分形简史

Fractal geometry is a branch of mathematics which was born in 1970s. It is another major revolution in geometry history after non-Euclidean geometry. It has important value and significance to study its development history. Based on the applicant's doctoral dissertation, this project is intended to use the methods of historical data analysis, literature textual research and comparative research, through the original literature and research literature. With the rigorous analysis in 19th century as the historical background, and the generation of morbid functions, curves and sets as the starting point, and the formation, development and perfection of fractional dimension theory and self-similarity theory as the context, and the specific application of fractal theory as the guide, and the promotion of the spread of fractal culture as the purpose, this project tries to sort out the relationship between fractal geometry and mathematical branches such as set theory, measure theory and analysis, from point to surface, external to internal causes, problem to root, to present a comprehensive historical overview of fractal geometry, and analyze the reasons for the creation of fractal geometry deeply, then answer two basic questions of “how fractal geometry comes into being ”and “why fractal geometry comes into being”. Since there is no systematic study on the history of fractal geometry in academic circles, the project will make up for the research gap in this field. The planned monograph "Brief History of Fractal" will popularize the knowledge of mathematical culture and let the advanced fractal theory fly into the common people's homes.

分形几何是20世纪70年代诞生的一门数学分支，它是继非欧几何之后几何学史上的又一次重大革命，研究它的发展历史具有重要的价值和意义。本项目拟在申请人博士学位论文的基础上，基于原始文献和研究文献，运用史料分析、文献考证和比较研究的方法，以19世纪的分析严格化为历史背景，以病态函数、曲线和集合的产生为切入点，以分数维数理论和自相似理论的形成、发展和完善为脉络，以分形理论的具体应用为指引，以推动分形文化的传播为导向，力图从点到面，从外因到内因，从问题到根源，精确梳理分形几何与集合论、测度论和分析学等数学分支的关系渊源，全面呈现分形几何的历史概貌，深刻剖析分形几何的创立原因，进而回答分形几何究竞是“如何产生”和“为什么会产生”两个基本问题。由于学术界尚无对分形几何发展历史的系统研究，项目的开展可以弥补此方面的研究空白。拟完成的数学史专著《分形简史》将普及数学文化知识，让高深的分形理论飞入寻常百姓家。

本项目以“为什么要研究数学”为指导思想，以分析严格化为历史背景，以病态函数、曲线和集合的产生为切入点，以分数维数理论和自相似理论的形成、发展和完善为脉络，以分形理论的具体应用为指引，以推动分形理论的传播为导向，梳理了分形几何与测度论、集合论、拓扑学等数学分支的渊源关系，剖析了分形几何的创立原因，较为全面地呈现了分形几何的历史概貌。项目出版的专著《分形简史》，将弥补国内外没有对分形历史系统研究的空白，《分形简史》通过对分形发展历程深入浅出的讲解，能够把分形理论带入寻常百姓家。项目发表的学术论文“盒维数概念的历史演变”“分数概念的产生”“自相似理论的形成和发展史实考源” 将有助于完善分形几何的发展历史。此外，项目得到的研究结果还将在数学文化、数学哲学、数学教育等学科的应用中产生积极的影响。