Mayan Geometry

Leonel Morales
Universidad de San Carlos de Guatemala

This article is a study of the geometry that is found in the various aspects of the daily activities of the Maya, such as design of cities, and the shape of buildings, ceramics and weavings. A geometry legacy is also found in the Mayan languages. Finally, an axiomatic geometry such as those of Occidental origin will be presented, but using Mayan elements, of a similar nature to those found in present-day indigenous weavings. Thus geometries of this type can be taught in elementary schools.

Cities

As happens with the study of other sciences developed by the Maya, in geometry we find that Mayan knowledge was integrated and developed for the collective good. In studying the layout of Mayan cities an impressive relationship to Astronomy is found. "The Mayan spatial orientation of the four corners of their universe is not based upon our cardinal directions..., but probably to either our intercardinal points..., or toward two directions in the east and two in the west, that is to say, sunrise at winter and summer solstices, and sunset at the same two solstices." (Vogt, cited in Leon-Portillo, 1988, p. 130). Also, there are many examples that show the alignment of the temples with celestial bodies. Some such examples are given by Vinette (1986), Morley (1968), and Morley and Brainerd (1983). The alignment of the Stelas 10 and 12 in Copán, Honduras, indicate the time of year in which fields are burned in preparation for the planting of crops. This also indicates that such monuments had a secondary function in addition to the primary function presented in their inscriptions.

In the oral tradition the priests declared that much of their knowledge came from the corn. It is from the ears of corn (la mazorca in Guatemala) that the shape of the temples is derived. And their grand staircases come from the rows of kernels. Corn also offers other kinds of knowledge. Many of the calculations on the Mayan calendar come from the period of cultivation and its various stages: planting (siembra), banking up of soil around the corn stalks (calza), weeding (limpia), etc.

Buildings

The great majority of Mayan temples are truncated tetrahedrons, rectangular prisms, or, in some cases, cylinders, such as those found in the archeological site at Ceibal.

The relationship that these architectural works have with celestial bodies indicates that they were carefully planned before their construction (Morley, 1983, p. 294), as does an observation of the evolution of the elements, such as the Mayan false arch, used in their architectural designs (Morley, 1983, p. 267).

There is also evidence that paintings were carefully planned. A good example is the symmetry in the murals of Coba (Vinette, 1986, p. 389). These planos (plans), as they are called today, were kept and in some cases served as property titles. Such is related in the book, Sobre los Indios de Guatemala (On the Guatemalan Indians), "and they showed you, for your interpretation, two paintings in which the natives of said town (Atitlán) have painted their houses and were antiquities of those who were caciques ... Paintings that were over 800 years old, by which I was able to find out information about the Quichés" (Carrasco, 1982, p. 72-73).

Ceramics

have left much information about their cultural development in their ceramics. Most archaeological excavations reveal the remains of ceramics, either as whole or reconstructible objects. These, generally, provide significant information to studies of geometry. Besides their shape, a collection of curves and other geometric figures are present adorning the exterior, and sometimes the interior, of the vessels. In Maya ceramics "three basic forms are found: jugs, bowls, glasses, plates and vessels with a restricted mouth" (Rubio, 1992, p. 6), and each of those categories is different from the other precisely because of its geometric shape.

For decorative curves the Maya used human figures, animal shapes, flowers, inscriptions and dates. Among the curves there was a predilection for intertwined curves. Spiral curves also appeared frequently. The concept of curves and lines seems to have existed naturally. The phrase "they were placed in a straight line" is found in verse 651 of the Popol Vuh, and in the examples that are presented below in the languages K'ekchÍ and ChortÍ expressions are found for line, align, row, in rows, side, edge of and many other terms.

Native Languages

Much of indigenous knowledge is transmitted in oral form. This method of study and conservation of indigenous culture is very well exemplified in the book, El Ladino Me Jodió (The Ladino "Harmed" Me) (a Ladino may be anyone who is culturally non-Indian). (Saravia, 1986). Given that even today the oral tradition is used to maintain the cultural heritage, it is undeniable that researchers also need to consider the use of that methodology. Thompson (1965, p. 123) pointed out that "...there is more, my contacts with our Mayan works of San Antonio and the long conversations with Faustino in the course of our travels, helped me realize that the modern descendants of the ancient Mayans still conserve many of the old customs."

Because the Maya are so conservative and equilibrated you can be well-assured that fundamentally they act today as they did a thousand years ago, and from there you can deduce much about the past by studying the present" (Thompson, 1965, p. 124).

In order to support this thesis, a study of geometric terms present in various Mayan languages was undertaken. The following geometric terms were taken from Nuevo Diccionario de las Lenguas K'ekchÍ y Española (1955):

rainbow	xoquik'ab
place horizontally	k'e'ebanc
short	ca'ch'in
cylindrical	bolbo
square	caxucut
to square	caxucutinc
quadrilateral	fumru, rok
dice	bul
to play dice	bulic, buluc
distance	najt, xnajtil
row	tzol
in a row	chitzol, tzoltzo
shape of a ball	t'ort'o
shape of a bundle	bolbo
egg-shaped	bak'bo
squashed in	pechpo
flattened	tz'artz'o
side	pacal, xpac'alil
one side	jun pac'al
various sides	q'uila pac'al
both sides	xca'pac'alil
long	nim rok
longness	xnimal rok
line	tzol
one row	jun kerel
align	tzolobanc
measure	bis, bisleb
measure	xbisul
half a measure	jun bas
measured	bisbo, bisbil
middle of two	yibej
in the middle	sa'xyi, yitok
two and a half	cuan rox
three and a half	cuan xca
to measure	bisoc
to measure by handspan	c'utu banc

The book, Método Moderno para Aprender el Idioma ChortÍ: Una Gramática Pedagógica presents some terms in the ChortÍ language that indicate the existence of a Geometry. It is apparently more metric and topological than that found in the K'ekchÍ language, which seems to be motivated more by shapes.

equal	t'isb'ir
what size?	cob'a?
on top of	tor
under	yeb'ar
little	chuchu
along side of	tuti'
big	nojta
very big	nixi

It can be concluded from the above examples, that given the great quantity of geometric terms that exist in these Mayan languages (here taken at random), it can be observed that these elements were used and continue to be used by the Maya people.

Weavings

The Popol Vuh, verse 237, the describes the tasks for children as being: "playing the flute, singing, writing, painting, sculpting, ...". Nowadays weaving and embroidery have been added to those tasks. It is in weaving that many of the designs, that were once present only in ceramics, can be found.

In Mayan weavings for both personal and domestic use a wide variety of mosaic designs can be found. The mosaics have many different interpretations. The work of Anderson (1978) provides a good guide to this area.

Let's take a look at a mosaic (Figure 1):

Figure 1

Notice the repeating triangles in rows or chains, either horizontally or diagonally.

Consider another example (Figure 2):

Figure 2

Broken lines seem to be repeated, but by analyzing those lines you will notice that they form the sides of rhombuses.

A final example (Figure 3):

Figure 3

The elements < and > are repeated in a horizontal row. These mosaics suggest the general idea of geometry in indigenous weaving. They are still present today and are a part of everyday clothing.

Geometry

From Paulos Gerdes' little book, Desenhos da Africa, suggested the possibility of a mathematization of the designs that appear in weavings. A generating element was sought to which various operators could be applied: translation, rotation, ***homostasis. Composition of that basic element is used to develop different shapes and the composed shapes are used to develop chains that are then used to form the mosaics.

The Element: The undefined element that serves as the foundation for this geometry was found to be a common denominator among the various shapes that appear in Guatemalan weavings. It is similar to the symbol for "less than" <

To this element various operators are applied, such as:

Dilation: Dilation acts on size, thickness, and positive or negative state:

Rotation: This acts on one or both sides, changing the angle or the orientation:

Shapes: A shape is defined as a set of two or more elements with a certain orientation. The elements used in the shapes can be simple or can be the result of the application of an operator, for example:

Two elements joined at their vertices:

A rhombus:

Two elements joined at their vertices, but negative:

 

Chains: A chain is defined as the union of one or more shapes, for example (Figures 4, 5, 6, 7, and 8):

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Mosaics: Mosaics are defined as the union of 1 or more chains. Let's considered a complete example:

We start with an initial element:

We define a shape:

We construct a chain:

 

With that chain we can form various mosaics (Figures 9 and 10):

Figure 9

Figure 10

Here are two more examples of mosaics (Figures 11 and 12):

Figure 11

Figure 12

As was indicated at the beginning, the objective of this article is to introduce the reader to the study of the geometry of mosaics that are found in Guatemalan weavings in order to raise the self-esteem and to praise this cultural treasure.

Bibliography

Anderson, Marilyn, Guatemalan Textiles Today, Watson-Guptil Publications, New York, 1978.

Carrasco, Pedro, Sobre los Indios de Guatemala, Seminario de Integración Social Guatemalteca, Publication 42, Editorial José de Pineda Barra, Guatemala, 1982.

de León, Carlos and López, F., Popol Vuh: Libro Nacional de Guatemala, CENALTEX, Ministry of Education, Guatemala, 1985.

Esparza, David, Cómputo Azteca, Editorial Diana, Mexico, 1976.

Gerdes, Paulos, Desenhos da Africa, Editora Scipione, Brazil, 1990.

Landa, Fray Diego de, Relación de las Cosas de Yucatán, Editorial Pedro Robredo, Mexico, 1938.

León-Portilla, Miguel, Time and Reality in the Thought of the Maya (2nd Ed.), University of Oklahoma, 1988.

Lubeck, John and Cowie, Diane, Método Moderno para Aprender el Idioma ChortÍ: Una Gramática Pedagógico, Instituto LingüÍstico de Verano, Guatemala, 1989.

Morales, Italo, U Cayibal Atziak: Imágenes en los Tejidos Guatemaltecos, Ediciones Cuatro Ahua, Guatemala, 1982.

Morley, Sylvanus, La Civilización Maya, Fondo de Cultura Económico, México, 1968.

Morley, Sylvanus and Grainerd, G. W., The Ancient Maya (4th ed.), Stanford University Press, California, 1983.

Rubio, Rolando, Introducción a la ArqueologÍa Maya, Cuaderno de Trabajo, Museo Popol Vuh, Universidad Francisco MarroquÍn, Guatemala, 1992.

Saravia, Albertina, El Ladino Me Jodió, CENALTEX, Ministry of Education, Guatemala, 1986.

Sedat, Guillermo, Nuevo Diccionario de las Lenguas K'ekchi' y Española, TipografÍa Nacional, Guatemala, 1955.

Thompson, J. Eric, Arqueólogo Maya, Editorial Diana, México, 1965.

Vinette, F., "In Search of Mesoamerican Geometry", in Michael Closs (ed.) Native American Mathematics, University of Texas Press, 1988.