【作者】 李成杰；

【导师】 刘弘；

【作者基本信息】 山东师范大学 ， 计算机软件与理论， 2010， 硕士

【摘要】 分形理论是现代数学的一个新分支,被誉为大自然的几何学,其本质是一种新的世界观和方法论。它承认世界的局部可能在一定条件下或过程中,在某一方面(形态、结构、信息、功能、时间、能量等)表现出与整体的相似性[1][2]。基于分形理论绘制的图案结构复杂,色彩斑斓,变化万千,给人以震撼的美感。分形图案既是数学的载体,也是艺术的体现,实现了二者的完美结合[3]。分形研究中发现大量精妙的图案,它们不但令人们思索柏拉图世界清纯、完美的构型,更让人联想到现实世界复杂多变的自然结构,这些图案在装饰艺术方面有广阔的应用前景。分形图案具有无限精细的特性,在一定色彩方案的支持下,某些局部的线条会呈现明显的层次感,具有抗复印功能,这使得分形图案可用于防伪设计。正因为分形图案有广阔的应用前景,所以研究分形构图过程,探索新的构图方法,以构造更多的分形图案,是十分具有意义的。文中提出了分形扩充图案的概念,它包含传统图案和非传统图案两类。传统图案是指由传统分形算法生成的分形图案;非传统图案是在传统分形构图的过程中加入了新的规则限制,构造而来一种分形衍化图案,它有传统分形图案的影子,但不同于传统分形图案,丰富了分形图案的门类。在绘制分形图案过程中,由于其自身新奇复杂、变化万千的特性[4],使得最终绘制的图案往往不可预知、不可控制。因此,在分形构图中引入构图规则的概念,从规则角度看待分形图案的构造过程,并利用规则对分形构图过程进行控制。虽然由此获得的图案依然新奇复杂,但是此类图案有章可循,符合一定的规则限定,并能按类型进行区分。由于规则是人为设定的,所以设计的分形图案能够在一定程度上体现人们的审美倾向和主观需求。本文将分形构图规则分为算法规则、布局规则、赋色规则三类,并从三方面分别进行深入研究。算法规则通过算法影响绘图效果,它既包括传统各类分形算法,又包括在传统分形算法的基础上进行的一些调整或改变。文中介绍了随机、布尔、条纹等具体算法规则,并给出了相应的构图公式和图案样例,这些规则从根本上改变了图案的结构形态。布局规则是在构图过程中,对图案的宏观结构进行控制,使目标图案呈现指定的结构形态。文中介绍了对称、形状、组合等具体布局规则,并给出了相应的构图公式和图案样例,这些规则使图案依照特定的外观显示。赋色规则是指图案的不同赋色方法,用于对图案的色彩进行控制。本文介绍了双色、过渡、融合等具体的赋色规则。采用不同的赋色规则,同样结构的图案会产生不同的艺术效果。本文致力于研究分形构图规则,旨在探索更好的构图规则和规则组合。我们对三类规则分别进行研究,并不是将三者割裂开来。我们既研究具体的构图规则,又研究不同规则之间的组合。将多种构图规则结合起来,灵活运用,用于分形构图,才是我们研究的真正目的之所在。更多还原

【Abstract】 Fractal theory, known as geometry of nature, is a new branch of modern mathematics, and its essence is a new world outlook and methodology. It recognized that partial of the world, under certain conditions or processes, may show self-similarity of the part and the whole in some respect (shape, structure, information, features, time, energy, etc.). Patterns obtained by fractal theory which are complicated, colorful and ever changing, give a strong heartshock to people. Fractal patterns which are not only a mathematics carrier, but also an expression of the art achieve the perfect combination of mathematics and the art.A large number of sophisticated patterns found in the fractal research, which not only make us think about pure and perfect structure in Plato’s world, but also guide us to think about the complex and ever changing structure in the real world, have broad application prospects. Fractal patterns have infinitely delicate structure and some local lines, with the support of some color scheme, will show strong layering and have the feature of anti-copying, thus fractal patterns can be used in security design. Because fractal patterns have so many potential applications, studies of fractal composition processes, exploring new methods of composition in order to construct more fractal patterns, is very meaningful.This paper proposes the concept of extended fractal patterns, including traditional patterns and non-traditional ones. Traditional patterns refer to fractal patterns which are generated by traditional fractal algorithms; non-traditional patterns refer to fractal-derived patterns which are generated by the way of adding new rules to traditional fractal algorithms and have the shadows of classic fractal patterns, but are different from traditional ones. It is non-traditional patterns that enrich fractal pattern categories.In the drawing process of fractal patterns, thanks to theirs complex and ever changing characteristics, the patterns finally drawn are often unpredictable and uncontrollable. Therefore, the concept of composition rule is introduced into the process of fractal composition, considers the procedure of fractal composition from the perspective of rules, and controls the procedure of fractal composition with rules. Although patterns obtained are still complex, such kinds of patterns have rules to follow, abide by some contraints, and can be divided into different categories. Because the rules are man-made, fractal patterns designed can reflect people’s aesthetic preferences and subjective needs. The rules of fractal composition can be divided into three categories including algorithm rules, layout rules and coloring rules, and intensive studies are carried out respectively from the three aspects.Algorithm rules influence the drawing results through algorithms, including traditional fractal algorithms and a number of adjustments or changes based on the traditional fractal algorithms. This paper introduces random, Boolean, stripes and other specific algorithm rules, gives corresponding composition formulas and patterns and these rules fundamentally change the structures of the patterns.Layout rules control the macro-structures of the patterns in the composition process, so that target patterns will present the specified structures. This paper introduces the symmetry, shape, composition and other specific layout rules, gives the corresponding composition formulas and patterns, and these rules make the pattern show in specified appearance.Coloring rules refer to different coloring methods for patterns, used to control the colors of the patterns. This paper introduces double color, transition, fusion and other specific coloring rules. Adopting different coloring rules, patterns with the same structure will produce different artistic effects.This paper is committed to the research of fractal composition rules, aiming at exploring better composition rules and combinations of rules. We carry out research on three types of rules respectively, but don’t seperate one from another. We study both specific composition rules and combinations of different rules. Combining a variety of compositions rules, using flexibly, in the process of fractal compositon, is the real pursose of our research.更多还原