Mandelbrotova množica in algoritem DEM : delo diplomskega seminarja

Ševerkar, Nejc (author), Kuzman, Uroš (mentor)

V nalogi je predstavljen algoritem DEM (Distance estimation method), ki omogoča učinkovito grafično prikazovanje fraktaličnih množic. Algoritem sodi na področje kompleksne dinamike, ki preučuje obnašanje iteracij kompleksnih preslikav. Začetek te veje matematike sega v obdobje med leti 1917 in 1919, ko so bile objavljene prve raziskave na temo iteracij kompleksnih racionalnih funkcij ene spremenljivke, s strani francoskih matematikov Gastona Juliaja in Pierre Fatouja. Sledilo je obdobje daljšega mirovanja teorije, ki pa ga je prekinil razvoj numerične matematike in posledično tudi fraktalne geometrije. Tako je področje znova postalo zelo popularno tako med matematiki kot med umetniki. V središču naloge bo eden izmed najbolj znanih objektov te teorije, tako imenovana Mandelbrotova množica. Gre za podmnožico kompleksne ravnine, ki na svojevrsten način ilustrira družino kvadratnih polinomov s povezano Juliajevo množico. V nalogi bomo podali njeno definicijo in dokazali nekaj njenih topoloških lastnosti. V ospredju bo dokaz njene povezanosti, ki nam bo v zadnjem poglavju omogočil izpeljavo algoritma DEM, s katerim bomo to množico tudi učinkovito grafično prikazali.

• Language: Slovenian
• Keywords: kompleksna dinamika, polinomi, Juliajeve množice, Mandelbrotova množica, algoritem
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-120178
• UDC: 517
• COBISS.SI-ID: 58367491
• Publication date in RUL: 17.09.2020

ŠEVERKAR, Nejc, 2020, Mandelbrotova množica in algoritem DEM : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 14 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=120178

Secondary language

• Language: English
• Title: The Mandelbrot set and the DEM algorithm

Abstract

In this paper we study the DEM (Distance estimation method) algorithm, which enables effective graphical representation of fractal sets. The algorithm is based on the theory of complex dynamics, which studies complex function's behavior under repeated iterations. This branch of mathematics had its beginnings between the years of 1917 and 1919, when the first research about iteration of complex rational function of one variable was published by French mathematicians Gaston Julia and Pierre Fatou. There followed multiple years of inactivity, until it was disrupted by the progress in the field of computer science and hence fractal geometry. Thus the field became immensely popular amidst mathematicians and artists alike. In the midst of this paper we will study one of the most recognizable objects of the theory, the so called Mandelbrot set. The name belongs to a subset of the complex plane, which in its own way illustrates the family of quadratic polynomials with a connected Julia set. We will define this set and prove some of its topological characteristics. The main proof being its connectedness, which will allow us to derive the DEM algorithm in the last section, using which we will be able to effectively represent the set graphically.

• Keywords: complex dynamics, polynomials, Julia sets, the Mandelbrot set, algorithm