Galileo said: The language of nature is mathematics, and its signs are triangles, circles and other figures. But geometry is an inadequate, ungeneral abstraction for understanding the complexity of nature.
In 1883, Cantor introduced a fractal to mathematics: Cantor set, take a straight line segment of length 1, divide it into three equal parts, remove the middle segment, leave the remaining two segments, and divide the remaining two segments into three equal parts, each removing the middle segment, leaving the shorter four segments in three equal parts... This repetition continues until infinity.
In 1895, Weierstrass proposed the first fractal function "Weierstrass function", and proved the existence of the so-called "pathological" function by virtue of the function curve characteristic that "everywhere is continuous, everywhere can be infinitely subdivided".
In 1906, Koch mentioned in his paper "On a Continuous, Tangentless Curve That Can Be Constructed from Elementary Geometry" a geometric curve like a snowflake, each of which can be infinitely subdivided into a similar shape, and this snowflake curve is the special Koch curve.
In 1914, the Polish mathematician Serpenski used an equilateral triangle to divide it into four small triangles along the line of the trilateral midpoint, removing the small triangle in the middle. Repeating the previous operation for the other three small triangles, we found the Sherpinsky triangle that could be infinitely subdivided. Two years later, he fractalized the square in a similar way and discovered the fractal of the square, the Sherpinsky carpet.
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Mandbro was born on November 20, 1924 in Warsaw, Poland, and was born a Lithuanian Jew. Mang's way of thinking is very special, and a preference for geometry is a feature.
He paid particular attention to the non-mainstream ideas of the time, especially those called "pathological" and "intuitive", such as: the Cantor trident, the Vierstras insignificant curve, the Piano curve that fills the square area, the Sherpinsky carpet and sponge, the Koch snowflake curve and the brownian motion of particles.
Long-term observation, analysis, collection and summarization have given Mendebro the impression that, in addition to the smooth Euclidean geometry, there should be a non-smooth geometry, which is more suitable for describing the true face of nature.
In 1967, the French mathematician Mendebro (1924-) asked the question in the American journal Science: "How long is the coastline of the United Kingdom?" His own answer surprised people, and the length of the coastline can be considered uncertain! This depends on the measure used for the measurement. For example, if measured down from an airplane, the value is a; if a person walks along the coast, the length of the coastline b is equal to the number of steps multiplied by the number of steps.
Since the bends passed on foot are much more detailed than those observed on the plane, b is greater than a; if it is a small one measured by crawling along the coastline, and the bends passed by more than when a person walks, the measured value c must be greater than b. This means that the closer and more carefully you look at the coastline, the more curved details you will find.
But when you compare two photographs of a 100-kilometer-long coastline taken from the air with an enlarged 10-kilometer-long coastline, they look very similar, which is self-similarity. Graphs with self-similarity exist in large numbers in the objective world. Mendebro named complex graphs of this property Fractal, translated as Chinese called "fractals".
In 1964, Mendebro attended the Congress of Logic and philosophy of science in Jerusalem, where he gave a report on the "tentative fractal declaration", but it was not officially published. He is also pondering that today's disciplines are very well differentiated and established, and if there is to be new achievements, it is necessary to create a new discipline.
In Mendebro's 1975 book, The Fractal Geometry of Nature, there are a few words: "Clouds are not just spheres, mountains are not just cones, coastlines are not circular, bark is not so smooth, and the path of lightning propagation is not straight lines." What are they? They are all simple and complex 'fractals'..." Fractals are proposed to better describe and explain the real nature. Because of this, Mendebro is known as the "father of fractals".
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Mendebrough broke with orthodox forms of thinking and used self-similar views to "fractalize" things that were irregular and extremely complex. The science of this fractal is called fractal geometry or fractal theory, in other words, the objects studied in fractal geometry have a commonality, that is, self-similarity, that is, the similarity of local forms and overall forms.
Fractal geometry, first founded by Mendebro in 1983, is a marginal discipline with extremely wide applications.
The basic idea of fractal geometry is that objective things have a hierarchy of self-similarity, and the part and the whole have statistical similarities in form, function, information, time, space, etc., and become self-similarity. For example, each part of a magnet has north and south poles like a whole, constantly divided, and each part has the same magnetic field as the whole magnet. This self-similar hierarchy, appropriately enlarged or reduced in geometric dimensions, remains unchanged throughout the structure.
Mendebro carefully analyzed Brownian motion and finally discovered the fractal problem hidden in irregular motion. Mendebro believed that the basic form of Brownian motion was random walking, and the basic mathematical knowledge used was gaussian normal distribution, which is a distribution in the shape of a bell (ancient cast bell) curve.
French mathematician Levy proposed in his research on this problem that the whole and part of the Brownian motion process sometimes have similarities. Although the common idea is a Gaussian process, i.e. a normal distribution, Levi showed that random walking (or flying) also has self-similarities. Levi's ideas had a great influence on Mendebro.
From Mans' book "Fractals" in 1977, it can be seen that he has freely applied "Levi Flight" to various occasions. Unfortunately, the scientific community did not recognize the importance of this part of the work until the 1990s.
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Hua Luogeng, a famous mathematician in China, once said: "As far as mathematics itself is concerned, it is magnificent, colorful, and fascinating... People who think that mathematics is boring and boring only see the rigor of mathematics, but do not experience the inherent beauty of mathematics. ”
Professor Zhou Haizhong, a famous Chinese scholar, believes that fractal geometry not only shows the beauty of mathematics, but also reveals the nature of the world, and also changes the way people understand the mysteries of nature; it can be said that fractal geometry is the geometry that truly describes nature, and the study of it has also greatly expanded the scope of human cognition.
There is a saying in the Buddhist scriptures: one flower is one world, one leaf is one bodhi
Now we see that the world itself is self-similar. It shows that our lives in the universe itself are guided by such simple rules, endless.