The
Mathematical Concept of the Mayan Universe
". . . Sonó la Primera Palabra de Hunab-Ku, allí donde no había cielo
ni tierra, y se desprendió de su Piedra y cayó al segundo tiempo y declaró su
divinidad. Y se estremeció toda la immensidad de lo eterno . . . y su palabra
fue una medida de Gracia . . ." (Chilam Balam de Chumayel).
"God made the integers, all the rest is the work of Man. (L.
Kronecker)
I believe it has not escaped the readers' penetration the fact that the
mathematical board on which the Mayas performed their operations is a
two-dimensional array scheme: a matrix. In fact, the most simple definition of
a mathematical matrix consists of a rectangular array of numbers that allows to
perform mathematical operations.
Already in 1911, Roso de Luna (1) made an interesting mathematical
analysis of the Cortes codex showing that the table of glyphs of the tzolkín on
pages 13 to 18 of said Maya anahté constitutes deduced permutations of a
fundamental determinant and those permutations present a cyclic-arithmetical
progression by constant difference, in a series or closed cycle.
It is convenient to remember that the techniques of matrix calculus were
recently developed in the past century while the Maya mathematicians used their
boards several millennia ago. Moreover, the discovery of the mathematical
matrices have created a revolution in the scientific methods of calculus
incorporating a branch in Mathematics such as the Theory of Determinants,
Vector Analysis and Tensor Analysis. All of these, in conjunction, is one of
the most valuable instruments for Astronomy, Mechanics, Electricity, Fluid
Mechanics, Elasticity, Plasticity and Rheology, Relativity, Nuclear Physics and
other fields of modern science. It is in actuality, thanks to matrix and tensor
calculus, possible to discover the parameters and variables of any phenomenon,
not only in Euclidean space but also in Riemannian and Finsler spaces, and even
others that we can not yet imagine.
Now, the starting point in the discovery of determinants, matrices and
tensors was the confirmation that it is possible to place a set of numbers in
columns and rows and perform diverse operations on them. For instance, adding
and subtracting along diagonals to resolve equations with "n" unknown
quantities or separating the isotropic and deviating components of deformation
or radiation tensors. It is admirable to think that the Mayas had already
discovered this method and performed operations of addition, subtraction,
multiplication, division and calculations of square and cubic roots on a matrix
array.
It is impossible to ignore the conclusion that in the current use of an
archaic technique to execute the four basic operations we have not recognized
that they are part of matrix calculus. We see, for example, the multiplication
of two any numbers as:
|
A = k |
|
B = k |
|
Where p = 1, 2, . . . . . m |
|
And q = 1, 2, . . . . . n |
For these parameters A and B can be
multiplied in a matrix, the condition is that the number of rows of the matrix
A must be equal to the number of columns of the matrix B. In this case, both matrices
are equal to 1:
The product will be:
Developing this matrix, explicitly, it is obtained the following:
Suppose we have chosen a vigesimal system that limits the range to 5, k
It can easily be seen that the matrix which would form the values of kpr
and krq is the following:
In explicit form, this matrix would have the following numerical values:
Therefore, if the addition of the subproducts apr brq is
made precisely along the diagonals parallel to the secondary diagonal, we will
be grouping integers of the same scalar order.
The previous operation can be visualized also as the addition of two
internal products of one linear matrix and a columnar matrix placed above and
to the side of a unitary matrix:
Use for example, a multiplication of 148 by 325. For better
comprehension, we will do it in the decimal system, that is, for k1
= 100, k2 = 10 and
k
In this case,
|
a1 = 1 |
b1 = 3 |
|
a2 = 4 |
b2 = 2 |
|
a3 = 8 |
b3 = 5 |
Substituting it in the general matrix, we obtain:
The addition along the diagonals parallel to the minor diagonal would be:
In the Mayan system, the unitary matrix would be represented with a dot
in the center of every square of the board.
With such matrix we could make all the mathematical operations but yet we
would obtain more versatility if we use other fundamental matrices of
serialized numbers such as the ones seen in the garments of the Mayan priests
on which empty squares are shown alternated with others having two or more dots
in a regular order.
On the Inca garments it happens exactly the same, with such frequency
that Guamán Poma de Ayala has marked the squares with Arabic numbers.
The most simple decomposition of the unitary matrix would give:
The first of Kronecker is necessary for calculation of the isotropic
component of a tensor:
The second is the deviating component of the same tensor.
The Mayas represented these mathematical squares in their monuments,
garments and paintings. It came to be something like a universal symbol of the
Mathematics of the Universal web.
The stelae of Copán, Quiriguá, and the crests of Yaaxchilán and Palenque
show in a uniform manner a grid design with squares on the clothing of the Maya
high-ranked personalities. In the interior of these squares numerals appear or
the geometrical symbols that I explained in the previous chapter. It would be
redundant to mention the numerous examples that until now have been
misunderstood. Archaeologists and other scholars have explained these patterns
are a sophisticated design of a turtle shell. However, these explanations are
highly incongruent with the appearance of the symbols in those areas of the
facades that we know represent atmospheric space and of the sidereal cosmology
or on the top of masks which these same archaeologists have identified as gods.
Notable examples of mathematical squares are on the internal facades of
Quadrangle of Las Monjas, in Uxmal as well as in the House of the Governor, the
masks of Xkichmook, the intercolumns of Kiuic, the arch of Labná, Las Monjas of
Chichén Itzá and other buildings and stelae of the New Empire.
The universality of the abstract numbers, a concept expressed with such
persistence for the Maya culture in their ornamental squares has came to be
postulated recently in our century as the Dimensional Analysis and the Laws of
Similarity. Science has recognized that there is in the internal mechanism of
any event determined mathematical relations which are independent of space,
time and mass. The discovery of this principle has made possible the deductive
process of numerous fundamental equations in every order of the human
knowledge. Thanks to this there are hydraulic models, structural models, analog
computers and a rational compilation of statistical data.
But to speak of Dimensional Analysis it is necessary to enter into
another aspect of the mathematical thinking of the Mayas. I mentioned in
relation to the units of measure that the Mayas did not seem to have the same
concept of time as ours. This would seem strange if we take into account an
overwhelming majority of inscriptions which have relation with records of
periods of time. However, in the Maya language there does not exist the word
"time." The most common expression is "kinil" meaning in
relation to the sun, or days. With a brief analysis of all Mayan words in
relation to time, it is convincing enough that there is no implicit notion of
what we actually experience as time. The applied words to the future mean only
"from here to more" as in UCHMAL or simply a hopeful "may
be" as in COCHOM. The expression for "ancient" (UCHBEN) does not
involve a meaningful root of time. However, there is a specific word for
"movement" in PEC and PECIL, which is such a highly significant fact
that the other use of this word is to count circular things or recurrent cycles
(turns, bows, etc.) There is a proper designation for "acceleration"
(ZEBAL), "velocity" (ALCABIL), and deceleration (NICIL), and all of
these show t hat the Mayas think fundamentally in terms of movement and not of
time. The previous implications have an actual scientific transcendence.
If we reexamine our concept of time, we can convince ourselves that it
may be an abstraction that does not exist. We talk about measuring the time
with a clock and, in reality, we are comparing movements with the hands of the
watch which are coupled with the apparent movement of the sun and the
translation of Earth and its orbit. None of these measurements is strictly time;
in fact, it may be as well an external general factor affecting equally all
these movements - including those that control our organic and conscious
processes - without being aware that time has been stretched or compressed. The
total mechanism of the Universe may be suffering accelerations and isotropic
pauses in group and we could never perceive them. When we want to know the mass
of a body, we weigh it or measure its inertia. To determine its volume, we
calibrate its dimensions. To know the contained thermal energy we use a
thermometer. However, to describe its movement, we use an equation where space
is involved and a conventional variable called "time", that is in
fact, movement. We can imagine units of valid universality for mass, space, energy,
and movement (i.e. speed of light), but not for time.
It is possible that the mathematical thinking of the Mayas forces us to
discuss the same bases of the dimensional analysis and we decide to abandon our
temporal postulates and replace them with kinetic principles. The dimensional
concept that the Mayas had about the Universe, which is based on four
fundamentals units, is traduced in the representation of matter by a square, is
an idea which has been confused with the affirmation that the Mayas believed
Earth was flat and square. The four elements are: MATTER, MOVEMENT, SPACE AND
ENERGY and correspond to known symbols of earth, water, air and fire. It
represents, besides, the four states of matter: solid, liquid, gas and plasma
transformation. But, I will not take this opportunity to conjecture that the
Mayas had intuitively known atomic fusion and fission. I reserve this
hypothesis for later confirmation of Project Ahau.
Finally, there is a third contribution of the Maya thinking that
justifies our hopes for those who think that through the study of our great
cultures of the past we can find solutions to our existential and philosophical
problems. I refer to Synergetic Arithmetic. It is interesting when I was doing
my Web research on matrix calculus, I came across Buckminster Fuller's F.A.Q.
page maintained by Christopher J. Fernley.
He directed me to Fuller's greatest publications to Mathematics, Synergetics
and Synergetics 2. Notice the titles of Fuller's work!
It is not easy to explain something that yet has not appeared in our
mathematical science and we can barely visualize. We attempt to understand and
give expression to factors intervening in the constitution of a functional
organism. Take a simple example: an automobile.
The pieces that constitute our vehicle need to satisfy certain
requisites. They must be able to perform a specific function and to work well;
they must be complimentary among themselves to be compatible. It is enough that
if a spark plug does not work correctly the car motor will not be able to
develop its full power. Moreover, we might have a good spark plug but the wrong
one for the model of car or one that is incompatible with the rest of the
components. All this would render impossible the harmonious working of the
group.
None of the branches of Mathematics have introduced the scientific field
of Synergetics. It is evident we can add in an algebraic form the components of
an organism to obtain the functional vector of the set. A bunch of iron parts
is not a car and a simple accumulation of organic tissues is not a human body.
Not even the vector or tensor sum allows us to differentiate between an
organism that functions and another that does not. There is a qualitative
change that operates instantaneously in which all the integrating elements
combine their efforts and perform harmoniously a new function, of superior
hierarchy, with an entropic reduction that represents a higher degree of
efficiency and transcends any individual possibilities of any of the components
of the mechanism. When this happens we have obtained the synergetic sum, but
for this to happen it is necessary to have a coordinating factor, a direction
that the Mayas expressed in their numerical symbolism. When we understand
Synergetic Arithmetic, and can express its precise formulas, additions,
subtractions, multiplications, divisions, integrations, and derivatives of the
factors and synergetic variables, we will be in a position to rationally plan
our human robots, and production machines, and our governments might appreciate
the magnificent synthesis of the Mayan philosophy relating the identity of Man
with the Cosmos.
I have started this work with the description of simple - almost childish
- techniques of calculus that the Mayas must have used. I can not conclude this
chapter without externalizing the conviction that that simplicity, far from
being a hint of our minuscule knowledge of a primitive culture in the Third
World, it represents a wisdom of knowledge we are far from reaching.
It disturbs me deeply to think that the great Mayan civilization
disappeared mysteriously three centuries before the Spanish conquerors arrived
leaving only their enigmatic stelae and deteriorated monuments. We ask as to
the cause of their disappearance and we think in terms of ecological,
psychological or historical determinants without finding a satisfactory answer.
We ask to the mute inscriptions, hieroglyphs, and scrutinize the pages of the
Chilam Balam's books and do not understand the mysterious reasons why the Maya
culture perished, leaving a pathetical scream carved in the rocks of a
Rainforest that could not devour everything.
And, suddenly, from my Mayan intuition, from my memories of Maya DNA
accumulated through long centuries, rises a light of comprehension reading the
Chumayel Codex and explained to me by our race that the Itzaes magi did not
disappear . . . they left!
And a hope crosses the Maya night when I hear the prophet saying in the
12 Ahau Katun the message will emerge from the white stones and there will be
scientists who will understand the language of Zuyua. As that date approaches,
I formulate an ardent desire that our Mayan wisdom nourish the roots of our
historic personalities as a nation and as an awakened giant behind a galactic dream
we extend to the rest of the world the formulas of our universal philosophy
that were once upon a time the solid columns of millennial greatness.
Pakal, Nexus Tzacol
(The author would like to express his appreciation to Bob Fritzius who kindly
edited the chapter.)
REFERENCES
Roso de Luna, Mario. La Ciencia Hierática de los Mayas. Librería de Pueyo. Madrid. 1911.
Copyright (c) 1997. All Rights Reserved.