摘 要：分形几何作为研究不规则形状和复杂结构的重要数学工具，为描述自然界中广泛存在的自相似现象提供了理论基础。本文聚焦于分形几何在自相似集构造中的应用，旨在通过引入迭代函数系统(IFS)方法，探讨其在生成具有特定维数的自相似集方面的有效性。基于Hutchinson理论框架，采用严格的数学推导与数值模拟相结合的方式，对不同参数条件下的自相似集进行构造，并分析其豪斯多夫维数特征。研究发现，通过精确控制IFS映射参数，能够有效生成具有预期分形维数的自相似集，且所得结果与理论预测高度吻合。特别地，本文提出了一种改进型IFS算法，在保持原有计算效率的同时，显著提高了生成自相似集的精度和稳定性。该研究成果不仅丰富了分形几何理论体系，更为相关领域如图像压缩、材料科学等提供了新的思路与方法，展示了分形几何在实际应用中的巨大潜力。
关键词：分形几何；自相似集；迭代函数系统
Abstract：Fractal geometry, as a significant mathematical tool for studying irregular shapes and complex structures, provides a theoretical foundation for describing self-similar phenomena widely observed in nature. This paper focuses on the application of fractal geometry in the construction of self-similar sets, aiming to explore the effectiveness of the Iterated Function System (IFS) method in generating self-similar sets with specific dimensions. Based on the Hutchinson theoretical fr amework, this study combines rigorous mathematical derivations with numerical simulations to construct self-similar sets under various parameter conditions and analyze their Hausdorff dimension characteristics. The findings indicate that by precisely controlling the parameters of IFS mappings, it is possible to effectively generate self-similar sets with expected fractal dimensions, with results showing high consistency with theoretical predictions. Notably, an improved IFS algorithm is proposed, which enhances the accuracy and stability of generated self-similar sets while maintaining the original computational efficiency. This research not only enriches the theoretical system of fractal geometry but also offers new approaches and methods for related fields such as image compression and materials science, demonstrating the substantial potential of fractal geometry in practical applications.
引言 1
一、分形几何基础理论 1
(一)分形维数的定义与计算 1
(二)自相似集的基本性质 2
(三)分形构造的经典方法 2
二、自相似集的迭代函数系统 2
(一)迭代函数系统的构建原理 3
(二)自相似集的生成算法 3
(三)参数对自相似结构的影响 4
三、分形几何在自相似集中的应用实例 4
(一)自然界中的自相似现象 4
(二)计算机图形中的分形模型 5
(三)物理学中的分形应用 5
四、自相似集的复杂性分析 6
(一)复杂性的度量方法 6
(二)维数与复杂性的关系 6
(三)应用中的复杂性考量 7
结论 7
参考文献 9
致谢 9
关键词：分形几何；自相似集；迭代函数系统
Abstract：Fractal geometry, as a significant mathematical tool for studying irregular shapes and complex structures, provides a theoretical foundation for describing self-similar phenomena widely observed in nature. This paper focuses on the application of fractal geometry in the construction of self-similar sets, aiming to explore the effectiveness of the Iterated Function System (IFS) method in generating self-similar sets with specific dimensions. Based on the Hutchinson theoretical fr amework, this study combines rigorous mathematical derivations with numerical simulations to construct self-similar sets under various parameter conditions and analyze their Hausdorff dimension characteristics. The findings indicate that by precisely controlling the parameters of IFS mappings, it is possible to effectively generate self-similar sets with expected fractal dimensions, with results showing high consistency with theoretical predictions. Notably, an improved IFS algorithm is proposed, which enhances the accuracy and stability of generated self-similar sets while maintaining the original computational efficiency. This research not only enriches the theoretical system of fractal geometry but also offers new approaches and methods for related fields such as image compression and materials science, demonstrating the substantial potential of fractal geometry in practical applications.
Keywords: Fractal Geometry；Self-Similar Set；Iterated Function System
引言 1
一、分形几何基础理论 1
(一)分形维数的定义与计算 1
(二)自相似集的基本性质 2
(三)分形构造的经典方法 2
二、自相似集的迭代函数系统 2
(一)迭代函数系统的构建原理 3
(二)自相似集的生成算法 3
(三)参数对自相似结构的影响 4
三、分形几何在自相似集中的应用实例 4
(一)自然界中的自相似现象 4
(二)计算机图形中的分形模型 5
(三)物理学中的分形应用 5
四、自相似集的复杂性分析 6
(一)复杂性的度量方法 6
(二)维数与复杂性的关系 6
(三)应用中的复杂性考量 7
结论 7
参考文献 9
致谢 9
