曼德博集合、朱利亞集合與演算法

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郭立峰(Li-Feng Kuo) 查詢紙本館藏

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數學系

論文名稱

曼德博集合、朱利亞集合與演算法
(Mandelbrot Sets, Julia Sets and Their Algorithms)

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★ 一維和二維的标准以及條件隨機遊走的性質

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摘要(中)

這篇論文研究碎形中的一大主題—曼德博集合。在了解曼德博集合之前我們需要從了解朱利亞集合開始，最終的目標則是利用程式來生成碎形圖像並希望能應用至藝術領域。
　　於是在這篇論文中首先介紹了朱利亞集合的定義及性質，並利用理論整理出可行的演算法來生成朱利亞集合的圖像。在對朱利亞集合有一定程度的理解之後便能開始研究曼德博集合，其原因來自曼德博集合的定義是蒐集所有另朱利亞集合連通的點。然而，在生成曼德博集合是會受到其定義的阻礙，如何有效的檢測朱利亞集合是否連通？這個問題的答案就是—曼德博集合基本定理，有了這個定理後便能生成曼德博集合。
　　最後也給了一些曼德博集合與朱利亞集合的例子，並且介紹了三維中的曼德博集合與朱利亞集合。

摘要(英)

In this thesis, we survey the big theme of fractals － Mandelbrot sets. We start to study Julia sets before study Mandelbrot sets, and the goal is generating figures of fractals and applying to arts.
Hence, we introduce the definition and properties of Julia sets firstly, and use this theory to arrange some useful algorithms for generating the figures of Julia sets. After we survey Julia sets, we can study Mandelbrot sets, since the definition of Mandelbrot sets is all of the points such that the Julia set is onnected. However, we obtain the obstacle when generating andelbrot sets, that is, how to check the Julia set is connected or not? The answer of this question is － the fundamental theorem of Mandelbrot sets, we can generate the figures of Mandelbrot sets by this theorem.
Finally, we give some examples of Mandelbrot sets and Julia sets, and introduce 3-dimensional Mandelbrot sets and Julia sets.

關鍵字(中)

★ 曼德博集合
★ 朱利亞集合
★ 演算法

關鍵字(英)

★ Mandelbrot set
★ Julia set
★ algorithm
★ Python

論文目次

摘要i
Abstract ii
Acknowledgement iii
1 Introduction to Fractals on a Complex Plane 1
2 Birth of Complex Fractals - Julia and Fatou Sets 3
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Coloring Fractal Sets on a Complex Plane . . . . . . . . . . . . . . . . . . 8
2.3 Julia and Fatou Sets within the Polynomials of Various Orders . . . . . . . 16
2.4 Other Variations of Julia and Fatou Sets on a Complex Plane . . . . . . . 27
2.5 Theoroms of Julia Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Fractal Revolution - The Mandelbrot Set 34
3.1 History and Concept of the Mandelbrot Set . . . . . . . . . . . . . . . . . 34
3.2 Geometrical Structure and Atoms of the Mandelbrot Set . . . . . . . . . . 37
3.3 Mandelbrot Set within the Polynomials of Various Orders and Other Variations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Fundamental Theorem of Mandelbrot Set . . . . . . . . . . . . . . . . . . 44
4 Searching for the Holy Grail: 3D Mandelbrot Set 52
5 What’s Next? Further Extensions of The Mandelbrot Set 56
6 Examples for the Julia Sets of Other Functions 58
References 66
iv

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指導教授

方向(Xiang Fang)

審核日期

2019-7-19

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