Fractal Geometry is a new discipline founded by American mathematician Mandebrot in the mid-1970s. It is ranked along with isolate theory and chaos theory as three major theories in nonlinear scientific research. I won't say that it is a mathematics subject for the time being because some people oppose it as a branch of mathematics. However, in fact, a large number of people think it is mathematics, but it is another new branch of geometry, which is different from Euclidean geometry. Its research object is the irregular, rough, self-similar geometry in nature. Practice also shows that it is a powerful tool for studying dissipative structure and chaotic dynamics .
Before introducing fractal geometry, we will first quote a concise summary and evaluation of the fractal he founded by Mandebrot when he won the Banad Award:
... This is a great ideological revolution that distinguishes modern mathematics from 20th century and classical mathematics from 19th century. Classical mathematics is rooted in the regular geometric structure of Euclid and the continuous evolutionary mechanics of Newton , while modern mathematics begins with the set theory of and Piano space filling curves.
… These new structures are considered "the devil's gallery", and they are as incomprehensible as the paintings and musical works of Cubism artists established almost inconsistently. The reason why mathematicians who created the ‘devil’ sets believed that these sets were important is that they reflected a mathematical view of people at that time: that the pure mathematical world is a creation of people's free thoughts, not just a reflection of the various simple structures seen in nature. Mathematics of the 20th century was developed on the basis of such a belief that it completely surpassed the limits imposed by its inherent origin.
As a discipline for studying geometric shapes, Euclidean geometry, which has dominated for more than 20 centuries, is well known. However, can it describe the shape of white clouds under the blue sky and the undulating mountains around us? Can it portray the boundary line between the beautiful sea and the archipelago? Can it also describe the randomly condensed structure and the graphics of viscous lipids in physics, the distribution of blood vessels and bronchial tree in the human body, the noise of information transmission lines, and commodity price fluctuations, etc.? It can't!
even in pure mathematics. Like function
This is an infinite series, and its sum is continuous function . The function diagram is as follows.
can be seen from the given analytical formula that it is not differentiated everywhere. From the image, you can find that it has no derivatives everywhere (i.e., it cannot draw tangents at any point on the curve). Can we make a good description of the geometric properties of this image using the traditional mathematical method ? No!
Another example is that in Swedish mathematician Koch constructed a curve as shown in the figure above in 1904. By observing the graph, you can find that it has the following characteristics:
- fractal chart shows the self-similarity of the whole and the part, and the similarity coefficient is 1/3;
- fractal chart contains the characteristics of the whole at any small scale. If the graph is enlarged to a certain extent, you will see the graph when the operation is started; the definition of
- fractal chart is very simple, but it has a very complex structure;
- fractal chart is obtained through a recursive (iteration) process;
- fractal chart is continuous and infinitely long.
Can traditional mathematical methods be used to describe it? The answer is no.
has many similar problems, which cannot be solved by traditional mathematics. Previously, as an individual counterexample of logical derivation in mathematics, people thought they were "weird" and "abnormal" and ignored them. However, the development of science and technology and the progress of society have raised many practical problems such as this that we cannot ignore.Such as word frequency distribution problems in intelligence, noise problems in information transmission, turbulence problems, coastline length problems, some phenomena in economics, etc.
is the "pathological" functions, curves constructed by logic in the history of mathematics, and the practical problems raised by the development of contemporary society, which provide materials for the birth of fractal geometry. It is in the study of these phenomena that Mandebrot created a new discipline with epoch-making significance and a new geometry.
So, what is fractal geometry? To borrow Mandebrot's own words, separating the morphology with "roughness and self-similarity" characteristics in the " geometry chaos" category is the research object of fractal geometry. From the brief description above, we can find that "rough and self-similar" is the core of forming fractal ideas . However, this is just a sensory intuitive understanding, and it also needs to be elevated to rational abstraction in order to have a deeper understanding of the essential characteristics of fractals. Because both "rough" and "self-similarity" are vague and cannot be operated, they must be defined theoretically. Of course, in fractal geometry, it is not a direct definition of "roughness" and "self-similarity", but a core concept of "fractal".
Initially, Mandebrot described fractals like this:
A fractal set is a set in which in the measurement space its Hausdorf-Basecovich dimension is strictly greater than its topology dimension.
where topological dimensions are always integers, and Hausdorf-Basecovich dimensions are often called fractal dimensions. In Euclidean geometry, the topological dimension is equal to the Sdorf-Basecovich dimension. But soon, people discovered the irrationality of this definition, because it excluded sets that were obviously fractal. Mandebrot himself realized this, so he later added that the mathematical definition of fractal is not strict, but is temporary, and it needs to be improved continuously.
The perfect mathematical definition of fractals has not yet been there. To improve the definition of fractals, many solutions have been proposed, but they all have their own shortcomings. Therefore, some fractal geometry html abandoned the pursuit of strict definitions and instead looked for some unique properties of fractals to judge whether a certain morphological structure can be studied as fractals. As the British fractometer Falcoine pointed out in his book "Fractal Geometry——Mathematical Foundations and Applications" that the definition of fractal can be just like the method of defining "life", and it is best to regard fractal as a collection of properties that most fractals have, rather than seeking the exact definition of fractals.
Because fractal geometry uses the spatial structure and properties of irregular sets as its research object, it attracted the interest and research of many natural scientists, especially scholars who study chaos phenomena, and became their powerful tool for studying chaos. At present, fractal geometry has been widely used in natural sciences, so Mandebrot has also made great contributions to science. But like all new things, its improvement and development requires a process, especially its conclusions come from computer experimental mapping results, lacking theoretical arguments, which leads to some controversy.
Controversy over fractal geometry
Since the foundation work of fractal geometry " Fractal Geometry of Nature " was published, some people have been arguing about whether it is mathematics. Indeed, when you read Mandebrot's works, you will find that there are almost no strict concepts, original axioms in fractal geometry, and there is no series of proofs based on basic definitions and concepts, according to strict deductive rules, but these are indispensable to traditional mathematics. What it has is a large number of intuitive images and conjectures obtained from computer experiments. However, from the discovery of the incomparable measurement of ancient Greek , it has been known that geometric intuition is unreliable. So it is no wonder that it has caused controversy among mathematical people on fractal geometry.Among the controversial sides, the opposition was represented by Krantz, while the supporters were of course represented by Mandebrot, the founder of fractal geometry.
Crantz believes that Glake's "Chaos Science——New Science" published in 1988 is "too bad". Because he turned difficult mathematics into something easy to understand, the "public understanding of the work that mathematicians are doing today" misunderstanding. In the eyes of the opposition, the popularity of fractals is just a temporary fashion, because it contains too many beautiful patterns and lacks theorems and proofs in the traditional sense.
In the face of the sharp criticism of Crantz and others, Mandebrot is also tenacious. He believes that proof is very important, but the imaginative and challenging work in fractal geometry is equally important. What is different from traditional proof that it will promote people's understanding of mathematics and nature more.
In addition to Mandebrot himself, Ian Stewart, the author of "Diet God Rolls: Mathematics of Chaos", believes that fractals and chaos are brothers of mathematics, and fractals are the language that describes chaos. In his own words,
In the 1970s, chaos and fractal were in the pioneering stage, and the two were incompatible. But they are all brothers in mathematics. They are all inseparable from irregular structures. Among them, geometric imagination is crucial, but geometry is attached to dynamics in the Mixed God, and dominates in fractals. Fractals provide us with a new language to describe chaotic shapes.
favor fractal geometry is more of a natural scientist. For example, Japanese physicist Hideki Gao An said that
denies differential, which is probably also epoch-making in history.
Physics has really spent so far on analyzing the large-limit universe and small-limit elementary particles. Although it has accumulated a lot of knowledge, it is difficult to say that it has conducted in-depth research on the medium-sized phenomena familiar to us in our daily lives. This is by no means because the medium phenomenon is meaningless, but because things that are easy to consider for large and small limits... Corresponding to this, medium-sized phenomena are essentially multi-system, with complex interactions between them. If only specific interactions are taken out, their important properties will often be invisible.
Theoretical physicist Wheeler evaluates fractals in this way. If a person cannot be equally familiar with fractals, he cannot be considered a scientific cultural person.
Academician of Chinese Academy of Sciences Professor Hao Bolin said:
The fractal geometry that Mandebrot has painstakingly promoted over the years is not only the geometry that is closer to natural phenomena, but also the geometry of chaos phenomena.
fractal geometry, symbolic dynamics and re-regularization groups, the trinity of the mathematical framework of chaos theory . The debate between the two factions was once somewhat stalemate, but with the development of practice, this debate is becoming increasingly weak. From its arguments, we can clearly see that they have opposing and contradictory opinions on how they view mathematics. The opposition believes that mathematics is mainly a problem of proof, while those who agree believe that mathematics mainly lies in understanding problems. Those who pursue understanding often ignore proof, while those who are obsessed with proof often do not value understanding. Should mathematics be proof or understanding? Is it a deduction or experiment of Euclid ? It seems that the husband is right and the mother is right. But the essence of the debate is the mathematical view problem of ", what is mathematics, ".
