52836833 Fractal Geometry in Mesoamerica

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This information has been digitized for use in the Ethnomathematics Digital

Library (EDL), a program of Pacific Resources for Education and Learning

(PREL). The EDL is sponsored by the National Science Foundation as a partof the National STEM Digital Library (www.nsdl.org).

FRACTAL GEOMETRY IN MESOAMERICA

by

Gerardo Burkle ElizondoUniversidad Autonoma de Zacatecas

Zacatecas, Mexico

Symmetry Foundation. Digitized 2004 by permission of publisher.

Elizondo, G. B. (2001). Fractal geometry in Mesoamerica [Special issue of Symmetry:

Culture and Science]. Symmetry in Ethnomathematics, 12(1-2), 201-214. Budapest,Hungary: InternationalSymmetryFoundation.

This product was funded by the National Science Foundation (NSF) as a component of the National Science,

Technology, Engineering, and Mathematics Education Digital Library (NSDL), award number DUE0121749. Any

opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do notnecessarily reflect the views of NSF.


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Symmetry: Culture and Science

Vol. 12, Nos. 1-2, 201-214, 2001

FRACTAL GEOMETRY IN MESOAMERICA

Gerardo Burkle Elizondo

Doctorado En Historia Del Centro Interinstitucional De Investigaciones En Artes Y Humanidades De La

Universidad Autonoma De Zacatecas.

Address: Colina de San Antonio #105, Las Colinas, CP 98064, Zacatecas, Zac. Mxico. E-mail:

[email protected].

Abstract: Fractal Dimension in different groups of mesoamerican artistic and

architectural works was quantified in 106 structures. To make the analysis we used the

Benoit program in order to calculate Box Dimension, Information Dimension,

Mass Dimension and Y value about Fractal Dimension. We found a general

average of 1.92 for fractal dimension with values by groups in a range between 1.883

and 2.038. These results are in a very good agreement with the fact that the values ofcomplex fractals are usually situated between 1.5 and 2.0. Fractal dimension shows the

efficiency with which an object fills the space that contains it, and is quantificated like

the morphology of their complexity.

Scaling properties exponent were proved, and specifically the fractal dimension

obtained looks like a possible pattern. We think that part of the simple rules that could

explain these complex dynamics findings is the fact that the artists and architects in

Mesoamerica used mathematics in their work. The finding of proportional systems

and golden units to measure in modular forms, like scaling rectangles and other

techniques to get harmonic and constant units in the case of Mesoamerica, had been

proved by some authors.

INTRODUCTION

The study of geometric analysis and mathematics that were used in art composition and

construction in Mesoamerica is not yet clearly established.


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202 G. BURKLEELIZONDO

Some approaches have been developed, and in general agree with the fact that there was

a basic form that fundamentally consisted in the use of golden unit measurements.

In her book, Margarita Martnez del Sobral (2000) speaks, as did before Beatrz de la

Fuente about the Olmec sculpture (1984), about the existence of proportional systems

consisting of rectangular standardized modules of standardized measurements. These

rectangles were in fact golden and harmonic units.

She found some constants in this way, basically different golden rectangle measures,

and a developing in a spiral harmonic method related to growing rectangles.

Therefore the figures appear like proportions in a module that can be geometrically

analyzed. The rectangles can also move and grow in a spiral way to get, with the

rotations, new and bigger rectangles but in proportion with its golden basic initial unit.

Also she shows in her book the Fibonacci serie and finds many turn angles that were

used in Mesoamerica in art and in architecture and urbanism.

Both, De la Fuente and Martnez had proven that in the prehispanic world a system like

this was used constantly by many cultures over the times, as a well developed geometric

scheme by scribes (Tlacuilos), artists, sculptors and architects making of this astandardized technique and a universal tool in composition, and that these rules were

transmitted like a tradition from one generation to another.

The fact that the use of mathematics was a conscious planning device is not discussed

now, but we need to try to understand better about the techniques that were used to

calculate, design and create aesthetic proportions, in order to know what the evolution

of art and history really was.

HYPOTHESES

Mesoamerican astronomers developed, as is well known, a deep knowledge about the

movement of stars and planets. They in fact made a connection between the cosmic

geometric scheme that learned from the sky, and their own lives. To them all were

related to sacredness and to the gods: their art, architecture and even the places and form

of construction and the orientation of the cities. The part of archaeology that studies this

matter is named archaeoastronomy.


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FRACTAL GEOMETRY ATMESOAMERICA 203

The Maya mathematical system was a vigesimal one. Eric Thompson (1950, p. 143)

found the uinic like a standard measure whose value was 20 units in a cord-length

used to measure the land.

Before him the linguist Daniel Brinton (1890, pp. 433-451) reported that he found

words for measurings in nahuatl and maya-yucateco languages that correspond to

some of the dimensions of the human body, for example the arm and the foot and

others.

Some authorities like V. Garth Norman had found a very probable mesoamerican

cultural marker in a codo equivalent, whose measurement is 49.5 centimeters or 19.49

inch, and that this and their multiples and divisions were used, for example, inTheotihuacan and in many parts of the Maya area.

The engineer Hugh Harleston (1974) who made measurements of the Theotihuacan

pyramids finally also found the smallest fraction represented by the number 19 and

other standards very close to a meter.

In a previous paper we reported the finding of Fractal Dimension in many

mesoamerican art and architectural works (Burkle 2001). Now we collected a greater

number of structures, especially a big and representative number of pyramids toincrease the value.

From the foregoing evidence, we think that the use in this field of fractal geometry

would make sense in two ways: first we hypostatize that mesoamerican art, sculpture,

aesthetics and architecture, with particular attention to the mayas, are fractals or at

least have fractal dimension in their basic form. This could mean that the designs, the

projects and their conceptions are easier to understand in a base of complex patterns

with up or down scale properties, that are characterized by a fractal exponents with

particular magnitudes. We found these values between 1.883 and 2.038 in the averages

of ten groups that we submit to analysis.

The second point is that we think that probably the mayas knew and used mathematics

that today science calls non-linear, scalant, boundaries, Fibonacci series,

extrange attractors, self-affinity, self-similarity, fractals, concepts that belong

to chaos theory.


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Confirmation about the discovery of a standard unit requires different ways of study and

precise and consistent use checking, and we think that the fractal and chaos theory

application could help to increase the understanding of the real dimension and grade of

mesoamerican cultural development.

METHODS

Today fractal analysis is a universal tool that had been applied to many sciences. It

seems that no previous study had been done about its application to understand ancient

mathematics, especially those that were used in Mxico during the prehispanic period.

We know that they were very good mathematicians, and in fact our findings in this

study confirm this, especially because in our fractal analysis we found a very close

geometric pattern has is shown in the results.

The study was divided in 10 groups:

1. Walls sculptures of Palenque temples

2. Maya stelas of different places

3. Maya hieroglyphs from Palenque

4. Pyramids and temples from various places of Mesoamerica

5. Different calendars in stones and codex (Tonalamatl)

6. Various codex pages, specially from the Dresde one

7. Murals of different parts of Mesoamerica

8. Great stone monuments like the Aztec calendar, Coatlicue, a Tula giant, stela D from

Copn and a colossal head from La Venta

9. Astronomic stones and

10. Figures in ceramic from the Maya culture.

The ten groups give a total of 106 figures that were analysed and that are described in

Table 1. Basically the images correspond first to Maya culture, developed at Chiapas

and Yucatn in Mxico, Tikal at Guatemala and Copn Honduras, and to the Aztec or

Mexica cultures developed at Mexican Central High-plains, and some to the Toltec and

Olmec cultures from Mxico.


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God Coyoxauqui of the Aztec culture at Mxico

Temple Pyramid of Tikal of the Maya culture at Guatemala


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Disc metal plate of the Cenote Sagrado of Chichen Itz, of the Maya culture at Yucatn, Mxico

Codex Fejrvry-Mayer page 1, Mixteca culture from Mxico


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Temple of the Sun Panel at Palenque Chiapas, Mxico

To make the analysis these figures were first scanned and saved as bitmap files.

Thereafter the images were analysed with the Benoit Program version 1.3 for FractalAnalysis System. Then we calculated three main characteristics about the fractal

dimension of each one: its Box Dimension, Information Dimension, and the Mass

Dimension with the Y value in each case with their respective standard errors, and

intercepts on log-log plots graphs.

The selection of the structures and images were non-random.

Table 1:Art and Architecture Analysed Groups

I. PALENQUE AND OTHER TABLETS (Milbrath 1999).

II. MAYA STELAS AND OTHERS (Aveni 1997).

III. MAYA HIEROGLYPHS (Milbrath 1999).

IV. PYRAMIDS AND TEMPLES OF MESOAMERICA (Marquina 1964).


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V. CALENDARS (TONALAMATL) AND ASTRONOMIC STONES

(Thompson 1950).

VI. CODEX PAGES (Harleston 1974).

VII. MURALS OF MESOAMERICA CITYES (Aveni 1997).

VIII. GREAT STONE MONUMENTS (Tyler 1995).

IX. ASTRONOMIC STONES (Broda Prucha 1991).

X. CERAMIC VASEES AND OTHERS (Schele and Grube 1997).

EXPLANATION ABOUT THE ANALYSIS

(BENOIT FRACTAL ANALYSIS SYSTEM)

To get our data we used the program (TruSoft Intl Inc. Benoit, version 1.3: Fractal

Analysis System. 20437th Ave. No. 133, St. Petersburg, FL 33704, USA) we had

mentioned before. This analysis system permits to get different kinds of fractal

calculations (fractal dimensions).

We chose three characteristics to be studied because it seems to be specifically useful to

the understanding of the dynamics we were studying. These three methods are:

a) Box Dimension;

b) Information Dimension and

c) Mass Dimension. In each case we get too the Y value.

The three methods belong to self-similar pattern x analysis, and can be better

understood if we pay attention to some of the relevant aspects: first, if we begin using

some important aspects of chaos theory, actually in this case the use of concepts like

order can help us to look at the fundamental tasks of this work and, second: fractals

are in fact the geometry of chaos, but a special kind of chaos in which order emerges

with laws governing the description of the system.

Starting from the properties of a fractal as the principles that make reference to its

own characteristics like:

1. The value of its dimension, that is between 1 and 2 depending on the occupied space

and the dynamic (Hausdorff dimension).


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2. Self similarity is another characteristic that means that the form does not change

when the scale does, but that becomes more and more complex (complexity) in the non-

linear fractals, especially the 2 nearest values.

3. The self-similarity in fact mathematically means a potential relation between the size

and the measurement or scale.

4. Its dimension is fractional or fractal and non-Euclidean.

5. They have an infinite longitude. The data set has scale invariance characteristics and

exhibits a fractional motion with statistical properties in a curve with a scaling factor

between 1 and 2 (Hurst exponent). This scale invariance can be tested studying the set

in many length scales, which can be done with the Benoit Fractal Analysis System

that we used to get the information about the figures to verify first if they were or were

not possible fractals, their fractal dimensions and get the exponent and try to see if this

is a pattern that can be may be recognized like a characteristic marker in the results.

About Dimensions - Box Dimension Estimation Method Interface shows the number

of boxes of linear size x necessary to cover a data set of points distributed in a two-

dimensional plane. In the Benoit program each box and grid rotates 90 degrees size

through dividing to get the minimum value of the x dimension.

Information Dimension Estimation Method Interface assigns weights to the boxes.

These boxes that contain more points or those that contains less, are analyzed from the

point of view of their masses contained in each box (information entropy).

Mass Dimension Estimation Method Interface implicates the number of points inside a

circle around the data set in a two-dimensional plane to define the mass; this can be

measured in circles of increasing radius.

These three methods will reveal first if the set have fractal dimension, and its

complexity degree.

About Y - Fractal dimension is expressed like the slope in the graph. The factor

represents the lagunarity or empty spaces distribution related to those non-empty, and

the exponent represents the fractal dimension.

The graphs a, band c correspond to Fractal Analysis of the Glyph 14 from Palenque

Palace tablet, respectively Box Dimension (Db), Information Dimension (Di) and Mass

Dimention (Dm).


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RESULTS

We could summarize our findings in the following order:

1. In all the 106 mesoamerican artistic and architectural images that were analysed we

found fractal dimension, and all were complex, with a Df between 1.8 and 2.0.

2. We find that this is a very close range between the different dimensions of the ten

groups of elements that were analysed. This could mean that this may be a fractal

marker for Mesoamerica measurement (1.92) with the method we used.

3. We find a pattern present in the average of each group, specifically in the more

complex groups. This probably reflects the use of golden units we had mentioned beforethat had been reported by other authors.

4. In fact this method of study, the fractals and chaos theory applied to art, history and

ancient scientific culture seems to be a good and interesting tool of complexity theory

for analysis to still studying.

5. We can see about the range we spoke about, that it is at the highest level of

complexity. This could be understood as order emerging from the edge of chaos

critically, which in fact define the concept of complexity.

In Table 2we can see the Box Dimension (Db), Information Dimension (Di) and the

Mass Dimension (Dm) averages by groups of study.

GroupNo. Db SD Di SD Dm SD

I 1.9180.010 1.9320.002 2.0180.111

II 1.9230.007 1.9400.001 1.8870.060

III 1.9100.008 1.9030.003 2.0360.088

IV 1.9260.011 1.9330.006 1.9750.010V 1.9210.008 1.9260.002 1.9370.051

VI 1.9180.009 1.9240.003 2.0380.269

VII 1.9190.006 1.9290.002 1.9640.058

VIII 1.9170.009 1.9140.003 1.9540.053

IX 1.9000.006 1.8770.003 1.9750.047

X 1.8830.013 1.8880.003 1.9660.214

Table 2


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The images of the next page belongs to some of the structures that were studied such as

a Jaguar sculpture from Teotihuacan Mxico with Db= 1.9170.003, Di= 1.9150.001

andDm= 1.9340.017.

The next image belongs to the Stela F from Quirigua Guatemala with Db =

= 1.9400.011,Di= 1.9300.009 andDm= 2.1420.276.

The analysis of the disc of Xochimilco from the Aztec culture at Mxico shows a Db=

= 1.90080.0075,Di= 1.8740.003 and aDm= 1.9940.048.

These are some examples about the results that we got in our study.

GENERAL CONCLUSIONS AND COMMENTS

It is very interesting to discover that for all the 106 cases that were studied of

mesoamerican culture images, all of them have Fractal Dimension. Some of these

belong to those kinds of fractals or fractal geometry that can be discovered at first sight

in its proportions and its dimensions, and are in general figures that imitate the natures

aesthetics in some way.

Others have greater complexity, maybe because its self-similarity or self-affinity is

statistic and anisotropic just as real objects are.

In both cases that we found out there could be some kind of metapatterns that need to be

understood better.

It is necessary to make more studies to get conclusions, but with the information we

gathered, we think that there undoubtedly existed a mathematical system and a deep

geometrical development in mesoamerican art and architecture, and that they used

patterns and golden units.

We also think that these wise men used complex mathematics to make their works,

maybe non-linear and chaotic, trying to imitate the movements in the sky and the

complexity of the natures phenomena to establish some kind of similitude from this

side, and the world were the gods live, trying then to understand in a holy way their

designs and random wishes.


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They needed to get the best possible information about the world, the universe, the

implied order that exists between them and the sacred cosmos that governed their

uncertain lives.

Dr. Joana Broda (1991, p. XII) says that in the archaic past, the science was historically

determinate in civilizations that, like now, assumed science like a part of a social

whole.

The use of fractals and chaos theory as a tool to study history for certain will open new

doors of understanding between social sciences and things that we perhaps can not

imagine now.

In his book Metapatterns, Tyler Volk (1995, p. 206) asks the following metaphoric

idea: Are boarding crossings in space the inevitable tool for portraying times breaks

in diagrams and myths?.

REFERENCES

Aveni, A. F. (1997) Observadores del Cielo en el Mxico Antiguo, Fondo de Cultura Econmica.

Briggs, J. (1992)Fractals. The Patterns of Chaos, Touchstone, Simon and Schuster.

Broda Prucha, J. (1991) Arqueoastronoma y Etnoastronoma en Mesoamrica, Universidad NacionalAutnoma de Mxico.

Burkle Elizondo, G. (2001) Valdz Cepeda Ricardo. Do the Mesoamerican Artistic and Architectural Works

Have Fractal Dimension?, In: Miroslav M. Novak, ed.,Emergent Nature. Patterns, Growth and Scalingin the Sciences, pp. 431-432. World Scientific.

Closs, M. P. (1990)Native American Mathematics, University of Texas Press.

Cowan, G. A. et al. (1999) Complexity. Metaphors, Models, and Reality, Perseus Books.

Daniel, B. G. (1890) The Lineal Measures of the Semi-civilized Nations of Mexico and Central Amrica ,

Essays of an Americanist, Poter and Coates.

De La Fuente Beatrz y cols (1999)Pintura Mural Prehispnica, Jaca Book/CONACULTA.

Fuente, B. de la (1984)Los Hombres de Piedra, Escultura Olmeca, Universidad Nacional Autnoma de Mxico.

Garcs Contreras Guillermo (1995) Pensamiento Matemtico y Astronmico en el Mxico Precolombino,Instituto Politcnico Nacional.

Harleston, H. Jr. (1974) A Mathematical Analysis of Theotihuacan, XLI International Congress ofAmericanists, October 3.

Kauffman, S. A. (1993) The Origins of Order, Oxford University Press.

Mandelbrot, Benoit (1997)La Geometra Fractal de la Naturaleza, Tusquets Editores.Mangino Tazzer Alejandro (1996)Arquitectura Mesoamericana. Relaciones Espaciales, Trillas.

Marquina, I. (1964)Arquitectura Prehispnica. Instituto Nacional de Antropologa e Historia, Secretara deEducacin Pblica. Mxico.

Milbrath, S. (1999) Star Gods of the Maya. Astronomy in Art, Folklore, and Calendars, University of Texas Press.

Peitgen Heinz, O. et al. (1992) Chaos and Fractals. New Frontiers of Science, Springer Verlag.

Schele, L. and Freidel, D. (1999) Una Selva de Reyes. La Asombrosa Historia de los Antiguos Mayas , Fondode Cultura Econmica.

Schele, L. and Grube, N. (1997) The Proceedings of the Maya Hieroglyphic Workshop, The Dresde Codex,March 8-9.

Schroeder, M. (1999)Fractals, Chaos, Power Laws, W. H. Freman and Company.

Sobral Margarita, M. del (2000) Geometra Mesoamericana, Fondo de Cultura Econmica.

Thompson, J. E. S. (1950)Maya Hieroglyphic Writing. Carnegie Institution of Washington, publication 589.

Tyler, V. (1995)Metapatterns. Across Space, Time, and Mind, Columbia University Press.

52836833 Fractal Geometry in Mesoamerica

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