Nepoualtzitzin

The Prehispanic Computing System

M. Ed. María Elena Romero Murguía *

Introduction

The significance that the Nepoualtzitzin has for those who love and believe

prehispanic culture is alive, is based on its accuracy and proximity to Nature's

wisdom.

This counting system is closely linked to the deepest feelings and understanding of

our Indian roots, with our own way to "see" things from our own place. In other

words: our cosmogonist vision.

Therefore, any proximity to its study should be based on those principles that have

given body and cohesion to our contemporary culture.

From all the great variety of the Indian colorful expressions, we can state that the

knowledge of the Nepoualtzitzin is the thread that joins the dispersed beads.

We have been trying to unearth from its origins, the memory that will reestablish

the spatial-temporal continuity that was so brutally interrupted by the conquest.

We know form Indian Philosophy that the ancient wisdom of sensible interaction

with the Cosmos is based on the concept of "The Creating Duality" and the four

directions of space, that in Nahuatl is expressed by the symbol of "Naui Ollin"

(Four Movement) as the emblem of the "Fifth Sun", the name of the era we live in.

(Figure 1).

Figure 1 The Naui Ollin Glyph of the Solar Stone

Nowadays, with all the development of science, we tend to fragment reality and

therefore, its knowledge, forgetting that everything is part of a whole, as Indian

Cosmo vision states. But why did Indian Cosmo vision develop in that particular

way? Where does it come from? What is the scientific basis of this belief?

We also know that the ancient Olmecs and Mayans were known as great

astronomers who developed a special and advanced computing system. But how

did they come to this knowledge?

Think, for instance: What do we count? What was the first thing that man

counted? Whether we think about the formation and organization of primitive

communities, or the child psycho motive, affective and cognitive development, we

will identify similar processes of body recognition. First of all: What do I have?

What do I count with? (In my body). This will give us the necessary point of

perceptual reference, in order to later identify the other members of the family

and the community, as similar fellows, participating in the social count.

The recognition of the environment and the natural phenomena, and its cycles set

up the basis of counting: the rhythm, the beat of life, the repetitive pulse.

So, based on these cycles and their meticulous observation is how science is born.

That is why, when we think about the things that are counted, we realize that

after being aware of SPACE, we develop the consciousness of TIME and its

relationship with the relevant natural phenomena in order to survive.

The apparent course of the Sun through the celestial vault, from its daily rising in

the east to its setting in the west, the lunar phases and the cycles of the seasons

are probably the most generalized observations in all humanity for development of

Culture: Agriculture.

That is the reason why, from the link of SPACE and TIME, MOVEMENT is the

maximum prototype of life, rhythm and events. That is why this ancient

computing system was based on the learning of cycles, the observation of Nature,

the rhythm of the seasons, the meteorological changes, the movements of the

Sun, the Moon, the planets and the constellations; through these observations is

how the human being develops knowledge, which from the Mesoamerican point of

view (particularly the Nahuatl Culture), is called: "NEPOUALTZITZIN": the

wisdom of Counting and its registry with every thing that it involves.

Etymology

The word Nepoualtzitzin comes from the Nahuatl Language and it means:

NE = impersonal suffix: Somebody

POUAL or POUALLI = to Count

TZI-TZIN = Small and similar elements

That is to say: "Somebody that counts with similar and small elements.

In this way, we realize that the root "POUALLI" = to count is the most important

element that we have within the word. So if we think about all we can possibly

mean by the concept "counting", we can ask ourselves some questions such as:

- What do we count?

- How long have we counted for?

- How did we start?

- Who counts?

- Why do we count?

- What was the first thing that was counted? How?; Where?: What for? And so

on.

Let's think about all the different meanings that the verb to count has: to number,

to tell, to refer, to relate, to include, to compute, to consider, to rate, to classify, to

rely on, to depend on, to think, etc. That is, either in the mathematical sense, or,

as in any other language, in a historical and literary sense, because we "count"

(tell): stories, tales, legends, history, gossips, jokes, etc. That is to say: the

complete exercise of oral tradition that takes all its meaning and transcendence if

there's a record of that account, a MEMORY that will remember the count.

We count as a parameter, a reference, to give things their place, to order them, to

classify them in our mind and in the practical world. We count to distinguish the

whole from the unity, to differentiate, to distribute.

But counting is only possible if it is based on a reference.

From the link between the earth and life, there is Culture, which is understood as

the "cultivation" of the earth. That is why, in the first stages of civilization,

agriculture requires, as any other way of counting, the measuring of space, time

and movement.

From the record of movements in a certain space and within a relative time, cycles

are deducted, and the whole computing system makes sense. Continuity in

changes and changes in continuity. Relationships of cosmic consonance from

which its clues speak: symbols, myths, legends, stones, crystals, mountains, a

reflexion of the sidereal view.

Geometry, astronomy and rhythm are the basis of the prehispanic computing

system: NEPOUALTZITZIN

Origin

The origin of this prehispanic computing system has been traced from its Olmec

roots.

The word Olmec within itself has the clues of the system: OLLIN which means:

movement and MECATL: ("mecate") a henequen rope, a line or "lineage", and

"measure" (because the rope is used to measure).

So the other meaning of "Olmecs" besides referring to the people that used to live

in the "Country of Rubber" (Ollin-Ulli = Hule) could be "the Measurers of

Movement" (cosmic movement).

So the registry of agricultural cycles stands out in their calculations, with a whole

spatial relationship with emphasis on VOLUME, rather than on plan, as proposed to

Euclidean geometry used in the Pythagoras school, which was based on land

planning, due to the social and economical system that laid its foundations on land

property.

In the meanwhile, Mesoamerican geometry comes from theocratic social systems,

based on the knowledge of the trajectory of the Sun and the main celestial bodies

in relationship with the seasonal variations and the cycles of agriculture, and its

registry through Architecture, codices, and stones.

In such a way, we can find actual astronomical observatories among prehispanic

buildings that were planned to point out the equinoctial and solstice axes, marking

the risings and settings of the main celestial bodies.

Based on the development of this knowledge we built up a whole system of

computing time, space and movement, and therefore, its reflexion on semiology.

Nepoualtzitzin is not only an abacus. It deals with varied aspects of the

knowledge of cycles, such as: Astronomy, Geometry, Mathematics, Biology,

Epigraphy, and of course heliocentric Philosophy.

We can find a good example of this in the "Sun Stone" (Aztec Calendar) in which

the main central glyph (fig. 2) refers to the legend of the Suns, which is narrated

in the Chimalpopoka Codex.

According to the Nahua concept, there has been Four Cosmic eras previous to the

present one, which is called "Nahui Ollin". (See the four squares around the face

of the Sun in figure 1, starting form the upper right square to the left).

The First one: The Sun of Earth ("Naui Ocelot"), symbolized by a tiger and four

dots around it.

The Second Sun: The Sun of the Wind ("Naui Ehekatl"), symbolized by a bird

and its four dots.

The Third Sun: The Sun of the Rain of Fire ("Naui Kiyouitl") with the glyph of

the rain and four dots.

And the Fourth Sun: "The Sun of Water" ("Naui Atl") with its glyph and dots.

Sun of Earth

Sun of WaterSun of Fire

Sun of Wind

Sun

of

Move-

ment

Figure 2 Naui Ollin and the Five Suns

The importance of the 4 phases in all of Nature is a main principle in this system.

All the cycles have 4 different and well-defined stages, such as: the four seasons;

the 4 quarters of the moon; dawn noon, dusk and midnight; childhood, youth,

maturity and oldness; etc.

On the other hand, and as a natural expression, prehispanic math bases its

correlations on the observation of the human body. The human being as the

measure of everything, as the representative of Unity: has 4 extremities (two legs

and two arms), each one with a value of 5 (the five fingers and toes).

That is why the human body is symbolized by a square with a value of twenty

("Cempoualli" in Nahuatl means: (CEM = one; POUALLI = Count).

The number 20 represents a complete unity. Each finger is equal to 1, but each

thumb is equal to 5, meaning the opposing force, by which means it is possible to

handle tools, because of its prehensile quality that made the evolution of the brain

possible.

The ancient system of counting with vigesimal basis is still present in the Binary

system used in contemporary Cybernetics.

If we relate this to child development, we will remember that the child goes

through a phase of discovering his body, as a way of self-recognition and his

fingers play a very important role in learning how to count.

So, in a certain way, we can say the Nepoualtzitzin is a representation of fingers,

because it is the first obvious countable reference of the body. We find it very

natural to count with them.

Now, talking about the zero, it is very important to point out its concept is very

different from the so-called "occidental math". The concept of numbers suggests

to us the presence of a quantifiable quality. Only by knowing the quality can we

make a detection of the absence (Theory of Sets). So the seashell as the symbol

of the zero refers to the absence of quantity, by establishing the quality. That is,

the seashell shows us that the animal is not there anymore; it is dead. And

therefore, in this sense, it means transmutation, passing from a manifested form

to another, leaving the testimony of its transformation. The spiral in the seashell

tells us about the concluded cycles; the rhythm of the nature drawn through the

course of time. In this expression, the zero has the idea of movement in itself, in

contrast with the European concept of zero as "nothing", a negation.

The Manual Computer

The presence of the Nepoualtzitzin as an abacus of prehispanic origin nowadays is,

in a certain way, due to the 33 years of research to which David Esparza Hidalgo

dedicated his life. Traveling through the mountains of Mexico, he rebuilt many

Nepoualtzitzin (abacus). Some of them were made of gold and jade (chalchiuitl)

and others of wood or seashell. (fig. 3)

Figure 3 Nepoualtzitzin as an abacus

Some had a mobile mechanism that allows us to use it with one hand or the other.

This fact points out, again, the importance that Duality has in all aspects of

Mesoamerican cultures. As the way that nature is expressed, everything is in

constant contrast, opposition and complementation: day and night, right and left,

up and down, sun and moon, the two sides of the brain, and the different

processes they control.

Some of the ancient abacus found, were in the shape of bracelets (fig. 4),

especially those from the Mayan zone. One of them appeared on a painted vase

in Guatemala known as the "Nejar Vase".

Figure 4 Nepoualtzitzin as a Bracelet

These instruments remind us of the Oriental abacuses such as the Chinese, the

Russian, the Arab and the Japanese that are taught and used in schools and

businesses. And this fact does not seem to interfere with the great benefits of

computer technology and the big production of electronic calculators.

The Nepoualtzitzin is used in many different levels and with different purposes.

For example, at the market, the women handle them in order to count what they

sell. On another level the great astronomers and priests called Tonalpouhke used

the instrument as an aid in counting the great cycles, such as the orbits of the

planets of the solar system, the main constellations and the cycles of eclipses

(Saros).

We know through some of the inscriptions, such as the Dresden Codex, that

among the Mesoamerican cultures the registry and prediction of the eclipses is a

common practice (fig. 5).

Figure 5 Dresden Codex

The astronomers used to be prepared at the "Kalmekak" (a kind of school) from

the time they were very young, they were elected by their qualities and taught by

the great wise men.

That is how an ancient instrument based on a profound understanding of the

universe ledge sets the grounds for modern science.

So it is not a matter of trying to go back to the past, neither can we ignore it. A

culture, just like a tree, develops strength by being nourished from its roots. The

validity of an ancient culture lies in its real application in the everyday life.

Other Indian cultures such as the Incas also developed a similar system of

measuring and counting. The so-called "Kipot" is used as a device to count by

making specific knots in a rope. (Fig. 6)

Figure 6 Inca counting system known as Quipus

Description

As we see in the diagram (Fig. 7) the Nepoualtzitzin has 13 rows of 7 elements

(beads, seeds, spaces, or keys), three on the right section and 4 on the left

section by row, divided by a central division in such a way that 39 of them stay on

the right side and 52 of them on the left side.

Figure 7. Nepoualtzitzin Diagram

From the combination of these elements, we can get a large amount of rhythmical

expressions (cycles) of nature, particularly the measurement and registry of time,

a fundamental aspect that developed culture.

The obvious question to be asked is: why these numbers?

If we take 4 times 13 we have 52, the number of the weeks of the year, or 52

years: a "New Fire" (a cosmic ceremony that celebrates a cycle of the "Pleiades",

a group of stars from the constellation of Taurus that is seen at the zenith of

Tenayuca pyramid every 52 years).

If we take the number 13 and multiply it by 28 (the rounded average from

the sidereal moon orbit of 27.3 and its synodic orbit of 29.52) we get 364.

If we multiply 7 by 13 (number of elements of the abacus) we will get 91

(number of days in a season).

If we take 2 times 91, we will get 183 days, which is the cycle of corn.

If we take 3 times 91, the result will be 273, which is approximately the

number of days needed for a complete human gestation (10 cycles of a

sidereal moon orbit: 273 x 10 = 273. Or in other words, a Ritual Cycle of

260 -Venusian’s calendar- plus 13 days).

And if we multiply 91 x 4 = 364. (Fig. 8)

Figure 8 Cycles inside the Nepoualtzitzin

The synodic orbits of the solar system are multiples of 7 and 13.

The human body has in the upper half of the body 7 major joints that allow a

larger expanse of movement. They are the 2 wrists, the 2 elbows, the 2 shoulders

and the atlas to-occipital articulation.

And if we consider the lower half of the body, we will find the other major joints:

the ankles, the knees and the 2 Cox femoral articulations. So, adding them

together, we have a total of 13.

The number of orifices in the head is 7: 2 eyes, 2 nostrils, 2 ears and the

mouth.

The normal proportion of an adult body is 7 times his/her head. In addition

for some Indian groups: there are not 4 cardinal points, but 7 (north, south,

east, west, the zenith, the nadir and the center).

So we could go on and on, showing evidences of these natural mathematical

constants: 7 and 13, Duality and the middle point -the basis of the Nepoualtzitzin-

a prehispanic computing system.

Geometry

We have all seen how Indian sculpture, architecture, codices, paintings, textiles

and basket weaving are full of geometrical designs. Mitla, in Oaxaca, is a good

example of this relevant characteristic in the development of Mesoamerican

cultures.

These figures that, to the superficial observer, have their roots in the same

computing system we are discussing.

These ingenious designs fulfill a logistic function in computing time. Taking

Duality as a base, they create a quadrille, crossing lines and inserting their figures

in this format developing a special color and geometrical code with the same

principles of astronomy, thereby, creating a mathematical and geometrical art.

The development of knowledge with a geometrical basis reached really amazing

proportions with such creations as the Sun's Stone (Aztec Calendar), which is one

of the clearest examples of this science, art, religion and philosophy of the

Mesoamerican universe.

The complete design of the monolith expresses the most sophisticated calculus

and the multiple relationships in which these geometrical divisions are expressed.

So, if we could imagine the stone to be dismembered in its elements, we could

think about it as an enormous model composed of 7 concentric circles just like one

of our modern slide-rules.

So, if we could make and abstraction of the elements of the stone and could take

off the central symbol ("Naui Ollin") and the two serpents that surround the stone,

we would see 7 circles designed with different elements. Figure 8

Figure 9. The complete Sun Stone

In this way it would be easier to imagine the possibilities of the combination of

each one of the 7 circles, taking as a basis our fundamental numbers 7, 13, which

would be useful as basic multipliers, getting from this a great and amazing number

of cycles, such as: combinations of the orbits of the solar system, cycles of

eclipses, comets, and so on.

An example of this is the combination of the second and the 7th. circle, which

when multiplied by seven gives us the Saros cycle, with an adjustment of 33 which

was commonly used in astronomical calculations.

Nepoualtzitzin and Biology

Another area where we can observe the usefulness of this knowledge is Biology.

Here we can see the strong relationship between the registry of cycles and the

phenomena of life reproduction, whether human, animal or plant.

In this area the Moon is one of the main protagonist since, as we know, it has a

definite influence on: fertility processes, promoting rhythms of growth and decay

in the planet, the ocean tides, the surface of the earth, the length of the day, and

even the movement of the axis of the earth called "precession", and of course the

eclipses.

Even though there is a more detailed way of using the abacus as a reference of

gestation and the cycle of crops, we will only deal with the Mecatl System at this

time.

Mecatl System, that we described before, the upper section of the abacus is used

as a moon cycle registry counting the beads of the perimeter so that we come to a

result of 28 (Fig. 10), and we must use another abacus in order to register any

other gestational cycle.

Figure 10. Upper part of the Nepoualtzitzin

As we have said, geometry is for many, the very first "science" that humans

developed, as a product of methodical and systematical knowledge that had its

origin in the observation of nature and its forms.

An example of this is the rattlesnake that lives in the Mayan zones.

In the species called "Crotalus Durissus" or the Mayan subspecies "Tzabcan" (L.

Kaluber, San Diego, California 1956), basic geometrical shapes can be observed,

such as the square and the circle, forms that were the basis of the development of

the Mayan culture.

We can see a total of 52 scales in the perimeter, in combinations of 7 in each

quadrant, or 13 on each side of the square (see Fig. 11)

Figure 11. Canamayte (skin scales of the Mayan Rattlesnake)

The significance of these proportions and designs is obvious. There are many

examples of the use of this design such as the so-called "False Arch" or "Mayan

Arch" in architecture that was modeled on this design.

The Nepoualtzitzin and Mathematics

Conclusions

As you could have noticed the scope of the system is very broad and what is more

important, the development of the Mesoamerican culture couldn't have been

possible in areas like: astronomy, architecture, geometry, mathematics,

agriculture, medicine, etc., if this system hadn't been developed.

And moreover, our links with other brother cultures will be fortified if we follow the

same principles that our great-grandparents followed: respect and learn from the

Mother Nature.

Bibliographical References

Códice Dresde, Fondo de Cultura Económica, México, 1988.

Engels, Federico: El origen de la familia, la propiedad privada y el Estado. El

papel del trabajo en la trasformación del mono en hombre, 4a.ed., Biblioteca

Marx-Engels núm. 1, Ediciones de Cultura Popular, México, 1973.

Esparza Hidalgo, David: Cómputo azteca, 2a. reimp., Editorial Diana, México,

1976.

-: Instuctivo para el manejo del Nepohualtzitzin. Computador prehispánico

D'Esparza, edición particular, México, 1982.

-: Los cómputos aztecas y la geometría, edición privada hecha a mano, México,

1976.

-: Nepohualtzitzin. Computador prehispánico en vigencia, 2a.ed., Editorial Diana,

México, 1978.

Lopez Austin, Alfredo: Cuerpo Humano e ideología. Las concepciones de los

antiguos nahuas, 2 vols. (Serie Antropológica núm.,39), Instituto de

Investigaciones Antropológicas, UNAM, México, 1980.

Macías Villada, Mario: Los Calendarios Indígenas Americanos, Sociedad

Mexicana de Antropología, México, 1956.

Martínez, Juan: La tabla maya de eclipses del Códice de Dresden. Mérida, 1975.

Morley, Sylvanus: La civilización Maya, 5a.ed., FCE, México,1965.

Romero Murguía: Nepoualtzitzin: Matemática prehispánica; su pedagogía y su

vigencia. Experiencia en la Sierra Norte de Puebla. Tesis. Colegio de Pedagogía.

Facultad de Filosofía y Letras, UNAM, 1985.

Romero Murguía: Nepoualtzitzin: Matemática Nahua Contemporánea. Dirección

General de Culturas Populares. SEP, 1988 México.

Sejourné, Laurette: El pensamiento náhuatl cifrado por los calendarios, 2 ed.

(Colección América Nuestra, América Indígena núm. 35), Siglo XXI Editores,

México, 1983.

Tonda y Noreña: Los Señores del Cero. El Conocimiento Matemático en

Mesoamérica, Pangea Editores, Consejo Nacional para la Cultura y las Artes,

México, 1991.

*International Program Teacher, Art History and Philosophy Coordinator

Social Sciences Department

Tecnológico de Monterrey, Campus Puebla.