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Fractais: Da Geometria À Videoarte UNIVERSIDADE ABERTA Fractais: da geometria à videoarte António Manuel Fonseca Domingos Dissertação orientada pelo Professor Doutor Adérito Fernandes Marcos Lisboa, 16 de dezembro de 2013 “Os trabalhos escolares são provas para o caráter, não para a inteligência. Quer se trate de ortografia, de poesia ou de cálculo, está sempre em causa aprender a querer.” Emile Chartier Alain (1868-1951), Les idées et les âges, 1927 Para a minha filha Fractais: da geometria à videoarte Agradecimentos Agradecimentos O trabalho que aqui se apresenta só foi possível graças à colaboração e apoio de algumas pessoas, às quais não posso deixar de prestar o meu reconhecimento. O presente trabalho foi realizado sob orientação do Professor Doutor Adérito Fernandes Marcos a quem me cabe exprimir um sincero reconhecimento pelo modo empenhado como me acompanhou e por tudo o que me ensinou. A todos os restantes professores do Mestrado em Expressão Gráfica e Audiovisual pelos seus ensinamentos. Aos meus colegas de Mestrado pelo encorajamento, caminhar juntos e troca de experiências. Um agradecimento especial à minha família que sempre me incentivou apoiou e esteve ao meu lado, vivenciando as minhas angústias e alegrias. Grato também aos meus colegas da Direção de Infraestruturas pela confiança e apoio que recebi. i Fractais: da geometria à videoarte Resumo Resumo A descoberta da geometria fractal é datada de meados do séc. XX. É conhecida como a geometria da natureza já que permite descrever muitos fenómenos naturais que apresentam irregularidades que são impossíveis de descrever através dos princípios da geometria tradicional. O aparecimento de novas tecnologias digitais veio alterar o modo como a arte é encarada. Este trabalho pretende demonstrar possibilidades de integração da geometria fractal e a Teoria do Caos na criação de videoarte, baseando-se em meios informáticos e tecnologias digitais. Visamos assim explorar as potencialidades artísticas oferecidas por software gerador de composições visuais como forma de expressão videográfica. Para a criação de vídeos desenvolveu-se e implementou-se um artefacto digital baseado no programa Apophysis 3D Hack, disponível gratuitamente, tendo como finalidade a criação de videoarte fractal. As características estéticas, as propriedades e inúmeras aplicações dos fractais, levaram-nos a realizar este trabalho de experimentação, baseado na geometria fractal e Teoria do Caos, que consiste na criação de vídeo através da aplicação informática. Para o efeito foi criado um blogue disponível em: http://galeriafractal.blogspot.pt, como forma de divulgação da arte digital e possibilidades tecnológicas na criação, partilha e globalização da videoarte. Palavras-chave: Fractais, Arte digital, Geometria, Caos, Videoarte ii Fractais: da geometria à videoarte Abstract Abstract The discovery of fractal geometry dates from the mid twentieth century. It is known as the geometry of nature due to many natural phenomena, which presents irregularities difficult or impossible to describe by the principles of traditional geometry. The birth of new digital technologies has changed the way we see art. This work aims at demonstrating the integration of fractal geometry and the chaos theory in the creation of videoart, based on digital media and computer technologies. We explore the potential for artistic creation offered by software generator of visual compositions as a form of videographic expression. A digital artifact was developed and implemented in order to create videos based on the program Apophysis 3D Hack, available for free, with the purpose of creating fractal video art. Thus, the aesthetic characteristics, the properties and numerous applications of fractals, led us to undertake this work of experimentation, based on fractal geometry and the chaos theory, which is the creation of video through the computer application. For this purpose we created a blog available at: http://galeriafractal.blogspot.pt, as a means of dissemination of digital art and technological possibilities in creating, sharing and widespreading fractal video art. Keywords: Fractals, Digital Art, Geometry, Chaos, Video art iii Fractais: da geometria à videoarte Índice Índice Índice de Figuras .................................................................................................................. vi Índice de Tabelas ................................................................................................................ viii Siglas .................................................................................................................................... ix 1. INTRODUÇÃO ............................................................................................................. 1 1.1. Contextualização do trabalho .................................................................................. 2 1.2. Justificação .............................................................................................................. 3 1.3. Objetivos e questões de investigação ...................................................................... 4 1.4. Resultados esperados .............................................................................................. 5 1.5. Metodologia ............................................................................................................ 5 1.6. Estrutura do trabalho ............................................................................................... 6 2. O QUE SÃO FRACTAIS? ............................................................................................ 9 2.1. Introdução ............................................................................................................... 9 2.2. Como tudo começou ............................................................................................. 10 2.3. O que é um fractal? ............................................................................................... 16 2.4. Características dos fractais .................................................................................... 18 2.4.1. Autossimilaridade ou autossemelhança ......................................................... 20 2.4.2. Complexidade infinita ................................................................................... 21 2.5. Dimensão .............................................................................................................. 22 2.5.1. Dimensão Euclidiana ..................................................................................... 24 2.5.2. Dimensão topológica ..................................................................................... 24 2.5.3. Dimensão fractal ............................................................................................ 26 2.6. Calcular a dimensão fractal ................................................................................... 27 2.7. Métodos geradores ................................................................................................ 29 2.7.1. Iteração .......................................................................................................... 29 2.7.2. IFS ................................................................................................................. 30 2.7.3. L-System ........................................................................................................ 32 2.8. Números complexos.............................................................................................. 34 2.9. Fractais na natureza............................................................................................... 37 3. FRACTAIS CLÁSSICOS ........................................................................................... 40 3.1. Poeira de Cantor .................................................................................................... 40 3.2. Triângulo e tapete de Sierpinski ........................................................................... 41 3.3. Curva e floco de neve de Koch ............................................................................. 43 3.4. Curva de Peano ..................................................................................................... 45 3.5. Esponja de Menger ............................................................................................... 47 3.6. Conjunto de Julia .................................................................................................. 48 3.7. Conjunto de Mandelbrot ....................................................................................... 51 4. SISTEMAS DINÂMICOS E A ARTE ....................................................................... 56 4.1. Introdução à Teoria do Caos ................................................................................. 56 4.1.1. Caos ............................................................................................................... 59 4.1.2. Sistemas não lineares ..................................................................................... 60 4.1.3. Atractores....................................................................................................... 64 4.2. Ciência, tecnologia e arte ...................................................................................... 65 iv Fractais: da geometria à videoarte Índice 4.3. A arte fractal é arte? .............................................................................................. 69 4.4. A autossemelhança e M. C. Escher ......................................................................
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