Saturday, August 10, 2013

Three videos of how to measure the Fractal Dimension of urban morphologies

Though the videos are in Spanish, readers will be able to understand them, as I am showing the methodology, the way I make it myself.
It´s my first time producing short videos, and maybe you´ll see some technical mistakes, I promise they will be better in the near future.
Here it goes, parts 1, 2 and 3. The last one is related to the selection of the scale.

Thursday, August 8, 2013

STRUCTURE OF CITIES INSIDE CITIES: Morphological analysis of Chan Chan



Mucho se ha avanzado en las posibles representaciones de una ciudad, y los cambios más significativos han surgido en la abstracción de imágenes que representan los complejos fenómenos urbanos en pos de una lectura más sencilla: el desarrollo de Google Earth de acceso público y gratuito, imágenes satelitales actuales, imágenes digitales en general. Todo el acceso a información que se nos permite, sumado a las múltiples herramientas digitales, hace que nos replanteemos cuál es la mejor forma visual de analizar la ciudad.
La aplicación de la geometría fractal en modelización espacial responde a  un método para analizar formas, que permite la comparación entre  modelos espaciales y estructuras empíricas. La metodología consiste en interpretar estas estructuras empíricas (ciudades, barrios, asentamientos, fachadas, etc) como fractales aleatorios o estocásticos. La geometría no es medición en sí misma, pero es interactiva entre la modelización y la medición.
La medición consiste en dar valores a los fenómenos que nos interesan, dentro de un marco establecido.

A los efectos de ilustrar estos conceptos, analizaremos las primitivas ciudadelas de Chan Chan, en Trujillo, Perú (aprox. Año 1000 D.C.), que tienen la particularidad de estructurarse como ciudades dentro de otras ciudades. Nos basaremos en el trabajo de campo y publicaciones de eminentes arqueólogos y antropólogos para todos los aspectos descriptivos, y a continuación, demostraremos que, a pesar de las concepciones generalizadas de clasificar el medioambiente construido prehistórico en la mera construcción y la arquitectura en intención artística, el modelo morfológico urbano de la cultura Chimu es la expresión  geométrica de los espacios cognitivos resultados de su experiencia como miembros de un grupo social. Veremos que en lo que desde la vista aérea se percibe como una forma de cuadrículas Euclidianas combinadas entre sí, existe un patrón morfológico no evidente de características cuasi literales de fractalidad, que otorgan una gran riqueza espacial. El tema es más sorprendente aún si consideramos que estos diseños urbanos son intuitivos y representan una gran evolución con respecto a las antiguas configuraciones urbanas de ejes ceremoniales.


A lot has been advanced in the possible representations of a city, and the most significant changes have arisen in the abstraction of images that represent the complex urban phenomena for a simpler reading: the development of Google Earth of public and free access, updated satellite images, digital images in general. The whole access to information that we are allowed, added to the multiple digital tools, makes that we reconsider which one is the best visual form of analyzing the city .  
The application of the fractal geometry in space modelization responds to a method to analyze forms that allows the comparison between space models and empiric structures. The methodology consists on interpreting these empiric structures (cities, neighborhoods, settlements, facades, etc) as random or stochastic fractals. The geometry is not measurement in itself, but it is interactive between the modelization and measurements. 
The Dimension consists on giving values to the phenomena that interest us, inside an established boundary.  
To illustrate these concepts, we will take the primitive citadels of Chan Chan, in Trujillo, Peru (approx. Year 1000 D.C.) that have the particularity of being structured as cities inside other cities. We will be based on the field work of eminent archaeologists' and anthropologists´ publications for all the descriptive aspects, and next, we will demonstrate that, in spite of the widespread conceptions of classifying the prehistoric built environment in the mere construction and the architecture in artistic intention, the urban morphological pattern of the culture Chimu is the geometric expression of the cognitive spaces of its experience like members of a social group. We will see that in what is perceived from the air view like a form of Euclidian grids combined to each other, a non evident morphological pattern exists of quasi literal characteristics of fractality that grant a great space richness. The topic is even more surprising if we consider that these urban designs are intuitive and they represent a great evolution with regard to the old urban configurations of ceremonial axes.


The Chimú Empire (950-1440 AD) extended along the Peruvian coast from Tumbez in the North to Lima in the South. Chan Chan, located in the Moche valley, was the capital of the Empire centralizing all the services. Built in adobe, was the biggest pre-Columbian urban center in South America, currently declared UNESCO World Heritage, since 1986.

The citadels are big settlements that contain a great number of monumental buildings. Six of them were named in honor of explorers and archaeologists that worked in the place: Squier, Bandelier, Rubber, Tschudi, Rivero, Velarde, and Tello. The other citadels are denominated Great Chimú, Chaihuac and Laberinto1. The small structures are the Huacas.
The civic constructions served to the aristocracy and the State, while the proletariat consisting in artisans, personal of service and farmers lived in dispersed neighborhoods outside of these monumental centers. The palaces articulated the urban space, since each king built his government place during his life and then this became monument after his death. 
Michael E. Moseley and Carol J. Mackey (1973, 1974), after their investigations in 1967 and 1969, formulated the hypothesis that once the Curaca (ruler) died and buried in the ¨funerary platform¨, all its citadel was transformed into an enormous catafalque where his servants, women, court characters and priests, were sacrificed and buried to accompany him and to serve him in “the other life”. In turn, the following ruler ordered to build his own palace citadel. Presumed habit, very similar to those of their successors, the Inca. 
In synthesis, Chan Chan shows three urban typologies:

1)     The slums, of agglutinated rooms, without surrounding walls, dispersed without apparent order in the Western suburbs. More complex than simple cabins and associated to cemeteries of adobe chambers with double walls.
 2)     An intermediate tipology located among the citadels, without a specific planning, represented by enclosures with niches, of regular geometries, also with lower surrounding walls and other similar attributes to those of the monumental architecture, but of smaller scale. The biggest ones show a more rigorous planning of patios, passages and storage rooms, in comparison with the smallest of more domestic characteristics. Seemingly the use was  residence of social classes
3)     The royal architecture of the citadels, of monumental character that, obviously did not reflect population's density, since many works were carried out by  non resident workers.

Each citadel was surrounded of adobe walls of approximately ten meters high  covered with a soft mortar in which intricate designs of birds were carved, mamals, fish, in two styles, one more realistic and the other one more stylized and abstract. 
An interesting aspect is that there are not openings to the North, although these are the walls more exhibited to the sun in detriment of the fog; the highest walls protect against the winds of the SO coast. The combination of these walls configurate a labyrinth, and, in turn, each inhabitable structure has multiple internal walls that form an intricate space of corridors, rooms, covered with the elite's artistic expressions.
The structures of these settlements are intimately bound to the economy of their characteristic activities and their social strata. The strata between noblemen and slaves did not resemble at all to the European models, since they were seen as products of separate creations, the noblemen derived  of two stars and the plebeians of two planets, what implies that both groups did not move among them. These conceptions of Andean social order were established in different material forms, the most evident are those belonging to the funeral landscape.2 Great part of the constructions is due to these beliefs. The sacrifices and other methods of ritual violence were imbued in the Andean cultural practices. The cosmological principles, the phenomena, the religion 3 and the social order were connected to the exercise of the power, and it was still maintained after the death of the Curaca who was confined in his palace so that the vassals could not  perceive his mortality. The lack of respect to the temples, the disobedience to the law was severely punished burying the culprit alive.

This concept where the physical  and the social world correspond each other, is denominated  doxa, term used by the sociologist Pierre Bourdieu in humanist instances that implies the limitation of the social mobility inside the social space; the imposition of limits for each individual like symbolic force of the political power. Some cultural objects are recognized as doxa to be inappropriate for a certain social position. Doxa contains the sense of "one's place", and the sense of ownership. The sacred landscape is also a symbolic way to establish social orders.

F1. Partial view of a citadel

F2. Niches

F3. Altar

F4. Decorated walls with the pattern of fishing nets


 Our first step consists on verifying if the citadels, taken on the whole, present fractal characteristics. We will use the method Box Counting to find the  fractal  dimension (D), the software is Fractalyse. 
For the complexity of the pictures, we will apply the filter ¨sharpen¨ to get more contrast. Then, the file. jpg is converted into binary to carry out the mensuration.
It also interests us to verify if, given a condition of fractality, it is also verified when we select sectors of the citadels. That is to say, if a part of the whole can be representative of the complete morphological urban structure. We will analyze the whole and three sectors of the same one. Of the main characteristics of a fractal, we will focus on the autosimilarity. Fractals define a complex hierarchical scale in each level, where the deterministic fractals show the same structure in each magnification. When the exact autosimilarity is lost, and some structural properties are only similar in each scale, we are before fractals of statistical autosimilarity more representative of the conditions of the real life, and they still continue presenting hierarchical connections of texture and/or shape in the different scale levels.

The concept of similarity explores a certain phenomenon, its immediate environment and its relationships with other phenomena. Generally, two phenomena are similar when they share attributes and circumstances. The similarity is governed by the context (function, users, reasons...), scale, techniques of recognition and mensuration. The grade of similarity is a resultant between a model and an objective; and not only of their attributes but also of their relationships. In our investigation field, the objective goes from pixels, to regions, in a context and a scale that we will define. From a perspective of the informatics, similarity can be described like a system that  allows  the data to be comparable under the criteria that the user specifies as similarity.  
There are different forms to approach to the concept of similarity.  
For example, an efficient study of objects that  share a similar form. 
Or that share similar areas. 
Or that derive in possible transmutations, like forms of analogical inference. 
Explicit or implicit similarities exist according to the context. 
The simplest way of images evaluation  is to compare its visualities, as measure,  shape, color, texture and other components that we can elucidate, either in its entirety or its segmentations in studies of fractal plot lanes. 
 The autosimilarity in urban morphology is interpreted as a design process - fractal property - and it is appropriate to the clinical interpretations of images, where a collection of interpretations of the sub-parts is part of a whole. Then, it is verified if this whole is similar to another image's whole and equally, as these sub-parts of the other whole is similar to those of the original pattern. 
The resolution in scales can affect  the comparisons, because different morphological structures could appear in the scaling variations.  We can apply the theories of fractal hierarchies but, sometimes, the environmental and habitat issues  are superimposed and we do not always find scaling autosimilarity. But yes we can find the tendencies. 

F5. The group of citadels. The slums are located on the West side, in our figure, along and to the left. 

F6. Sector 1

F7. Sector 2

F8. Sector 3

Table 1. Table of results with Box Counting method

In the following figures, we analyze the audiences in the context of the dominant architecture. Notice  that the texture denotes the irregularities of the desert landscape and some constructions that are not completely excavated.

F9. Sector of Ciudadela

F10. Sector of Ciudadela

F11. Sector of Audiencia

F12. Sector of Audiencia with storages

Table 2. Results with Box Counting method

For both groups of mensurations, we verify that: there exist a tendency to fractality and autosimilarity through the different urban scales, which a Fractal Dimension (D) fluctuating between 1.70 and 1.75.

What still makes  the  theory even more interesting,  is the progression in complexity in the U structure as time goes by, of the audiences to the variants of audiences combined with trocaderos that in turn,  are related to the three major type of adobe brick proposed by Alan Kolata (1982). That is: 
1) high, which dimension in height is bigger to that of the width. 
2) cubic, with similar dimensions on high, wide and depth. 
3) wide, more similar to our contemporary brick.

The formal evolution becomes from a simple U structure with a chamber to the typical insert of six chambers, that is a geometric recursivity of fractal type that is more complex as these cells of six chambers become primary and they take their own recursivity. The example of  figure 13 is one of the so many variations in the evolution in the U structures. In general terms we are before designs that share the same pattern of cells, (read it as structure, model) that seen as a group,  look like  squares inside squares. From our point of view, it is also very suggestive  the disposition of the carvings of dragons in Huaca of the Dragón, where the dragons with their fires contain Moche snakes  of two heads. We have  counted a succession (or conceptual iteration?) of three in a wall partition.  

F13. Conceptual iteration of the U structure

F14. Formal articulation between two U structures

The concept of boxes inside boxes has taken us to compare the mensurations of D with the Sierpinski carpet. Nevertheless, this is more compact, with a fractal dimension of approximately 1.8928 against 1.75 of Chan Chan. Logical results if one keeps in mind the quantity of patios and chambers. 

F15. Sierpinski Carpet

A more precise methodology consists on looking for a second underlying structure, subjected to a primary one that is visible to the expert eye. So, we intend to verify the Fourier transform applied to one of the audiences. The software utilized is Fractalyse.

F16.  Fourier Transform applied to an Audiencia

The spectrum of gray  is uniform and a structure arises in diagonal, with other approximately parallel lines that, understood from our previous experience with applications of Fourier for blocks in the neighborhood of La Boca (Buenos Aires), we conclude that there is another pattern of space covering, different to that of boxes and more appropriate to that of curves filling the plane. 
The articulation in the U structures  takes us to the consideration that these representative urban forms of the Chimú royalty, go covering the plane with a similar development to those of the Peano curves. The Peano fractal ¨comienza con un segmento unitario que se divide en tres partes iguales y el camino recorrido incluye nueve segmentos, cada uno de los cuales tiene una longitud de 1/3¨. (V. Spinadel, J. G y J. H. Perera. Geometría Fractal, p. 23, 2007).
(¨it begins with a single segment that is divided in three equal parts and the path includes nine segments, each of which has a length of 1/3¨) (My translation)

For the following recursivity level, the  fractal dimension is D:1.7491, extremely similar value to the one taken in the overall urban analysis of Chan Chan and that of its sectors. What demonstrates a tendency to the autosimilarity, characteristic property of fractals, very close to that of the selected analogous fractal.

F17. Peano fractal

Table 3. Aplication of the Box Counting method in the Peano fractal

The interpretation about iterations of the curves of Peano is ours, and applicable to uniform structures  of Audiencias. This constitutes an advance with regard to Kevin Lynch's opinion who defines the citadels like ¨boxes inside boxes¨, for the used resource of patios and chambers. (K. Lynch, p. 15, 1994). It is very important to point out that this disposition of ¨boxes¨ is reinforced by the absence of streets and directional axes. However, the movement and accesses were not irrestrictive, the movements were structured in independent routes, so much in Chan Chan as in the provincial centers. (Moore, p.219)

This standardization in the shape does not seem to be random, but rather it has been a  consequence of an important economic change, of the society progress, based on a basic agrarian economy to an economy of crafts type goods of production. The complex system required of a bigger economic control that manifested in efficient standard forms, in other words, a functionality based on a common system  of interpretation for different administrators, like a  rudimentary computerized  database of the epoch. The paradox, is that the higher standardization in the shape,  the more complex the resultant morphological pattern of the possible compositions becomes, until supporting a fractal structure of astonishing analogy with those digitally created.4 
It is also important the relationship with similar structures in the rural neighborhood; it implies an absolute concentration of control and power in the agriculture resources and irrigation. If the U structures  were specifically for control and storage of goods,  is a theory discussed by John Topic (2003) since he sees this system like process of flow and saving of information, in a way similar to the quipu 5, where the architectural elements reflect the transmission processes from an official to another. For example, beans, stones could have been stored in recipients and/or bags inside the containers and niches, to keep non written  quantitative information, but as a mnemonic rule, based on the memorization of facts represented in an object.


At plane level the U structures show an uniform hierarchy and we have already seen that they follow laws of similar intuitive design to the curves of Peano. However, analyzing in detail, the spatial architectural resolutions, added to the ritualized practice of the burials under the zero level are indications that some of these structures have a higher status, specifically the U structures  that belong to the royal enclosures, while the rest denotes solely administrative purposes. (J. Piekarski, 2007).
Andrews (1974) proposes a constructive classification that we should call scaling hierarchy but not spatial because the height do not always grow linearly as complexity does. This concept of scaling  supports our thesis of tendency to fractality in its high concept of recursivity. From smaller to bigger, we summarize:

.- The auxilios, linear structures of low walls without niches, but with carvings.
.- The trocaderos, of smaller span whose name makes allusion to the Spanish word ¨troca¨ or spaces that go through.  Also with their formal variants. With three or four niches.
.- The arcones, small constructions of low niches that, as the word indicates are like bins.
.- Audiencias and their variants in U and C shape.

To each one of these classifications it corresponds a cross section that becomes more complex as the structure increases in size. In the following figure, we schematically show the corresponding cross sections (up to down) of: auxilios, trocaderos, arcones and audiencias. Although the sketches are not in scale, our intention is to point out the relationship among the U structures, the quantity of niches, the height and excavations for burials, that could be seen in the trocadero. We could infer that to our first relationship of size of the structure and complexity of the cross section, it corresponds a graph of linear growth. But to this idea, the funerals incorporation is opposed. Let us see it this way : the second graph should exclude the funeral, because the excavation is full with gravel, earth. However, conceptually the  Chimu culture has had the intention of working the earth in niches, excavations, mounds, not only in the urban constructions but also in the landscape and to the purpose of  cultivations. A clear example is the sunken gardens, called puquios. From our point of view, all intentional work on and underground should be considered in the morphological analysis, independently of the fillers that take place afterwards.
Making an abstraction of the cross sections, we will see that from the simple line that indicates a wall, we would pass to another broken line that implies the wall with carvings (auxilios), plus the successive similar iterations to the first stadiums of the curve of Minkowski that include niches and funerals.

F. 18. Schematic floor plans and cross sections of typical structures. Drawing made by the author based on the ones published by Andrews.

F. 19. Example of the first iterations of the Minkowski curve


The form of an urban settlement is the result of how its creator glimpses its use, and the social frame sustains its sense. 
The urban morphology of the citadels of Chan Chan has been born of abstract geometries  product of deliberate actions of the human logical-political thought, with a concrete purpose: the one of being adjusted to the growing productivity, social separation and control of both, production and society. 
And what the archaeologists and anthropologists wisely point out  is a geometric evolution that neither is accidental nor it arises of the application of an intentional mathematical model. We will fix the concept in the following quotation of R. Doberti:
¨La transformación, la seriación, la mutación, no son accidentes que les acontezcan a las formas, más bien parece que su ley es la inestabilidad, la metamorfosis. Uno de los grandes aportes de la geometría al mundo de las formas es haber establecido muchas de las leyes de la transformación. Y entonces la geometría se convierte en una de las destrezas para la construcción y el manejo de la forma¨. (Roberto Doberti, 2º Jornadas Internacionales de Mathematics & Design, Special Edition, Vol. 5, No 2. p.61,2005)
¨The transformation, the seriation, the mutation, are not accidents that happen to shapes, but it seems that their law is the instability, the metamorphosis. One of the great contributions of the geometry to the world of the shapes is having established many of the transformation laws. An then, the geometry is converted into one of the achievements for the construction and the management of shape¨. (My translation).

We have seen that the citadels possess a certain autosimilarity whose approximate value of D is 1.75 and its morphological structure contains two basic concepts:  
1) the composition of boxes inside boxes in a spatial sense;  
2) the tendency to fill the plane (the desertic landscape) with labyrinthine articulations constructions of passages, niches, rooms that follow similar iterations to the fractal of Peano, hiding the true accesses to the visitors. 

And a higher level of abstraction exists, that of the non apparent hierarchies, given by the burials that accompany the royalty and reflect an organizing principle based on the division of classes. For example, the Inka buried  the deads in crypts or caves following the patriarchal family kins, and the Chimú according to its social class. Although the governors were placed in constructions with platforms inside the citadels, the location of the plebeian deads in the suburbian cemeteries, helps to make the urban form even more complex, the conceptual fractal continues being generated in other underground planes, and it reaches the tridimensionality, even for cultivations.
We are before a  geometry of non apparent , spontaneous order, whose capacity is the one of generating diversity starting from simple rules. This diversity is not arbitrary, because of  having non desirable results, it would have been discarded.
The methodology of analysis in urban morphology by means of the fractality is a very wide field and a single ¨narrative¨ will not be enough. We have been interested in the built environment that is evidenced today in Chan Chan, but the advance in the archaeological excavations,  will guide us to more complete evaluations by means of the application of mathematical models based on the theory of the Complexity and fractality; we will be able to achieve complete reconstructions and recognition of urban morphological models that include the social order in their relationship with the dimension of the sacred landscape, the symbolism and the ritual of the deads.  And the results will be even more surprising if we understand the built environment like a great adobe modeling space  that as a fractal, has autosimilarity in the provincial constructions, passing from the monumental and domestic scale  in the capital to the minimum scale of the gardens plantations. 


Andrews, Anthony P. The U Shaped Structures at Chan Chan, Peru. In the Journal of Field Archaeology, Vol. 1, No 3-4. pp. 241-264. University of Arizona, Tucson.1974. 

Doberti, Roberto. Construcción de la Geometría y Geometría de la Construcción. In Journal of Mathematics and Design. Proceedings de las segundas jornadas de MyD. P. 49-66. 2005

Lumbreras, Luis G. The Peoples and Cultures of Ancient Peru. Smithsonian Institution Press. City of Washington. 1974

Moore, Jerry D. The Archaeology of Plazas and the Proxemics of Ritual: Three Andean Traditions. In American Anthropologist, New Series, Vol. 98, No. 4, pp. 789-802 Blackwell Publishing on behalf of the American Anthropological Association (Dec., 1996)

Moore, Jerry D. Architecture and power in the ancient Andes. The archaeology of public buildings. Cambridge University Press. 1996.

Moore, Jerry D. The Social Basis of Scared Spaces in the Prehispanic Andes: Ritual Landscapes of the Dead in Chimu and Inka Societies.En Journal of Archaeological Method and Theory, Vol. 11, No. 1, March 2004

Moseley, Michael E. Chan Chan: Andean Alternative of the Preindustrial City. In Science. Vol 187, no. 4173, pp. 219 – 225. Harvard University, Cambridge, Massachusetts 02138. Enero 1975

Gorin, Abbye A. Chan Chan, Perú AD 950-1440.  The Latin American Library. Tulane University.

Lynch, Kevin. Good City Form. MIT press. Cambridge. 1994.

Piekarski, Joe A. The applications of Chimú Administrative Architecture: An analysis of architectural form and function in the Moche Valley, Peru. Senior Thesis. University of Wisconsin. La Crosse. 2007

Spinadel Vera W. De, Perera Jorge G., Perera Jorge H. Geometría Fractal. Nueva Librería S.R.L. Ciudad Autónoma de Buenos Aires. 2007.

Throop C. Jason, Murphy Keith M. Bourdieu and Phenomenology. A critical assessment. P. 188-189. SAGE publications, California, 2002.

Topic, John R. From Stewards to Bureaucrats: Architecture and Information Flow at Chan Chan, Peru. In Latin America Antiquity. Pp. 243-274. Society for American Archaeology. Vol. 14, No 3, 2003.

Pictures downloaded from these web sites:

1 Some authors do not consider Laberinto, Tello y Squier as exact citadels, as they lack some of the typical attributes of the rest. For example, Tello does not possess a burial platform, Laberinto is smaller and could have been a nexus. To the purpose of this publication, we are not making distinctions.
2 Though there are no precise Spanish documents on the Chimu rituals, there are pieces of myths that emphasize the differences between nobility and plebeians. Social strata arose since the beginning of times.  One of the legends says that the sun sent three eggs, one of gold, from which caciques and noble men were born; another one of silver, from which women were born and the last one of copper, from which plebeians and their families were born. (See J. Moore, p.103)
3 Religion serves as an excellent mechanism of social control, as human behaviour is linked to rewards and natural punishment.
4 John R. Topic, in his publication From Stewards to Bureaucrats: Architecture and Information Flow at Chan Chan, Peru (september 2003), establishes comparisons between Chimu administrative patterns and those of its predecessors Huari and Tiwanaku and the successor culture, the Inka. Topic demonstrates that the compiling and saving information technology affects the political economy and the State expansion strategies. Here is a clear example of how urban shapes affect the future of States.
5 The best known Andean system to save statistic and narrative information is the quipu, that consists on a string from which another long strings with knots are pending. The knots characteristics, their location, colors, indicate an informative code. Ex. Quantity of animals in a herd.