Research On The Early History Of Fractal Geometry

Fractal geometry is a new branch of mathematics which was born in 1970 s.It is another significant revolution in the history of geometry after the creation of non-Euclidean geometry.As the geometry of nature,it has a very wide range of applications in real life.Therefore,it is very important to study the early history of fractal geometry.Based on reading the original literature and related research literature,through the methods of historical analysis and textual research,guiding by " why mathematics" ideology.This dissertation studies the contents and thoughts of the early history of the fractal geometry comprehensively and systematically,and analyses the reason of the creation of fractal geometry in depth.The research results obtained are as follows:1.This dissertation investigates the background,causes,process and influence of early classical fractal sets such as Weierstrass function,Cantor set and Koch curve comprehensively under the background of rigorous analysis.In order to find out the relationship between the continuity and the differentiability of the function,Weierstrass constructed a continuous but everywhere non-differentiable pathological function.Cantor constructed a complete but everywhere non-dense pathological point set in the unit interval.By using the thought of recursion,Koch constructed a continuous but everywhere non-tangable pathological curve which can be expressed geometric intuitively.The appearance of these pathological functions,curves,and sets has become the internal cause of the creation of fractal geometry.2.The generation process of the concept of fractional dimension is combed systematically by this dissertation.In order to measure the size of Cantor set accurately,Cantor,Lebesgue,Borel and other mathematicians have proposed the solutions and ideas successively,but the result is not satisfactory.It is not until Carathéodory defined the p dimension measure set in q dimensional space to make some progress.Hausdorff extended the dimension from integer to fraction based on Carathéodory's work,and solved the problem of measuring the size of Cantor set accurately.Besicovitch perfected the definition of the fractional dimension of Hausdorff,and gave the exact concept of fractional dimension.3.This dissertation discusses detailedly the contribution to fractional dimension theory by Besicovitch,Bouligand,Kolmogorov and other mathematicians.Besicovitch studied the density properties,calculus,specific applications in real number theory and etc related to the fractional dimension set.The box dimension is a kind of important fractional dimension.The initial model of box dimension was established by Bouligand.Pontrjagin and Schnirelmann defined the box dimension with mathematical expressions,but lack of rigour.Kolmogorov and Tihomirov gave the strict definition of box dimension.Falconer defined the box dimension in the modern sense.4.This dissertation elaborates detailedly on the contribution of Lévy,Moran,Mandelbrot and other mathematicians to self-similarity theory.The idea of self-similarity can be traced back to the ancient Greek era.There are some ideas about self-similarity in the work of Democritus,Aristotle and the book of Mathematics,philosophy and medicine in ancient Chinese,but no strict theoretical system was formed.Lévy is the first mathematician who studies the self-similarity systematically.He introduced parameters,order and other basic mathematical concepts.Moran combined the theory of set with self-similarity theory,then defined the concept of self-similar set,which gave an embryonic form of self-similarity theory.Mandelbrot integrated the Statistics into self-similarity theory,and described statistical self-similarity,then solved the problem of the length of the coastline which perplex people for a long time.5.The creation process of fractal geometry is carefully explored,and the causes for the creation of fractal geometry are deeply analyzed.The creation process of fractal geometry is carefully explored through the paper "How long is the coastline in the Britain" and the book "The Fractal Geometry of Nature".It is pointed out that the main causes for the establishment of fractal geometry are the incentive of the pathological functions,curves and sets,the promotion of the development mathematical theory,the encouragement of practical problems,and the advantages of founders based on original literatures and related research literatures.