Blurry Frontiers Between Arts and Architecture

Building-sculpture of "La Mona" in Tijuana. by “Derrick”. From his blog http://derrikchinn.blogspot.com/

As we can see in the following definitions, architecture is completely related to Arts. In the history of modern Art, the collage is an approach to architecture. Collage was invented by the Cubists as a means of exploring the differences between representation and reality. It gave the canvas a 3D dimension.

The first sign of this approach was the experimentation with materials and textures in Post-painterly abstraction; the pure painting seemed to be exhausted, and artists who wished to find their way forward were inclined to abandon temporarily the idea of painted canvas as a vehicle for what they wanted to do or say. This resulted in a new attention to sculpture and an increasing number of experiments with mixed media (E. Lucie-Smith, p. 112), where architecture and art shared a blurry frontier.

Artist Joseph Cornell's box: the 3 dimensions of Art. Web download.

There are some different definitions for the term “Architecture”:

The art and science of designing and erecting buildings based on a functional program and employing the technology available at that time. (Arts and construction)

A style and method of design and construction. (Style and methodology)

Orderly arrangement of parts; structure (Composition and techtonics).

The result of all the constituent elements originated in the relationship between the space that a building produces and its surrounding space. (Architecture conforming and transforming space).

Traditionally, architecture has been considered one of the six Beautiful Arts; in the best of its manifestations, Architecture conjoins beauty and utility, to a degree that both interactive characteristics become essential. It is said that it is only Art when the construction is the expression of a society’s spiritual will in a certain period of history.

The Roman architect Marco Vitruvio (SI BC), in his treaty L’Architettura, says that architecture rests in three principles: the beauty (Venustas) the building possesses; the firmness (Firmitas), or supporting structure; the utility (Utilitas) or function to which the building is destined. In SXV, the architect and artist Leon Battista Alberti sustained that architecture was a construction where the movement of the weights and the union of the bodies, was adapted, of the most beautiful form, to the own needs of the human beings. In SXIX, the architect and theorist Eugene Viollet-le-Duc considered that Architecture or the art of build, had two parts equally important: theory and practice; the art critic and social thinker John Ruskin (1849), who was concerned about socio-cultural and economical issues, defined the architecture as the art of decorate and compose buildings of any destiny which contemplation should contribute to health, strength and pleasure of the human spirit. In modern times, the historian Sigfried Giedion stipulated that the architectural creation was the correct application of materials and of the economic principles to the creation of spaces for the man.

Resuming, in theory, while a building complies with conditions of firmness, function and beauty, it will be considered a work of architecture. And Art, by definition, is the quality, production, expression, or realm, according to aesthetic principles, of what is beautiful, appealing, or of more than ordinary significance. If Art means to satisfy human spirits, then, it has no need for “Utilitas”, though it needs Venustas and Firmitas.

In the last fifty years, beginning with pop art, these considerations are not like black and white. In some cases, Art has been eclipsed by non-art and architecture eclipsed by non-architecture. Here, there is no way to find a frontier between both disciplines, both of them are joined in one discipline, maybe I can call it “artchitecture”.

In “artchitecture”, I find four different resolutions –for now-:

1) The bizarre buildings, closer to vernacular architecture, where everything is possible, any shape or alien image that was never related to architecture.

Ladder to access inside "La Mona". The building-sculpture has an access at the back and a window at the breasts location. It is possible to see the neighborhood around once inside the "statue". Picture by “Derrick”. From his blog http://derrikchinn.blogspot.com/

A bizarre house in Baja California, Mexico.http://www.flickr.com/photos/xdisone/3094148962/

2) The buildings as a clear manifestation of a recognizable artistic style. In this category, I would add the appropriate technology constructions that also belong to landscape and natural art.

Coca Cola building in Las Vegas. Once you approach it, it is possible to see the standard interior floor plans. The sculpture consists of a glass liner, with the shape of a Coke bottle. It is a reminder of pop art with a clean technology. From commons.wikimedia.org/wiki/File:Coca_Cola_bui...

3) Buildings where one or two of the three Vitruvian standard principles are missing, in consequence, the building could be considered a work of art, though it also contains the function….

House in Poland challenging "Firmitas". Created by Polish businessman and philanthropist, Daniel Czapiewski.The house represents the Communist era and the state of the world.http://quazen.com/arts/architecture/crazy-and-bizarre-architecture/

Cantilevered house. This one defies gravity. Cantilever houses use a beam supporting on only one end. The beam carries the entire load of the building without any other external help. Though "Firmitas" is correct, our perception of lack of gravity makes it appear as an object of art. http://quazen.com/arts/architecture/crazy-and-bizarre-architecture/

4) Buildings where the materiality is not conventional and is closer to the collages or installations ideas. The structure (Firmitas) cannot be seen as typical as it is a consequence of utilizing innovative materials due to lack of the standard ones. It is close to vernacular architecture.

Glass bottle house. From Weburbanist.com

More definitions of architecture will be written in SXXI, as there are new dimensions involved. As the scientific dimension, the one that Vitruvio never dreamt of. Genetic manipulation applied to construction materials, virtual spaces, robots building perfect digital designs in the middle of the New York streets, designs that do not contain the imperfections of human labor. So impeccable they are, that any artistic form is developed, for any function and material: "We marry the digital reality of the computer with the material reality of a building," (Science Daily.com, 2009) say the two architects in charge of my NY’s example, Fabio Gramazio and Matthias Kohler, who both teach and research at ETH Zurich. They call this approach "digital materiality", term that is not found in any of the historical treaties.

REFERENCES

http://www.almendron.com/arte/arquitectura/claves_arquitectura/ca_01/ca_011/arquitectura_011.htm
http://www.myetymology.com/encyclopedia/Architecture.html
Robot Builds Brick Wall In New York City. In Science News. October 26, 2009
http://www.sciencedaily.com/releases/2009/10/091026133016.htm
Lucie-Smith Edward. Movements in art since 1945. Singapore, 1995.

The City in Physical Layers

Sierpinsky fractal en 3D, by Johnny Layne. From www.graynod.org/johnny/screenshots.html

The fractal dimension of cities in 2D will be always a value between zero and two. If we calculate the fractal dimension of Buenos Aires, we will see that its value does not differ - at least significantly - of the values reached in other cities. For example, at the neighborhood scale, La Boca (Buenos Aires) is in the range of D=1.80, the same is for East Los Angeles in California. Considering that they are two very different cities, this reduccionist analysis would not be enough to understand the morphological patterns of the city.

One of the important aspects is the constitution of the city in layers. And this time I do not refer to socio-cultural questions, but material constructions in layers.

In this regard, I recall a comment from the architect-archaeologist Dr. Daniel Schavelzon, who, on the Mayan cities, said "........ of NONE historical (excavated) city we have the real floor plan, it is not even possible with the techniques and methods of archeology itself, to have a floor plan of a particular moment in time ....... But our culture works on destruction-construction, which never existed before, it was physical overlay architectures. One above the other, with the abandonment, the contemporary stages between each building were destroyed. Yes, it is complex ,........."

What is the objective to reach in our spatial scale of analysis? Isn't it that cities exist in layers like Jericho and Troya, a city, superimposed to other, and other.....? In the comparison of cities, this issue will be of supreme importance.

Returning to the case of Buenos Aires-Los Angeles, the second, for being located in a seismic region, lack the multiple undergrounds and mysterious tunnels of Downtown Buenos Aires. This outline is to understand that the question does not finish here.

Buenos Aires also has aerial cables and signs that make very different the pedestrian's perception. Los Angeles appears like a city visually ¨cleaner¨. Therefore, depending on the scale selected, these visual elements should be kept in mind.

Two different layers, both graphics are ready for the fractal D calculation. By M. Mahiques

Visualization in 3D of the concepts discussed here. Picture: Internet download.

There is a methodology, for the physicists, and it is the analysis by means of ¨fractal cheeses¨, called this way because of their complex three-dimensional morphology, similar to gruyere cheeses of fractal dimension D between 2 and 3, being 3 the value of the solid cube. These graphs bear extremely complex and unreachable formulas for architects. It is for this reason that I propose humble fractal and lacunarities mensurations in 2D, but in different levels. And the comparative analysis will be simpler, we can draw graphics in 3D but these will be the expression of the results of the layers. Should we wonder then, do we study buildings or a summation of buildings in the space-time? in which period do we produce the temporary cut to carry out the analysis, if everything is constantly transforming? Do we have to be satisfied with contemplation and reach conclusions from the mere observation?

La Ciudad en Capas Físicas

Carpeta de Sierpinsky en 3D, por Johnny Layne. Bajada de www.graynod.org/johnny/screenshots.html

La medición fractal de ciudades en 2D siempre será un valor entre 0 y dos. Si tomamos la medición fractal de Buenos Aires, veremos que su valor no difiere –al menos significativamente- de los valores alcanzados en otras ciudades. Por ejemplo, en la escala barrial, La Boca (Buenos Aires) está en el rango de 1.80, lo mismo que el Este de Los Angeles en California. Considerando que son dos ciudades muy distintas, este análisis reduccionista no sería suficiente para comprender las características morfológicas de la ciudad.

Uno de los aspectos importantes es la constitución de la ciudad en capas. Y esta vez no me refiero a cuestiones socio-culturales, sino construcciones materiales en capas.

Al respecto, recuerdo un comentario del arquitecto-arqueólogo Dr. Daniel Schavelzon, quien al respecto de las ciudades mayas, me decía “........de NINGUNA ciudad histórica (excavada) tenemos su planta real; ni siquiera es posible por las técnicas y métodos de la arqueología misma, tener una planta de un momento determinado en el tiempo....... Pero nuestra cultura funciona por destrucción-construcción, cosa que nunca existió antes, se hacía superposición física de arquitecturas. Una encima de otra; con el abandono se destruyeron las diferentes etapas no contemporáneas entre uno y otro edificio. Sí, es complejo,.........”

Hasta dónde hemos de llegar en nuestra escala espacial de análisis? Acaso no existen ciudades en layers como Jerico y Troya, una ciudad, sobre otra, sobre otra.....? En la comparación de ciudades, este ítem será de suma importancia.

Volviendo al caso de Buenos Aires-Los Angeles, la segunda por estar en una región sísmica, carece de los múltiples subsuelos y misteriosos túneles del Centro de Buenos Aires. Este esquema es para comprender que la cuestión no termina ahí.

Buenos Aires cuenta además con cables aéreos y carteles que hacen muy distinta la percepción del transeúnte. Los Angeles aparece como una ciudad visualmente ¨más limpia¨. Por lo tanto, dependiendo de la escala de trabajo, incluso estos elementos visuales debieran tenerse en cuenta.

Hay una metodología, para los físicos, y es el análisis por medio de ¨quesos fractales¨, llamados así por su compleja morfología tridimensional que asemeja a quesos gruyere de dimensión fractal D entre 2 y 3, siendo 3 el valor del cubo sólido. Estas gráficas conllevan fórmulas sumamente complejas e inalcanzables para arquitectos. Es por ello que propongo humildes mediciones fractales y de lacunarios en 2D, pero en distintos niveles. Y el análisis comparativo será más sencillo, podemos dibujar gráficas en 3D pero éstas serán la expresión de los resultados de las capas. Debiéramos preguntarnos entonces, estudiamos edificios o una sumatoria de edificios en el espacio-tiempo? En qué momento producimos el corte temporal para realizar el análisis, si todo se transforma constantemente? ¿Hemos de contentarnos con observar y sacar conclusiones de la mera observación?

Population and Dissipative Dynamical Systems

Airplane vortex. courses.gnowledge.org/lessonsview?ssid=3394

Dissipative systems are those that use the energy to maintain their form, as the atmospheric vortexes or alive organisms. If we consider for ex., a pendulum, under ideal conditions it would not stop, but under normal conditions, it would arrive to its point of minimum energy to continue moving and then it would stop. Now, if we take a multidirectional system of pendulums, small interferences could cause an avalanche in the system. These interferences can grow and spread through the system with very little resistance. As the energy vanishes in this process, this should be fed so that the avalanches continue.

Landslide induced by an earthquake over a neighborhood near Santal Tecla, El Salvador. 2001.http://carletong.files.wordpress.com/2009/04/landslide1.jpg

Another simple example would be that of the pile of sand. If we slowly add sand to a pile full with sand, a re-accommodation of particles will only take place. The individual grains of sand (or grades of freedom) will not interact in big distances. Continuing the process, we will end up increasing the pile achieving a critical value, with a slope that will produce an avalanche. The pile cannot already be described by means of local degrees of freedom. The distribution of avalanches follows a law of power.

The dissipative systems act opposing to the intent of displacing them of the state where they are. That is to say, they contain regulatory properties. In the dissipation, in the irreversibility, in the lost of equilibrium, is the key of evolution.

Mechanically speaking, dissipative systems are dynamical systems that are characterized by an “internal friction” that tends to contract phase space volume elements. Phase space is a mathematical space spanned by the dependent variables of the dynamical system. In our example of the pendulum, it has a one-dimension movement with a two-dimensional phase space spanned by the position and momentum variables.

These theoretical aspects are not exclusive of the physics, they are also applicable in urban morphology to population's limitless growth, to the ¨urban sprawl¨, and its search of organization levels; to cities and buildings’ overcrowding; to conventillos generation; to the multiple illegal constructions; “hot beds” (camas calientes) habits, usually the result of continuous arrival of immigrants. Severe analogical cases are the refugees coming massively to a country, local habitants being the original “pile of sand”. The system will reach a critical point where the internal regulatory properties will intend to look for the equilibrium. These properties may be reflected in the law, the “razzias”, the deportations, or the provision of dwellings and jobs opportunities, among other situations.

Refugees from Rwanda arrive in Tanzania. Photo by UNHCR/ P. Moumtzis. http://nocameranointervention.files.wordpress.com/2009/03/refugees9.jpg

The energy models provide us helpful analytic tools that explain the development of the societies, their collapse, the wars, the religions. It is considered that the human communities are dissipative systems, because they are in constant fluctuation, and life is in itself an expansible process.

Further readings (for glossary)
http://www.cna.org/isaac/Glossb.htm#Dissipative

Morfología Urbana de Buenos Aires y Geometría Fractal

Analogy between a block in La Boca neighborhood and the Sierpinski gasket.

Analogía entre una manzana de La Boca y la carpeta de Sierpinski. Personal archives.

The following is an excerpt of my article for the proceedings at SEMA Morphology Symposium.

El siguiente texto es parte de mi artículo para el Congreso de Morfología SEMA, 2003.

No reproducir el texto sin permiso.

“De hecho geometría y orden son en más de un caso sinónimos. Para ello basta pensar que así como hay geometrías en plural, con sentidos dialécticos, complementarios y/o antagónicos... también hay niveles de orden en plural y nadie puede fijar frontera exacta entre orden y desorden. Una sociedad humana en tanto grupo constituido sobre la base de un orden de sus interrelaciones implica una geometría de hombres”. (G. Breyer, 2000)

La forma es relativa a nuestro acto de registro de la misma. Hay muchos diseños que son estrictamente euclidianos, pero puede suceder que terminen en versiones fractalizadas, un ejemplo interesante es el quincunx que representa la “luz de Alá”, en las culturas africana y afro-americana. Este símbolo religioso parte de cinco cuadrados, con uno en el centro y uno en cada esquina, finalmente, luego de varias iteraciones, el patrón geométrico es recursivo, por lo tanto es un fractal. Y a veces, el fractal está encubierto, se descubrirá en el cambio de escala, por ejemplo, una aspirina de forma cilíndrica se ve como un hermoso fractal multicolor bajo el microscopio.

Un ejemplo de fractal no intencional, puede encontrarse en los procesos de dinámicas sociales de crecimiento urbano –inconsciente-, que denotan bordes sumamente irregulares. A este tipo de fractal se lo clasifica como DLA (“difussion limited aggregation”), debido a su analogía con el comportamiento de partículas en una solución, atraídas por un electrodo.

En escalas más pequeñas, podríamos suponer, que las ciudades planificadas, -donde el sujeto impone su geometría sobre el medio-, son típicamente euclidianas, pero esta postura se desmorona cuando cambiamos nuevamente la escala y comenzamos a estudiar el tejido de las manzanas típicas de Buenos Aires. Análogamente al quincux, las figuras euclidianas se entremezclan en una urdimbre hasta conformar un fractal, que será plausible de análisis por sus bordes o sus superficies, en plural, ya que cada sección del mismo mostrará distintas conformaciones que resultarán en diferentes valores de dimensión fractal.

-Un fractal tiene una estructura fina; esto es, detalle en escalas arbitrariamente pequeñas, y tiene la propiedad de auto-semejanza, quizás aproximada o estadística. También se caracterizan por su propiedad de ser invariantes ante cambios de escala, aún en la más pequeña.-

El ejemplo de las distintas iteraciones de la Carpeta de Sierpinski hace evidente que las propiedades fractales se refieren a la organización espacial de un patrón, a través de distintas escalas –niveles- de observación: satelital, regional, áreas metropolitanas, pueblos, secciones de pueblos y ciudades, manzanas o conglomerados. Estos patterns pueden consistir en subconjuntos con diferente dimensión fractal, que muestran un comportamiento multifractal.  Pero si un fractal tiene una estructura fina, también deberíamos verificar qué sucede cuando continuamos nuestra observación para escalas menores. Amos Rapoport sostiene, por ejemplo, que la arquitectura doméstica muestra claramente su carácter compositivo: “edificios dentro de edificios”, el techo de una vivienda puede tener una evolución independiente relativa, incluso considera al fuego –ya incorporando significados simbólicos- como un tipo primordial de “edificio” de evolución independiente aunque integrado a la casa; también podemos llegar a la escala del material componente y su textura, etc. Cada forma, sin embargo, no es vista como una entidad aislada sino en función de sus antecedentes: los caracteres de individuación se relacionan a los de la entidad mayor.

“A primera vista nada parece semejar menos a Eudoxia que el dibujo del tapiz, ordenado en figuras simétricas que repiten sus motivos a lo largo de líneas rectas y circulares, entretejido de hebras de colores esplendorosos, cuyas tramas alternadas puedes seguir a lo largo de toda la urdimbre. Pero si te detienes a observarlo con atención, te convences de que a cada lugar del tapiz corresponden un lugar de la ciudad y que todas las cosas contenidas en la ciudad están comprendidas en el dibujo, dispuestas según sus verdaderas relaciones que escapan a tu ojo distraído por el trajín.....”(de “Las Ciudades Invisibles”, Italo Calvino)

Pareciera ser que la búsqueda del conocimiento es perpetua, y ya en una concepción más abarcadora, en niveles metafísicos, difícilmente podamos separar la comprensión de la forma de las sensaciones y los efectos físicos; incursionar el diseño en el espacio gravitacional implica ya un conocimiento de tales efectos, incursionar en el espacio subterráneo también, tal vez sea lo que llevó a algunos vanguardistas a buscar los sonidos de la descomposición subterráneos de las tumbas de personajes famosos –como Borges-, en una curiosa forma de “arte”?

“El estudio de la espacialidad es clave en la producción del conocimiento, en la auto-referencialidad del ser humano, en la construcción del sujeto” (Doberti, 2003).

The Chaos Game: Some Considerations About Urban Patterns

The Chaos game. IFS Fern structure generated by Myriam Mahiques. See the difference for both iterations.

Michael Fielding Barnsley is a researcher mathematician who has worked on fractal compression. By the time Mandelbrot had resolved his set, Barnsley was taking a different road. He was thinking about nature’s images and patterns generated by living organisms. He experimented with Julia sets, always trying to achieve greater variables. Until he realized that randomness was the basis for modeling natural shapes. He used the term “chaos game” to refer the global construction of fractals by means of iterated function systems.

Julia sets, that are outcomes of deterministic processes, proved to have a second valid existence as the limit of a random process. Barnsley suggested an analogy for this situation: one could imagine a map of Great Britain drawn in chalk on the floor of a room. A surveyor could find it complicated to measure this awkward shape with fractal coastlines. But, if grains of rice are thrown into the air, allowing them to fall randomly on the floor, these grains inside the map could be counted. As times goes on, the result begins to approach the area of the map. That is the limit of the random process.

Barnsley and his co-workers produced images of nature by writing down a set of simple rules randomly iterated; the question was how to reverse the process: given a particular shape, how to choose a set of rules. The answer, which he called a “collage theorem” was pretty simple. One can begin drawing the shape to be reproduced. Then, using a computer terminal and a mouse as a pointing device, small copies of this shape would be overlaid on the original image, not exactly on top of it, if needed. A highly fractal shape could easily be tiled with copies of itself and at some level of fractal approximation, every shape could be tiled. However, in Barnsley’s technique, the chances serve as a tool, because the results are deterministic and predictable.

To generate a fern leaf with the chaos game, each new point falls randomly, but gradually the fern emerges.

I take this theory as a reflection for urban patterns. To define a shape, Barnsley had two ways, working in superimposed layers or using falling random points. Applying the concept to urban morphologies, if we use a fractal as an analogical model, we need to understand if the fractal generation is consistent with the urban fabric generation. If not, the fractal will be just a simple drawing in 2D with a similar shape. Nothing else. A fishbone structure, is an abstract simplification of the fern pattern. An analogy for the city, would be an axis (an avenue) and lateral oblique streets. But if the city was planned and not generated by random aggregated houses, Barnsley fern is not a good comparison. Or, we can consider the superimposed layers and find out if it works.

This example of an aerial picture of a neighborhood in China, shows us the different criterion on the subject. Though the regional scale is arranged with cross streets –not exactly a quadrille- the houses are located in straight lines perpendicular to main axes. The first approximation to urban morphology analysis is the Euclidean pattern of lines (houses alignment) across an axis.

Houses alignment. Personal archives.

A fractal pattern discovered through filters. Personal archives.

All patterns together in overlaid layers. Personal archives.

If I decide to work on the image, and see what is behind those lines, I sharpen the edges, make it binary and now I have a fractal pattern with the streets already there, it is not Euclidean any more, as the urban fabric is emphasized over the streets. And if I keep on working, I can see the possibilities of overlapping layers, as Barnsley’s example. That is a more complex fractal image that contains all the information of streets and urban fabric. Compared to Barnsley’s pictures of nature I would say this is a “superfractal”. All three concepts are correct, the difference is which aspect of the urban form will be studied to accomplish our objectives. It is only a matter of professional experience to choose the best one.

REFERENCES

Gleick, James. Chaos. Making a new science. USA. 1987

Barnsley web site:

http://www.superfractals.com/default.htm

Teoría del Caos y Fractales

Wave 1, by Myriam Mahiques

This is a Spanish translation of a previous post (adapted).
For further reading/ Para más lecturas
http://myriammahiques.blogspot.com/2009/10/what-is-fractal.html

Los fractales son la representación geométrica de la teoría del caos. Llamamos caos a todo aquello que no somos capaces de sistematizar.

En 1970, desafiando la física clásica, unos pocos científicos de Estados Unidos y Europa comenzaron a buscar un camino a través del desorden: fue el comienzo de la teoría del caos. Matemáticos, físicos, fisiólogos, economistas, químicos, biólogos, intentaron buscar conexiones entre diferentes tipos de irregularidades; reflexionaron que si bien hay fenómenos que pueden ser descriptos linealmente, es decir que el resultado de una acción es proporcional a su causa, la mayor parte de los fenómenos en la naturaleza son no-lineales, “incontrolables”, como el clima, las turbulencias, el tránsito en una gran ciudad, fluctuaciones en la bolsa, la física del cuerpo humano, etc.

Todo lo investigado se relacionaba directa y formalmente con el mundo natural –las formas de las nubes, las bifurcaciones arteriales, la textura pulmonar, las agrupaciones de estrellas, etc. Al mismo tiempo que el físico matemático Mitchell Feigenbaum comenzaba a pensar en la teoría del caos en Los Alamos, EEUU, en Berkeley se formaba un pequeño grupo de matemáticos que se dedicaba a estudiar “sistemas dinámicos”. En Berkeley, California, se estudiaba el complejo comportamiento de modelos biológicos simples; en Francia se estudiaban turbulencias en fluídos y atractores extraños y en IBM, Benoit Mandelbrot, descubría modelos de comportamiento del precio del algodón a través de todas las escalas e inventaba un neologismo –fractal- para describir una familia de formas dobladas, fracturadas, plegadas que él consideraba un principio organizador en la naturaleza.

La búsqueda de una explicación a los fenómenos complejos e irresolubles mediante modelos matemáticos, configuró la Teoría del Caos, de carácter interdisciplinar, que no niega la ciencia clásica sino que propone dejar de lado el reduccionismo, aplicando otros métodos de estudiar la realidad en una visión de todo. La principal ley de la teoría del caos es que hasta el desorden tiene sus reglas. El sustento de esta nueva ciencia, que algunos consideran disciplina, se debe al matemático, físico y filósofo francés Henri Poincaré, quien a fines del S. XIX, destruyó la imagen clásica de la naturaleza al dudar de la estabilidad del sistema solar y considerar la extraña posibilidad de la existencia de órbitas erráticas y caóticas, preguntándose qué pasaría si al sistema ideal de dos cuerpos añadimos el movimiento de un tercer cuerpo?. Poincaré demostró que el caos es la esencia de un sistema no lineal y que aún un sistema completamente determinado, como los planetas en órbita, podrían tener resultados indeterminados. El sistema, de pronto podría romper en una inquietante complejidad.

La teoría del caos sugiere un mundo fluído e interconectado, concebido como un todo. Contrariamente a los postulados científicos tradicionales, que toman a los seres humanos y la naturaleza como objetos individuales, la teoría del caos considera que todo tiene un valor intrínseco, como el arte. Esto se debe a los efectos no lineales o de retro-alimentación: por ejemplo, los planetas no pueden ser tratados como si sus efectos fueran independientes y se pudieran sumar: si la débil atracción entre un planeta y otro comienza a retro-alimentarse y a acumularse, algunos planetas podrían cambiar su órbita y hasta salir despedidos del sistema solar. Bajo la teoría del caos, los eventos no suceden al azar, las condiciones iniciales son determinantes, pero el producto, por ser dinámico y complejo, implica un resultado impredecible.

Los fractales representan los sistemas dinámicos, la geometría de la naturaleza, las infinitas retro-alimentaciones, en síntesis, lo que no puede ser medido en términos Euclidianos. Un fractal es un modo de ver el infinito.

El término que en latín significa fragmentario o interrumpido, fue presentado por el matemático polaco Benoit Mandelbrot (1924-) por primera vez en su libro “Les Objets Fractals: Forme, Hasard et Dimension” (1975).

Aún no se ha definido un fractal, sino se enumeran sus propiedades características:

Un fractal tiene una estructura fina; esto es detalle en escalas arbitrariamente pequeñas.

Un fractal es demasiado irregular para ser descripto con la geometría euclidiana tradicional, tanto local como globalmente.

Con frecuencia un fractal tiene una cierta forma de auto-semejanza, quizás aproximada o estadística.

En general, la “dimensión fractal” es mayor que su dimensión topológica

Se pueden generar fractales geométricos en forma muy simple, por lo general recursiva. (Spinadel V., Perera J. G, Perera J.H. p. 2, 2000)

Muchos objetos en la naturaleza son mejor descriptos geométricamente como fractales, con caracteres de auto-semejanza en todas las escalas. El universo consiste en racimos de galaxias, organizado en racimos de racimos de galaxias, y así siguiendo. Un excelente ejemplo de fractal es el coliflor: la flor grande contiene florcitas más pequeñas, que a su vez contienen otras, todas con la misma estructura.

Si bien se ha hecho un esfuerzo por caracterizar los fractales geométricamente, no ha habido mucho progreso en comprender su origen dinámico. Tenemos una tendencia a pensar que el universo se forma a partir de estructuras estáticas porque la dinámica que forma estas estructuras tiene mayor escala que el período de observación, que puede ser la vida de un ser humano. Los terremotos que observamos duran unos pocos segundos, y la formación de la falla parece estática, pero se construye en millones de años.

Por lo tanto, el origen de los fractales es más bien un problema dinámico, no geométrico. Las leyes de su física son locales, pero los fractales nunca se organizan en las mayores distancias, a no ser que sean absolutamente deterministas, como el Fractal de Mandelbrot.

Rites of Intensification and The Day of the Dead

Three beautiful girls in typical Mexican skirts celebrating the Day of the Dead. Picture by Myriam Mahiques

Rites of intensification mark crisis in a social group. The crisis could be a plague, war, a severe lack of rain, etc., and mass ceremonies are performed to mitigate the danger. These rites are carried out by a group of people and the effect is to unite people in a common effort in such a way that fear and confusion yield to collective action and a certain degree of optimism until the natural balance is restored. The performance of rites of intensification are not limited to times of crisis. In regions when seasons differ and human activities change accordingly, annual ceremonies develop; this is common among horticultural and agricultural societies. “Participation in such ceremonies cultivates the habit of reliance on supernatural forces through ritual activity, which can be activated in other stressful circumstances”.(Grant, Gorin and Fleming, p.163)

Religious manifestation for the Day of The Dead in Olvera st., Los Angeles. Personal archives.

The day of the deads is a religious celebration in all Mexico and California, on November 2nd, and it surrenders honors to the memory of the members of the family that "have left". The ceremony is tied to the prehispanic agricultural calendar, since this was the only celebration that took place when the crop began.

The precise origins of The day of the Dead (Dia de los Muertos) are not clear and possibly they go back to the Olmecs, the first Mesoamerican civilization of south-central Mexico that originated the Aztecs, Toltecs, Mayas, Zapotecs, etc.

The Aztecs, commemorated the dead for the entire month of Miccailhuitontli which was presided over by the goddess Mictecacihuatl, the Lady of the Dead and Huizilopochtli, the god of war. After the Spanish conquest, when Catholicism became the dominant religion, the customs intertwined with the Christian commemoration of All Saint’s Day on November 1st.

On the left, Mictecacihuatl, the Aztec godess of death. Archivo de Miguel Covarrubias, Universidad de las Americas, Puebla.

Funerary ceremonies, though being rites of passage, can be regarded as rites of intensification too. The difference is that in the rites of passage the individuals have to adapt themselves to a new pattern of interaction, and in the intensification rites, the existing pattern of interaction is periodically reinforced. The death of a person might be a crisis for the entire group, specially if the group is small; the survivors must readjust and restore the balance. In some cultures, the sadness and emotions can take extremes. For instance, one of the parts of the funerary rites of Melanesians was the eating of the flesh of the dead person. This ritual cannibalism, witnessed by anthropologist Bronislaw Malinowski, was performed with “extreme repugnance and dread and usually followed by a violent vomitting fit. At the same time it is felt to be a supreme act of reverence, love and devotion.” (Haviland, p. 371). This ambiguous attitude shows at the same time the disgust and fear to death and the desire to maintain the tie to the dead person.

Celebration of the Day of the Dead in Plaza Olvera, Los Angeles. Personal archives

Mariachis playing typical Mexican songs in Olvera st., celebrating the Dead of the Dead. Personal Archives.

But for Aztecs it was a joyous occasion, and the same is for Mexicans now, as the belief is that spirits of the dead come back to Earth to be with their families and loved ones, they eat, drink and celebrate with them for the good memories. It is a celebration of both life and death. Regarding the urban aspects of these ceremonies, altars of the dead are installed in houses, plazas and cemeteries, tourists and inhabitants join the celebrations and for two days some areas of the city are transformed by means of the religious installations.

For Native American cultures, both life and death are two dual aspects of the eternal cycle. Let me offer a tribute to the celebration of the day of the dead with a beautiful poem by Mary Elizabeth Frye: Do not stand at my grave and weep.

http://en.wikipedia.org/wiki/Do_not_stand_at_my_grave_and_weep

Do not stand at my grave and weep.

I am not there, I do not sleep.

I am a thousand winds that blow.

I am the diamond glints on the snow.

I am the sunlight on the ripened grain.

I am the gentle Autumn’s rain.

When you awaken in the morning hush,

I am the swift uplifting rush,

Of quiet birds in circled flight.

I am the soft stars that shine at night.

Do not stand at my grave and cry:

I am not there, I did not die.

REFERENCES

Grant, Jim; Goring, Sam; Fleming Neil. The archaeology coursebook: an introduction to study skills, topics and methods. Routledge, New York, 2005

Haviland, William A. Cultural Anthropology. University of Vermont. USA 1990

http://customsholidays.suite101.com/article.cfm/origin_of_the_day_of_the_dead

http://myriammahiques.blogspot.com/2009/10/religious-altarcitos-in-chicanos-houses.html