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ART AND/OR SCIENCE? ANTÓNIO PEDRO A. N. L. LIMA F.C.T. - Fundação para a Ciência e a Tecnologia C.I.A.U.D. - Centro de Investigação em Arquitectura, Urbanismo e Design Universidade Técnica de Lisboa - Faculdade de Arquitectura António Pedro A. N. L. Lima, Ph.D. E-mail: firstname.lastname@example.org INTRODUCTION Geometry is measure, is concept and is, above all, order. Along with other areas of mathematics, or other sciences such as physics, geometry, particularly in the last hundred years, has been extending its horizon as the result of the development of new theories, like relativity, non-linear dynamics and chaos, providing art and architecture with new instruments of reflection, new languages, new concepts, new methods and techniques, fundamentally based on the development of informatics and freeing itself from the understanding of form and space defined by its Cartesian dimensions or strictly based on Euclid’s postulates and common notions. Mandelbrot’s Geometry In his work, The Fractal Geometry of Nature, Benoît Mandelbrot, understanding the richness and importance of patterns and techniques common to the work of mathematicians like Cantor, Hausdorf, Koch and Sierpińsky, among others, created the foundations of a new geometry able to describe many of the irregular and fragmented patterns that surround us. This new geometry was coined fractal geometry. A few years later, Falconer [2; 3] gave a clear definition of the objects generated by this new geometry: “When we refer to a set F as a fractal, therefore, we will typically have the following in mind: (i) F has a fine structure, i.e. detail on arbitrarily small scales. (ii) F is too irregular to be described in traditional geometrical language, both locally and globally. (iii) Often F has some form of self-similarity, perhaps approximate or statistical. (iv) Usually, the ‘fractal dimension’ of F (defined in some way) is greater than its topological dimension. (v) In most cases of interest F is defined in a very simple way, perhaps recursively.” Self-similarity, I. F. S. and Fractal Geometry One of the most recognizable attributes of a certain class of fractal objects lies in their recursive character, i.e., in the fact that they are generated by an infinite repetition of a same rule over the result of a previous operation where this same rule has already been applied, fig. 1, 2 and 3.
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Figure 1: First four iterations of a gone dimensional h fractal object.
Figure 2: First four iterations of a htwo dimensional h fractal object.
Figure 3: First four iterations of a â€œthree dimensionalâ€? fractal object. Self-similarity emerges as a consequence of this recursive character and it refers to a class of objects that, in the whole or partially, strictly or statistically, fig. 4, are composed by parts similar to the entire set.
Figure 4: Strict self-similarity on a Koch curve and statistic self-similarity on a natural fractal. This characteristic evokes an iterative process of object generation named Iterated Function System, or I. F. S., and it starts with an initiator, in many cases a segment of straight line, and a generator, composed by a predefined number of reduced copies Figure 5: The initiator, in black, and the generator, in red, of a Sierpinsky Dragon. (also see figure 1)
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of the initiator, submitted to a predefined order, fig. 5. A specific rule determines how the initiator is replaced by the generator, and how, in any iteration, each segment is replaced by reduced copies of the generator, also giving their respective orientations, if, and when, applicable. Fractal Geometry in Art and in Architecture A few characteristics of fractal geometry, like the repetition of a same base shape at several scales, have been used in art and in architecture throughout history and among many different cultures, from America, with Jackson Pollock´s paintings , to Asia, in some indian temples , and from Europe, with Vieira da Silva´s paintings, to Africa, in a Ila village, southern Zambia , as an example. Before the theorization of this new geometry, such characteristics were used in an intuitive way. Only after the publication of Mandelbrot’s first book on fractal geometry we can declare the intentional use of fractal characteristics in the arts and in architecture. The work of Ashton Raggatt McDougall, on the RMIT Storey Hall project , Daniel Liebskind, on the Victoria & Albert Museum extension , Peter Eisenman, on the Cannaregio Town Square project , Steven Holl on the Woningbouwvereniging Het Oosten building , in architecture, as well as the work of Bathsheba Grossman  and of Koos Verhoeff , in Sculpture, and of Michael A. Coleman  and of Miguel Arzabe , in painting , are a few examples of the intentional use of fractal objects and of fractal characteristics. Thus, if in the first examples, fractal characteristics, related to the shape repetition at different scales, were used in an intuitive way, these contemporary architects and artists made an intensional use of fractal objects, or even fractal patterns, to generate architecture and art. With the research we started for our PhD thesis , new instruments, that enable the effective utilisation of concepts directly connected to fractal geometry, were presented through the implementation of specific algorithms written in AutoLISP, under AutoCAD environment, and working as an automatic shape generator tool. Starting with a deterministic fractal object, the Menger sponge (fig. 3), and supported by the idea that simple rules can generate infinitely complex structures and behaviors, new instruments were developed by stages, from the understanding of how to write the specific deterministic fractal algorithm, to the introduction of multiple probabilistic characteristics. If, in a second stage, the introduction of probabilistic characteristics was carried out over this object’s generative process, maintaining the proportions of its respective parts, fig. 6. Figure 6: 3rd iteration In a third stage, changes were introduced in of a probabilistic Menger its relative proportions in order to obtain a new sponge, preserving its relative object possessing more complex and dynamical proportions. 22
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characteristics, fig. 7, and, in a final stage, using the AutoCAD´s 256 colors palette, we introduced probabilistic color characteristics to the last version of the algorithm, where a large variety of color intervals is available, from a single color to a maximum interval of 255 colors, fig. 8.
Figure 7: 3rd iteration of a probabilistic cubic sponge with different relative proportions. Figure 8: 3rd iteration of a multicolored probabilistic cubic sponge. However, and due to the complexity of the instructions and symbols used on all these objects definition, the present paper is far from being an extended presentation of the way how these algorithms were written under AutoLISP programming language. Nevertheless, it is important to understand some basic features and the generative mechanism invariants there used, somewhat similar to the above presented example for the I.F.S. generative process, fig. 5. Fractal Objects, Turtle Graphics and AutoLISP Linear fractal objects are usually drawn using turtle graphics. This method of programming vector graphics makes use of a relative cursor, named turtle, embedded in a three dimensional Cartesian coordinate system where a dimensional unit and two angles define the turtle position in the Cartesian space. In these conditions, the turtle can perform four basic functions: (i) Draw a one unit lengthen line and update the turtle’s position (F – forward). (ii) Go ahead one unit without drawing (G – go). (iii)- Rotate to the left or to the right by the first angle, in the 2D plane (+ or -). (IV) Rotate to up or to down by the second angle, the elevation angle (/ or \). If we use a von Koch snow flake as an example, fig. 9, we have the axiom (F (- 120.0) F (- 120.0) F), corresponding to the drawing of an equilateral triangle, and the set of rules ‘((F F (+ 60.0) F (- 120.0) F (+ 60.0) F)), corresponding to the set of segments that will replace each side of the triangle to obtain the first level of this object generation and every resulting segments of the subsequent levels of this infinite generation process. In this specific case, the set of rules has one single rule where F is an instruction, an operator, which acts also as a symbol for substitutions, and (+ 60.0), or (- 120,00), are rotation instructions, exclusively processed by the turtle graphics drawing part of the algorithm. 23
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Figure 9: First four iterations of a von Koch snow flake. The turtle graphics interpreter was also extended to support a wide range of other features, including stacks and mesh drawing, used to define the three objects presented in fig 6, 7 and 8. This way, specific algorithms had to be written to define them. However, these algorithms are always composed by an axiom and a list of rules, whose complexity depends on the specific characteristics of each object. The object presented in fig. 7, was geometrically defined by each cubic component’s faces. For that, there was used the algorithm [ K ] G [ K ] G [ K ] (+ 90) G [ (- 90) K ] G [ (- 90) K ] (+ 90) G [ (- 180) K ] G [ (- 180) K ] (+ 90) G [ (- 270) K ] G (+ 90) (/ 90) G (\ 90) [ K ] G G [ K ] (+ 90) G G [ (- 90) K ] (+ 90) G G [ (- 180) K ] (+ 90) G G (+ 90) (/ 90) G (\ 90) [ K ] G [ K ] G [ K ] (+ 90) G [ (- 90) K ] G [ (- 90) K ] (+ 90) G [ (- 180) K ] G [ (- 180) K ] (+ 90) G [ (- 270) K ]) (K K) ((K D) M[ [ G (+ 90) M[ [ G (+ 90) M[ [ G (+ 90) M[ [ G (/ 90) G M[ ] (/ 90) G M[ ] (/ 90) G M[ ] (/ 90) G M[ ] (MD (7 6 4 5)) (MD (6 5 0 1)) (MD (7 6 3 0)) (MD (7 4 3 2)) (MD (5 4 1 2)) (MD (3 2 0 1)) M]]) ) l) to write the list of rules that define the object., where K is the axiom and where (MD (7 6 4 5)) (MD (6 5 0 1)) (MD (7 6 3 0)) (MD (7 4 3 2)) (MD (5 4 1 2)) (MD (3 2 0 1) represents a position of each square face, fig 10. From there, in order to define the other two sponges, fig. 7 and fig. 8, a totally different approach was used based on the definition of its composing polygons instead of defining their components by the respective faces. For that, there was used the algorithm ( (CUBE (E setq q (u3 1)) (E setq rx (/ (* (- 1 q) dx) 2) ry (/ (* (- 1 q) dy) 2) rz (/ (* (- 1 q) dz) 2) dx (* q dx) dy (* q dy) dz (* q dz)) PUSH [ (E setq dx rx dy ry dz rz) CUBE RESTR ^ (+X rx) (E setq dy ry dz rz) CUBE RESTR ^ (+X rx) (+X dx) (E setq dx rx dy ry dz rz) CUBE RESTR ^ (+X rx) (+X dx) (+Y ry) (E setq dx rx dz rz) CUBE RESTR ^ (+X rx) (+X dx) (+Y ry) (+Y dy) (E setq dx rx dy ry dz rz) CUBE RESTR ^ (+X rx) (+Y ry) (+Y dy) (E setq dy ry dz rz) CUBE RESTR ^ (+Y ry) (+Y dy) (E setq dx rx dy ry dz rz) CUBE RESTR ^ (+Y ry) (E setq dx rx dz rz) CUBE RESTR ] (+Z rz) [ (E setq dx rx dy ry) CUBE RESTR ^ (+X rx) (+X dx) (E setq dx rx dy ry) CUBE RESTR ^ (+X rx) (+X dx) (+Y ry) (+Y dy) (E setq dx rx dy ry) CUBE RESTR ^ (+Y ry) (+Y dy) (E setq dx rx dy ry) CUBE RESTR ] (+Z dz) [ (E setq dx rx dy ry dz rz) CUBE RESTR ^ (+X rx) (E setq dy ry dz rz) CUBE RESTR ^ (+X rx) (+X dx) (E setq dx rx dy ry dz rz) CUBE RESTR ^ (+X rx) (+X dx) (+Y ry) (E setq dx rx dz rz) CUBE RESTR ^ (+X rx) (+X dx) (+Y ry) (+Y dy) (E setq dx rx dy ry dz rz) CUBE RESTR ^ (+X rx) (+Y ry) (+Y dy) (E setq dy ry dz rz) CUBE RESTR ^ (+Y ry) (+Y dy) (E setq dx rx dy ry dz rz) CUBE POP ] (+Y ry) (E setq dx rx dz rz) CUBE) ((PUSH D) (E tupush dx) (E tupush dy) (E tupush dz) (E tupush q) (E tupush rx) (E tupush ry) (E tupush rz)) ((POP D) (E setq rz (tupop)) (E setq ry (tupop)) (E setq rx (tupop)) (E setq q (tupop)) (E setq dz (tupop)) (E setq dy (tupop)) (E setq dx (tupop))) ((RESTR D) (E setq rz (turest 0)) (E setq ry (turest 1)) (E setq rx (turest 2)) (E setq q (turest 3)) (E setq dz (turest 4)) (E setq dy (turest 5)) (E setq dx (turest 6))) ((CUBE D) (E tqbx 24
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dx dy dz)) (PUSH PUSH) (POP POP) (RESTR RESTR) ) l) to represent the list of rules needed to generate the object. In all cases, the sponge construction sequence is identical, fig. 11.
Figure 10: Scheme used for positioning the faces of each cubic sponge component. Figure 11: The three levels of a sponge generation sequence (from left to right).
Conclusions To the questions raised by complexity and by a family of patterns, many of them natural, that gstandard h Euclidian geometry wasn ft prepared to describe, Mandelbrot answered with the creation of a geometry that, introducing a few concepts, like self-similarity and non-integer dimension, extended Euclidian geometry fs ability to describe and generate such kind of patterns, showing its potentiality to be implemented in many different fields from art to science. Fractal geometry and its connection to chaos theory can, through the application of some of its concepts, like self-similarity, allied to I. F. S. generative mechanisms, establish a new paradigm of complexity in art and in architecture as a dynamic operative instrument able to define, and generate, new models and new ways of perceiving form and space. Consequently, and from the point of view of intentional, and conscious, application of such concepts to art and to architecture, our main intention was to create new instruments and new encouraging mechanisms of approach and reflection about the integration, and unity, of the compositional elements where also the issues raised by complexity can have a coherent and effective answer. References  Mandelbrot, B. B. (1977). The fractal geometry of nature. New York, W. H. Freeman and Company.  Falconer, Kenneth J. (1985). The geometry of fractal sets. Cambridge University Press.  Falconer, K. J. (1997). Techniques in fractal geometry. Chichester, John Willey & Sons Ltd.  Ouellette, J (2001). Pollock Ls fractals. Discover magazine, published online 25
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November 1st.  Sala, N. (2001). Fractal models in architecture: A case of study. in Novak, M. M. (editor). Emergent nature - patterns, growth and scaling in the sciences, 273282. Singapura: World Scientific Publishing Co. Pte. Ltd.  Eglash, R. (2004). Fractals in African Material Culture: Applications to Design and Education. In Fractarq 2004, Proceedings of the first international conference on foundations for 21st century architecture and environmental design, Madrid: Inphiniart, 25-27 March (CD copy).  McDougall, A. R.: http://www.a-r-m.com.au/projects_ storeyhall.html  Liebskind, D.: http://www.designboom.com/eng/interview/libeskind.html  Eisenman, P.: http://www.eisenmanarchitects.com (Cannaregio Town Square project).
Published on Nov 28, 2012
Published on Nov 28, 2012
Geometry is measure, is concept and is, above all, order. Along with other areas of mathematics, or other sciences such as physics, geometry...