put into practice the concept of the number zero. Secondly, they calculated where in the positive and negative numbering system that the
zero should go. It is actually unknown if the Mayas originated the concept of
the number zero, or if they borrowed it, or even developed it further from the earlier Olmecs. Much of the culture of the Mayans was passed down from the Olmec civilization of Mexico. The Mayas were a specific group that “came out” of that culture and continued
to advance in every area known at that time.
Although the Hindu also
used the zero very early in the onset of modern mathematics, it did not appear in their culture until three centuries
later. Mayan mathematics are the most sophisticated that have ever been developed in this
hemisphere. No doubt early man counted on his fingers and toes (as one friend of mine still does). Perhaps this is the reason that the Mayas
used the vigesimal system (20, ten fingers + ten toes) of mathematics
in lieu of the decimal system (10) that is utilized today. This means
that instead of out system which uses 1, 10,
100, 1,000, and 10,000, the vigesimal would have used 1, 20, 400, 8,000, 160,000, 3,200,000, et cetera.
Their system permitted them to express very large numbers by using a limited number of characters. For example,
dates on their “long-count”
calendar which used a date span of over fifty centuries, could be expressed simply, accurately, and concisely.
The Mayan numbers consist basically of only 3 symbols.
A dot which represents the number 1, the bar which represents 5, and the horizontal shell which represents zero. For
additional clarification, it must be said that a shell with one dot above it represents 20.
The bars may be drawn either horizontally or vertically, side by side. A dot, or up to 4 dots may then be placed either on top of the horizontal bar(s), or to the left of the
Here are some examples. The number 3 would
be written by placing 3 dots side by side horizontally. If a bar were placed
under the three dots, it would then become the number 8. If three bars are placed
in a parallel fashion, either horizontally or vertically, this would be how the number 15 is written. The maximum number of bars and dots together would be the number 19.
It would be written as three horizontal (or vertical) bars placed one
over the other (or to the left if using the vertical positioning method); and above these three bars, in a horizontal manner
there would be four dots. In counting this number 19, one would see three bars
(15) plus four dots (4), totaling 19.
number would be represented if one were to see 2 horizontal bars, with one dot over them?
That is correct. It would be the number 11. Would you ever see four bars? No. Why not? That might be 20, correct? Yes, but it would not be
written like that. Twenty is written as a shell with one dot above it. Do you think that the number forty would be written as a shell with 2 dots above it? Yes it would.
Let’s mention the calendar
numbers. One day is called a kin. There
is a symbol for the kin. Twenty days, which is a Mayan month is called a uinal. There is a symbol for uinal. Eighteen
uinals is a tun, which is a solar year. This is 360 days. An additional short month of 5 days of unlucky days are added to the tun to give 365 days. There is also a symbol for the tun.
Now, for larger time counts there
are other symbols. Twenty tuns (solar year) make up a katun (20 years). Twenty katuns make up a baktum (400 years).
Why all these twenties? Remember that their mathematics is a vigesimal
system, not a decimal system. Thirteen
baktuns is 5,200 years. 5,200 years is the length of their “long count”
calendar, marking time from the creation, to the final sunset. But what is with this thirteen?
Isn’t it somewhat out of place when we are talking twenties?
The Mayas had some sacred numbers
such as 13, 20, 52, and 400. Thirteen was the number of original Mayan gods. Twenty being the number of mankind (remember the fingers and toes?) Fifty-two years are a “bundle”. This was similar
to our “century”, and each 52nd year was special occasion, as is discussed in another topic. Four hundred was a sacred number because is was the number of Mayan gods of the night.
What if the Mayans wanted to count in much larger number of years? Did they have such a system? What if they wanted to refer
about the period of the dinosaurs which happened some 65 million years ago? Could
they do that? Yes, they actually could do that.
It has been mentioned that a baktum was 400 years, and that 13 baktums was 5,200 years.
Other time spans were the pictun, calabtun, kinchiltun, and the analtum.
pictun was 8,000 years; a calabtun 160,000 years; a kinchiltun 3,200,000 years, and an analtun was 64,000,000 years. They could almost talk about the dinosaurs which existed c. 65,000,000 years ago.
The Mayas were every bit as interested in the planets as were the Egyptians.
While it seems that the Egyptian’s interest was more in the deities alone, the Mayas were also steeply engaged
in actual measurements and alignments of the heavenly bodies. Their precision
is mind-boggling even today. The solar orbit of a planet is the time that a planet
takes to go from a specific point in its orbit, revolve around the sun, and return to the point from which the measurement
began (a solar year for that planet). A synodical orbit is different. The synodical year (orbit) of a planet (other than the Earth) is the time a planet takes to travel from
a specified point, “as observed from the Earth”, around the sun, and
return to the same point of earthly observation. The ancient Mayas calculated
the annual orbit (synodical year) of the planet Venus to be 584 days. With the
precision of measurements we are able to make today, using the largest telescopes and computers, etc, we now know that the Mayas were in error. The synodical orbit
of Venus is not 584 days, but rather only 583.92 days. They missed it by
one hour and 55 minutes. Or do you think they merely rounded it off to
the nearest (vigesimal) point? It takes 2,922 days (8 years) for the Earth, Venus,
Sun, and the stars to return to the same alignment. It may be significant to
note that Venus appears as out “evening Star” for 260 days, disappears under the horizon for 50 days, then re-appears
as our “morning star” for another 260 days. Is it coincidence that
their ritual calendar has 260 days in it?
The Mayans were intensely interested
in the planet Venus. It has been noted that major battle were planned according
to positional points of Venus in the sky.
were made a t the time that Venus’ appearance was the dimmest. This would
be immediately after the plants Superior Conjunction.
Most of the Mayan cities were located south of the latitude 23 degrees. This means that twice a year, the summer and winter solstice, that the Sun passed
directly overhead, casting no shadows at noon.
This did not go un-noticed. This is reflected in the congruence of their
260 day calendar cycle with their 365 day calendar cycle. This tropical congruence
has an annual error of only 19 minutes.
Minute details were recorded by
the astronomer-priests in the observatories, no doubt using devices casting shadows.
A number of these books of recordings have been lost. A few survive
until today. These books of records
are called “codices”. The Dresden Codex which survived, is
similar to a present day almanac. In it is recorded the full cycle of Venus.
In practically all of the Mayan cities, one finds that the ceremonial temples
are accurately aligned with “true north” and south compass directions. Note
that there is a slight difference between “magnetic” north and “true”
In the same way that the Egyptian Pharaoh Ramses had his great monument constructed
and aligned so that on his birthday, the rays of the sun at sunrise shot into a deep cave and fell impressively upon the chest
of an elegant statue of himself, the Mayas too, did something similar.
In the Yucatán Peninsula, in the great Mayan city of Chichén Itzá, they constructed
the steps ascending to the temple dedicated to the feathered serpent god Quetzalcoatl; and aligned it in such a way that on
the mornings of both the Spring and the Fall equinoxes, as the sun arises, a shadow like a serpent appears at the top the
steps. Then, with its head downward, as the sun rises, the shadow serpent slithers
down the sacred “mountain” (temple). Many people arrive each year
at these times to see this phenomenon.
Why did they go to so much trouble to align their temples? No doubt is was their way of honoring their gods, especially the sun god.
The king was viewed as the embodiment of the sun god just as was the Pharaoh of Egypt.
Whether his name is Ra, Quetzalcoatl, or Kulkulkan, the reason is the same – to honor their principal god.
In the Mayan site of Palenque, in the main
temple, the Temple of the Inscriptions, not too many years
ago, the tomb of its king Pacal was found.
As a matter of fact, this temple happens to be aligned with the sun in a
way that at the winter solstice, as the sun treks across the sky, it enters a doorway in the temple. It then makes its way to the rear of the chamber, and as the sun sinks lower on the horizon, the rays of
the sun seem to go down the temple stairs into the tomb of its deceased king, Pacal.
Astronomical alignments were also a point of importance to the Mayans. Speaking of Palenque, King Pacal’s
son, Chan-Bahlum, dedicated a group of temples on July 23, 690 AD. On this date,
a conjunction of the Moon, Jupiter, Saturn and Mars took place.
In Copán, I believe that all of the huge stelae, which are Mayan Kings, have
their hands in front of them holding something. What they are holding is a ceremonial
“bar”. This bar which has a symbol of the sun denotes their command
of the heavens. The Mayans were “big on jaguars”. Kings and priests often dressed themselves in jaguar skins.
The jaguar’s spots represented the stars of the heavens. Every 52
year period brought a fear the end of this period might be the last. It
was sort of a reoccurring Mayan Y2K.