Landscape digital paintings

Folliage

Leaf

Violeta de los Alpes

Landscape

Orchid

All digital paintings by Myriam B. Mahiques.

Urban Simulation From Newton Fractals

Rural settlement of Nahalal in Palestine founded in 1921, shows a restricted growth. Published in National Geographic, february 2008, page 33. On the right side, its corresponding Fourier Transform.

First of all, a warning. We must not be tempted for the beauty of fractals, the intention in my exercise is to find the hidden pattern of a fractal urban simulation, originated in the Jewish rural settlement of Nahalal. Alan Mairson, author of the text accompanying the picture in National Geographic, says the settlement´s shape is rigid, with limited growth. The concept of this center with rays is a reflex of egalitarianism: the circle allows everybody to have equal access to neighbors and facilities.The center contains public buildings surrounded by a residential ring, the farms are beyond and around. When the cooperativist spirit was dissipated in the 90´s, the urban shape was not so strict any more. "The village layout in Nahalal, devised by architect Richard Kauffman, became the pattern for many of the moshavim established before 1948; it is based on concentric circles, with the public buildings (school, administrative and cultural offices, cooperative shops and warehouses) in the center, the homesteads in the innermost circle, the farm buildings in the next, and beyond those, ever-widening circles of gardens and fields. initially to 80 equal parts, 75 parts to the members and 5 parts for the agricultural school (the first two parts and the last three parts contain the agricultural school). This equal parcelling of the land became the trademark geometric shape of Nahalal." (Excerpt from Wikipedia.org)Newton fractals are helpful for simulations of growth for urban morphologies with this particular pattern. To find an analogical Newton fractal is not enough, we still need to find hidden patterns consistent with urban simulations.

Here we have two analogical fractals originated in generalized Newton´s fractals. On the right side, the hidden structure is shown in the FFT image (Fourier Transform).

We have found the rays and the center; The first fractal is a representation of a growing center, the avenues are still empty, they can be associated to freeways. The second fractal is showing the loss of egalitarianism, the urban fabric is extended between the rays, in a chaotic pattern.

The next step is to analyze the Fourier Transform image, to take a decision about the best simulation of urban growth.

On the left, the binary image of the first FFT is shown. On the right, I show a binary image of the second FFT with the chaotic pattern. Definitely, the second image is more accurate for Alan Mairson's explanation. The pattern of rays is lost, as it will be in the future, if farms were mixed with residential use. But the main center, origin of the primary idea is still kept.

Nahalal between 1939 and 1945. Picture by Frank Hurley. http://nla.gov.au/nla.pic-an23565289

All simulation images by Myriam B. Mahiques

Fractals From Newton Method

Generalized Newton, by Myriam Mahiques

Newton's method, also called the Newton-Raphson or Newton iteration method, is a root finding algorithm of a real function. It can be used to find the minimum and maximum of a function.

Isaac Newton discovered what we now call Newton's method around 1670. Although Newton's method is an old application of calculus, it was discovered relatively recently that extending it to the complex plane the result is a boundary set, a very interesting fractal pattern.

My interest in this type of fractal is its analogy with urban morphogenesis with a center and radial avenues. It can be applied to urban morphology simulation exercises.

All images generated by Myriam Mahiques.

REFERENCES
http://www.chiark.greenend.org.uk/~sgtatham/newton/

http://facstaff.unca.edu/mcmcclur/mathematicaGraphics/Newton/index.html

Cálculo de la Dimensión Fractal

Membrane 2, by Myriam B. Mahiques

Teóricamente, las formas que tienen la misma rugosidad, debieran tener un comportamiento similar. Y si podemos visualizar la forma, podemos comprender el sistema.

La geometría fractal cuantifica la rugosidad de los objetos mediante un índice llamado “dimensión fractal”. Normalmente consideramos que los puntos tienen dimensión 0, las líneas 1, las superficies 2 y los volúmenes 3. Es lo que llamamos “dimensión topológica”.

Sin embargo, una curva rugosa, que recorre una superficie puede ser tan rugosa que casi llene la superficie en la que se encuentra. El follaje de un árbol o el interior de un pulmón pueden ser entonces tridimensionales. Podemos así pensar que la rugosidad es un incremento de la dimensión: una curva rugosa tiene una dimensión entre 1 y 2 y una superficie rugosa entre 2 y 3.

Líneas y planos son autosimilares y pueden cortarse –por ejemplo- en 4 segmentos de intervalos autosimilares, cada uno de la misma longitud, y cada uno de los cuales puede ser magnificado por un valor 4 del segmento original. Pero también podemos cortarlos en N partes autosimilares, cada uno con un factor de magnificación N.

Tomemos un cuadrado. Lo descomponemos en 4 sub cuadrados autosimilares y el factor de magnificación es 2. Si lo cortamos en 9 piezas, el factor de magnificación será 3, y si se corta en 25, el factor de magnificación será 5.

Estos esquemas han sido bajados de Internet.

Esta es una forma alternativa de especificar la dimensión de un objeto autosimilar. La dimensión es el exponente del número de piezas autosimilares con un factor de magnificación N, en el que la figura puede ser partida. O sea,

D= log (número de piezas autosimilares)/ Log (factor de magnificación)
D= log N2 /Log N
D= 2 Log N/Log N=2
La fórmula matemática más común es la que corresponde a Hausdorff-Besicovitch:

donde N es el número de partes idénticas en que se ha dividido el objeto auto-semejante y r es la relación de las partes con el todo.

Similarmente, la dimensión de un cubo es:

D= log N3 /Log N

D= 3 Log N/Log N=3

Hay varios métodos manuales para el cálculo de la dimensión fractal; si el fractal es muy complejo, se utilizan computadoras.

Un dibujo, un esquema, un plano, una fotografía pueden ser útiles. Y ayudarán a desarrollar nuestra intuición en búsqueda de pruebas.

Comparativamente, si un fractal se mide en un número de iteraciones, para una estructura empírica tomaremos progresivas variaciones de medida (con la aplicación de un factor de escala). Un fractal determinista, en cualquier escala de observación, dará el mismo valor de Dimensión Fractal, o sea, es perfectamente autosimilar. Pero, una morfología urbana no es un fractal determinista; con lo que la comprobación de autosimilitud no registrará valores de D idénticos, pero podemos inferir, al encontrar valores similares, que “la forma muestra propiedades análogas a la autosimilitud fractal”.

Beginning the fractal analysis of simulation of Urban Fabric Regeneration

These graphics are analysis of fractal dimension, surface plots and plots of the images I simulated for Nagasaki. My objective -today- is to find out if I can get closer to the original fractal dimension of Nagasaki, before the Atomic bomb's impact. Then, I'll see if a similar fractal Dimension is representative of the original characteristics of the urban pattern.

This is Nagasaki before the bomb. Fractal Dimension D is in the rank of 1.80

After the bomb, the fractal dimension D is approximately 1.55. From now on, we'll see how we approach to the original fractal dimension. Surface plots will be showing the rugosity of the urban fabric. The land will be filled slowly following the main ruins concentration.

The problem I find here, is that I can get very close to the first fractal dimension  value, but the urban fabric, as shown on the left side plot, still does not have the complexity shown on the first image before the bomb. This one appears softer. It means, probably lacunarities are Ok, but the urban morphological pattern is still not representing the inhabitants' main urban structures. Fractal Dimension, is not enough for this exercise, and more adjustments of the urban fabric will be needed.

To be continued.....

Simulation for the regeneration of urban fabric of Nagasaki

Nagasaki ruins. http://invitadodeinvierno.files.wordpress.com/2009/08/nagasaki.jpg

This post is the very beginning of my paper 

Nagasaki along the years: An urban fractal analysis

REFERENCE:

Journal of Mathematical Modelling and Application

2011, Vol 1, No 5

ISSN: 2178-2423

You can read it also in the blog where I'm compiling my publications 

(still working on this blog)

http://urbanfractality.blogspot.com/2013/02/nagasaki-along-years-urban-fractal.html

Catastrophe theory, was originated with the work of the French mathematician René Thom in the 1960s, as a branch of bifurcation theory in the study of dynamical systems. Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation. (Wikipedia.org). Resuming, it means the loss of a stability in a dynamic system. As I've stated before, cities are dynamic systems and the loss of stability is  reflected in the geometry of their urban patterns.

I have made these simulations in order to prove that it is possible to regenerate the old urban fabric from total destruction. The pattern will not be exactly as the original one, and of course this is not the intention. My objective is to find a pattern that shows the same basic fractal (?)pattern before the destruction, to maintain the inhabitants' cultural manifestations. My goal is to find a methodology of reconstruction that could be applied after different catastrophes as volcanoes eruptions, landslides, floods, earthquakes, wars. Maybe it is implicit that I do not support the reconstruction by catalogue, as done with Berlin. Though, it was an ideal solution for long years ago. Now, the technology allows us new ways of understanding the city as an organism, and as such entity, it can reconstruct itself.

This is a binary file of Nagasaki, before the atomic bomb destruction. There is no urban quadrille, its urban morphology looks like a fractal pattern. The background jpg image had to be converted into a binary file to work with different tools.

This is a binary file of what was left after the bomb. Notice that no urban pattern is seen. Just a few ruins surrounded by emptiness. These ruins will be the seed for urban regeneration.

My first simulation indicates a new urban growth beginning in the areas of more -former- densification. The pattern is dispersed as the original one. The white empty areas will be growing more slowly.

I have generated this one from a process of erotion-dilation, alternating the tools. I did not want more agglomerations, the original urban fabric was not so compact, but, it was many years ago, and now I need to consider a certain urban sprawl to be closer to reality.

I have more density now, and the process continues. I've emptied some small areas to include new plazas-open spaces. It is important to clarify that no center was considered, it is only a natural regeneration of the old urban fabric. The impact of a a new commercial center, for example, would change this morphogenesis. Please disregard the circle, it is part of the aerial picture I've been working with and does not belong to the simulation.

Breves Conceptos Sobre Arquitectura y Cultura

Angkor Vat, Cambodia. http://www.dfaweb.de/image/kbjavt3.jpg

La arquitectura existe en la relación entre la construcción, el pensamiento, y el habitar. Por ende, se produce un diálogo entre la cultura y el edificio.

¨Por definición es arquitectura todo lo que concierne a la construcción y es con las técnicas de la construcción que se instituye y organiza en su ser y en su devenir, esa entidad social y política que es la ciudad. La arquitectura no sólo le da cuerpo y estructura sino que la vuelve significante con el simbolismo implícito en sus formas; así como la pintura es figurativa, la arquitectura es representativa por excelencia. En la ciudad todos los edificios, sin excluir a ninguno, son representativos y con frecuencia representan las malformaciones, las contradicciones, las vergüenzas de la comunidad¨. (Giulio C. Argan, 1980)

Los asentamientos y lugares se definen culturalmente, los grupos de usuarios son parte de una función de la cultura; el cómo del comportamiento y las estructuras sociales es altamente variable y puede ser visto como una expresión de la cultura. Nadie puede “ver” la cultura, sólo sus efectos, significados, expresiones, o productos: “Si los términos teóricos son sólo modos abreviados de aludir a hechos observables, entonces ellos podrían ser eliminados del lenguaje de la ciencia sin pérdida alguna. Pero no es así. Los términos teóricos tienen un contenido inobservable, y sin embargo, no por eso carente de sentido”. (J. Samaja, 1993)

Machu Picchu, Perú.Foto de http://www.pasaporteblog.com/tag/machu-picchu/

A la tendencia generalizada de realizar inferencias de una entidad inobservable que nace en la mente (la cultura) a través de la pregunta ¿Qué hacen ellos?, el arquitecto antropólogo Amos Rapoport ha agregado la pregunta ¿Qué son ellos? y consecuentemente, ¿Cómo hacen ellos?

En este sentido amplio, la cultura:

Permite mantener la identidad de los grupos.

Actúa como mecanismo de control , llevando información al comportamiento y a los artefactos que serán creados.

Metafóricamente es el ADN, un diseño para lo viviente.

Actúa como estructura que da significado a lo particular.

Queda asumido entonces, que la cultura y el ambiente diseñado son unidades no equivalentes en escala, y la cultura es un vasto dominio donde la forma construída es una pequeña parte de ella y se subsume a ella.

Como no es posible relacionar cultura y ambiente en este nivel de generalidad y abstracción, tampoco es posible restringir los conceptos y considerar la posibilidad de “diseñar edificios para una cultura particular”. Esta transición Rapoport la encuentra en el término “lifestyle” (estilo de vida), como el resultado de elecciones acerca de cómo distribuir los recursos.

El estilo de vida implica los sistemas de actividades que realizan los grupos, que a su vez tendrán sus criterios según su edad, sexo, raza, religión, profesión, ideología, historia, influenciados por la información y el marketing.

El concepto de sistemas de actividades es por lo tanto más concreto y los diseñadores-investigadores estamos familiarizados con él y se puede utilizar en los análisis urbanos, siempre recordando los niveles por los que pasamos hasta llegar a él.

Se trata finalmente, de dar una clara idea de la relación entre los edificios y el hombre, como oposición a las posturas donde se define al objeto arquitectónico como producto del arquitecto poseedor de un rol autocrático, erigiéndose en genio creador. En el campo del Diseño, en su aspecto artístico, es imposible evitar el subjetivismo y una cierta cuota de narcisismo. Ernesto Sábato, a propósito del arte, en su libro “Abaddon el Exterminador”, escribe lo siguiente: “Objetivismo en el arte! Si la ciencia puede y debe prescindir del yo, el arte no puede hacerlo, y es inútil que se lo proponga como un deber. Esa “impotencia” es precisamente su virtud...... Esas misteriosas grutas que habitan las criaturas de Leonardo, esas azulinas y enigmáticas dolomitas que entrevemos, como en un fondo submarino, detrás de sus ambiguos rostros, qué son sino la expresión del espíritu de Leonardo?”. (E. Sabato, 1974)

Esfinge de Giza, Egipto. http://www.argentinamatch.com/portal/uploads/giza-esfinge.jpg

Una visión más global es la que desarrolla Amos Rapoport, en su libro “House, Form and Culture”, ya que interpreta a la arquitectura desde la antropología y la etnografía, de manera constructiva, a partir de la forma y concepciones de espacio activo e inactivo, en paralelo con la evolución humana y cultural. La investigación se sigue desarrollando mediante una red mundial multidisciplinaria.

Rapoport encuentra tres componentes en los estudios de comportamiento ambientales:

Asentamientos y lugares (concreción de la cultura)

Grupos de usuarios (función de la cultura)

Fenómenos de comportamiento social (expresión de la cultura)

....¨la idea que tenemos de la ciudad y que no se ha modificado por el momento, es la de un acumulo cultural que da al núcleo la capacidad de organizar una zona más o menos extensa de territorio. Sin estos puntos de concentración e irradiación cultural no es concebible, en la actualidad, ninguna forma de organización del ambiente¨. (G. C. Argan, 1980)

LECTURAS PROPUESTAS

Giulio Carlo Argan. Arquitectura y Cultura. 1980

http://www.revistacontratiempo.com.ar/argan1.htm

Architecture, Urbanism and ¨Cosmovision¨

Indigenous mysticism. www.stumbleupon.com/stumbler/anphibius/all/

It was Geoffrey Chew who introduced the term Bootstrap in 1968; from a philosophical perspective, Geoffrey's vision included a strong relational cosmology which included consciousness as a fundamental part of the universal bootstrap. Fritjof Capra's The Tao of Physics (1975) made liberal use of this idea to support a Zen based model of physics. In his book, Capra makes a particular contribution on the parallelism between Physics and mysticism, bringing the science to a more popular approach; what has become the belief of the New Age movement. Capra’s claim is that Oriental misticism provides a consistent and relevant philosophical background to the theories of contemporary science, though the parallels strictly apply to the verbal formulations. In this theory, it is implicit the criticism of current non-holistic (reduccionist) assumptions about the nature of reality. (W. Hanegraaff) What is important, indeed, is that higher understanding can be built on deeper roots with a holistic framework; a possibility that cannot be considered in Western scientific paradigm because its presuppositions are directly refuted by the evidence of advanced physics (W. Hanegraaff). Resuming, Oriental philosophical points of view make sense of science, but do not explain it. Professor Geoffrey Chew’s theory of Bootstrap, states that there are no fundamental entities (laws, particles, fields, principles, equations) in nature, which in turn cannot be reduced to its fundamental entities, it can only be fully understood through the autoconsistency of its elements: there is no entity of main law, since the Universe is seen as a dynamical web of interrelated events. The global consistency of its interrelations determines a spontaneous process of self-organizing emergence that conforms the total structure of the web. Capra considers bootstrap as the culmination of his hyphotesis, and he finds the metaphor in the Avatamsaka Sutra idea of penetration, expressed in the 2500 years old metaphor of the Hindu god Indra’s net: his heaven is portrayed as a network of jewels arranged in such a way that looking at any of them you can see all the others reflected in it. (Sal P. Restivo)

Indra’s net artistic interpretation.

http://suejames.com/wp-content/uploads/74443.jpg

“Since motion and change are essential properties of things, the forces causing the motion are not outside the objects, as in the classical Greek view, but are an intrinsic property of matter. Correspondingly, the Eastern image of the Divine is not that of a ruler who directs the world from above, but of a principle that controls everything from within:
He who, dwelling in all things,

Yet is other than all things,

Whom all things do not know,

Whose body all things are,

Who controls all things from within-

He is your Soul, the Inner Controller,

The Immortal.”

(excerpt from the Tao of Physics)

These concepts are tied to Eastern thought, in clear opposition to the Western thought. Philosophies such as Hinduism, Buddhism, Taoism, Zen, practiced by Chew and other physicists and Western thinkers have contributed to other scientists perceive the physical world differently with a new view of reality based on the territory in harmony with spiritual traditions. Traditional Western science, working with questions formulated with total clarity and experimentally verified, did not accept the ambiguity of Chew’s theory and hence did not assigned to the Bootstrap approach the character of science.

My reference to this theory is to emphasize the contribution of Eastern thought on the idea of a new paradigm closest to reality, where knowledge is a network without a solid foundation, where the man lives in connection with mysticism and religious beliefs that are often manifested in humble offerings to beings “from beyond”.

In urban morphology, for example, the anthropological studies of the 70’s, focused in the theory of “Central Place¨ postulated by the German geographer Walter Christaller, have been replaced by current theories competing for an explanation of fractal morphologies related to more domestic shapes of smaller scales, especially in workers’ homes of Mesoamerican cities. Current trends have been to link all the cities, or parts of them, with the elements of a “cosmovision”, cosmology or cosmogony, documented through ethnohistoric writings and images.

A local “Radha” throws flower petals during celebrations ofHoli in Khatraj, India. Photo: Ajit Solanki

http://humanflowerproject.com/images/uploads/radha-throwing-flowers.jpg

Nowadays, it is imperative to have additional research to help clarify the morphologies of the cities inhabited by indigenous people or their descendants, and their relationship with the physical environment and the objects of their daily veneration, including the creation myths. We cannot complete our knowledge of the social practices and the consequent architectural-urban morphologies utilizing only mathematical models borrowed from the Physics, without taking into account the representational spaces derived from the history and spirituality, led by the gestures and actions of those who inhabit it. These representational spaces can be analyzed from mathematical models in combination with other disciplines such as history, sociology, anthropology, among others.

“In the Penal Colony” by Oscar Grillo. 2007

In a way, this application of cultural-historical concepts in architectural and urban theories takes us back to Kafka’s story, ¨In the penal colony”, where the meaning of the sentence was not understood by the prisoner until the words literally penetrated his skin with needles, and were slowly clarified in his mind. An outsider to the process, could not read the words embellished with several mannerisms, therefore, was not part of the prisoner’s knowledge of the sentence, as we are not part of the knowledge and inner meanings of others’ imaginary. But we can try to be.

REFERENCES

Hanegraaff, Wouter J. New Age Religion and Western Culture. Esotericism in the Mirror of Secular Thought. E. J. Brills, The Netherlands. 1996

Capra, Fritjof. The Tao of Physics. 1975

http://www.bibliotecapleyades.net/archivos_pdf/taophysics.pdf

Restivo, Sal P. The social relations of physics, mysticism and mathematics. D. Reidel Publishing Co. Holland, 1985