Fraktali

ROJKO, MOJCA (author), Virk, Žiga (mentor)

V 18. in 19. stoletju se je razvila veja matematike, ki je postala znana kot fraktalna geometrija. Ideje in dela Benoita Mandelbrota so krepko pripomogla k njenemu hitremu razvoju in od objave njegovih del so se našli različni načini praktične uporabe tako fraktalov, kot bolj pogosto fraktalne dimenzije. Fraktalna geometrija je bila uporabljena na različnih področjih, kot so informacijska teorija, ekonomija, nevroznanost, medicina, fizika, akustika, analiza slik in drugih. Diplomsko delo pokriva zgodovino in osnove fraktalne geometrije, opiše najbolj temeljne definicije in izreke, ki so potrebni za razumevanje področja. Očrtani so koncepti kot so mere in fraktalna dimenzija, predvsem dimenzija po metodi "štetja škatel". V nadaljevanju so opisane različne tehnike za generiranje fraktalov, med njimi je opisan tudi moj praktičen pristop k razvoju programskega orodja za generiranje fraktalov. Podala sem tudi pregled že obstoječe prosto dostopne programske opreme s to funkcionalnostjo. Naredila sem program, ki je zmožen generirati veliko različnih tipov fraktalov: IFS sistem, Mandelbrotova množica, L-sistemi, algoritem Fractal flame, fraktalna pokrajina, naključni fraktali, ...

• Language: Slovenian
• Keywords: generiranje fraktalov, fraktalna grafika, fraktalno programiranje, IFS sistemi, L-sistemi, fraktalna pokrajina, naključni fraktali, prepisovalni algoritmi
• Work type: Bachelor thesis/paper
• Organization: FRI - Faculty of Computer and Information Science
• Year: 2015
• PID: 20.500.12556/RUL-72290
• COBISS.SI-ID: 1536524227
• Publication date in RUL: 10.09.2015

Secondary language

• Language: English
• Title: Fractals

Abstract

In 18th and 19th century a new branch of mathematics developed, today known as fractal geometry. The ideas and work of Benoit Mandelbrot have been of great importance to its fast development and since the publication of his work, many practical uses of fractals and even more often fractal dimension have been found. Fractal geometry is used in many different fields such as information theory, economy, neuroscience, medicine, physics, acoustics, image analysis and other. This thesis covers history and basics of fractal geometry, describes fundamental definitions and theorems that are necessary for understanding of the field. Concepts like measure and fractal dimension are described, especially the "box counting" method. In continuation different techniques for fractal generating are described. Along with them my practical approach to development of fractal generating software is also described and an overview of existing fractal generating software is given. I have developed computer software that is capable of generating many different types of fractals: IFS system, Mandelbrot set, L-systems, fractal flame algorithm, fractal terrain, random fractals, ...

• Keywords: fractal generating, fractal graphics, fractal programming, IFS systems, L-systems, fractal terrain, random fractals, rewriting algorithms