1 Who created a fractal geometry? In 1973, Mandelo (Mandelbrot) lectured in the French University, the idea of the division and fractal geometry was first proposed. The word fractal is created by Mandelo. It is willing to have irregularities, branches, etc. Since irregular phenomena is common in nature, the fractal geometry is also known as the geometry describing nature. After the fractal geometry, it will soon attract many disciplines, which is because it is not only theory, but also has important value in practical. 2 Dividation geometry is what characteristics compared to traditional geometry: (1) From the whole, fractal geometry is irregular. For example, the coastline and the mountain shape are observed from a distance, and its shape is extremely irregular. (2) The rules of graphics are the same on different sizes. The above-mentioned coastline and the mountain shape, from a close range, and its local shape is similar to the overall morphology, and they are from the whole to local, they are self-similar. Of course, there are also some fractal geometry, which are not entirely similar. Some of them are used to describe the general phenomenon, and some are used to describe chaos and nonlinear systems. 3 What is division? In the European space, people are used to seeing space as three-dimensional, flat or spherical appearances as two-dimensional, while seeing the straight line or curve as one dimension. It is also possible to promote tip, it is considered to be zero, and it can also introduce high dimensional space, but usually people are accustomed to the dimensions of the integer. The fractal theory regards the dimension as a score, which is an important concept that physicists need to introduce when the theory of chaotic attractors is studied. In order to quantify the "non-rule" level of objective things, in 1919, the mathematician introduced the concept of dimension from the angle of the measure, expand the dimension from integers to the score, thus breaking through the general topology set elements. The concept of dividing dimension we can build from both aspects: On the one hand, we first draw a line segment, square and cubes, and their edges are 1. At this time, the line degrees of the original map are reduced to the original 1/2, and the original map is divided into several similar graphics. Its line segment, square, and cubes are separated into 2 ^ 1, 2 ^ 2 and 2 ^ 3 similar sub-patterns, and index 1, 2, 3, exactly equal to the corresponding empirical dimension corresponding to the graph. Generally speaking, if a graphic is composed of a similar B graphics that reduces the original map, there is: A ^ D = B, D = logb / loga relationship is established, the index D is similarity Dimension, D can be an integer or a score. On the other hand, when we draw a straight line, if we use the 0-dimensional point to it, the result is infinite, because the line contains infinity multiple points; if we use a flat content, its result is 0, Because there is no plane in the straight line. So, how to use what scale will be limited? It seems that it will only be limited to the small line segment of the same dimension, and the number of different dimensions here is 1 (greater than 0, less than 2).
Similarly, if we draw a Koch curve, it is an unlimited long line folding, obviously, with a small line segment, its result is infinite, and uses a flat volume, its result is 0 (not this curve Contains a plane), then only find a size quantity of the Koch curve dimension, it will receive limited value, and this dimension is obviously greater than 1, less than 2, then it can only be a decimal (ie, the score), so there is a tight dimension . In fact, the dimension of the KOCH curve is 1.2618 ...... 4 Fractal (fractal) The word "Professor Mandelo said by Professor Mantra, Fractal term is a quiet night in the summer of 1975. He suddenly thought of when he turned his son's Latin Code during the meditation. This word derived from the Latin adjective Fractus, the corresponding Latin dynamic word is Frangere ("broken", "producing non-fragment"). In addition, FRACTION ("Defragments", "Scores") and Fragment ("Defragments") have the same role. Prior to the mid-1970s, Mandelo has always used the word English FRACTIONAL to express his fractal thinking. Therefore, to pick up the head of the Latin word, fractal in the tail of English, is intended to be irregular, broken, score. Mandelo is a large class of complex random geometric objects that cannot be described in the natural European Sri Germany. For example, the curved coastline, the undulating mountain range, the rough cross section, the fantasy float, the river of the nine songs, the vertical and horizontal blood vessels, the eyes of the eyes, the full day, the sky, etc. They feature that extreme irregular or extremely light. It is roughly straightforward, and these objects are fractal. 5 Dismissal definition Mandelo once two definitions under fractal: (1) Meeting a collection A, which satisfies the following conditions DIM (a)> DIM (a), called a fractal set. Among them, DIM (a) is the HAUSDOFF dimension (or number of differential numbers) of the collection A, and DIM (a) is its topological dimension. Generally speaking, DIM (a) is not an integer, but the score. (2) The portion is similar to the overall shape, referred to as fractal. However, through the theory and application test, people found that these two definitions have difficulty including restrictive content. In fact, for what is fractal, there is no exact definition so far, as there is no strict definition of "life" in biology, people usually list a series of characteristics of the living body to explain . The definition of fractal can also be processed. (I) The fractal sets have a proportional detail at any minor scale, or it has a fine structure. (Ii) Dividation sets cannot be described in traditional geometry, which is neither a trajectory that meets some of the conditions, nor is it university that is university. (Iii) The fractal set has some kind of self-similar form, which may be approximately self-similar or statistical. (Iv) Generally, the "fractal dimension" of the fractal set is strictly greater than its corresponding topology dimension. (V) In most interesting situations, the fractal set is defined by a very simple method, which may be generated by the transform iteration.
