In the first part of the paper, Mandleberg discusses how the length between the coastline measured by Louis Frey Richardson and other natural geographical boundaries depends on the measurement scale. Richardson observed that the length L(G) measured at the borders of different countries is a function of the measurement scale g. He collected data from several different examples, and then guessed that L(G) could be estimated by the following form of function:
L(G)=MG 1-D
Mandleberg interprets this result as indicating that the coastline and other geographical boundaries can have statistical self-similarity, and the index d calculates the Hausdorff dimension of the boundary. From this perspective, Richardson's research example has a dimension from the South African coastline 1.02 to the British west coast 1.25.
In the second part of the paper, Mandleberg describes different curves about Koch snowflake, which are standard self-similar graphs. Mandelberg showed the method of calculating their Hausdorff dimensions, all of which are between 1 and 2. He also mentioned peano curve, which is full of space and has a dimension of 2, but did not give its structure.
This paper is very important, because it not only shows Mandleberg's early thoughts on fractals, but also is an example of the relationship between mathematical objects and natural forms-the theme of many subsequent works by Mandleberg.
How long is the coastline of Britain?
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The research object of Euclidean geometry is a geometric object with characteristic length;
One-dimensional space: line segment, with length and no width;
Two-dimensional space: parallelogram with perimeter and area;
Three-dimensional space: sphere, surface area and volume;
Many objects in nature have characteristic lengths, for example, people have heights and mountains have altitudes.
There is a special kind of problem. Mandelbrot asked the question: How long is the coastline of Britain?
Maybe you think this problem is too simple, and it is not easy to measure the coastline. You can get the answer by using maps or aerial surveys.
However, in 1967, an epoch-making paper entitled "How long is the British coastline?" was published in the internationally authoritative American magazine Science. In Statistical Self-similarity and Fractal Dimension, the author Beoni blauth is a contemporary French-American mathematician and computer expert. At that time, he was working in the research center of IBM in new york, but his answer surprised you: he thought that no matter how carefully you did it, you could not get an accurate answer, because there was no accurate answer at all. The length of the British coastline is uncertain! It depends on the scale used in the measurement.
It turns out that the coastline has formed large and small bays and promontories due to the long-term erosion of seawater and the movement of the land itself.
If you fly along the coastline at an altitude of 10000m and take photos of the coast at the same time, and then calculate the total length of the coast shown in these photos in an appropriate scale, is the answer accurate? Don't! Because, you can't tell many small bays and small straits from high altitude.
If you take a small plane instead and repeat the above shooting and measurement at a height of 500 meters, you will see many details that you have not seen before, and your answer will greatly exceed the last one.
Now suppose you are on the ground. If the length is measured in kilometers, the bending of several meters to several hundred meters will be ignored and cannot be counted. Let the length be l1; If the length of the coastline is measured with a gauge with the length of 10m, then those curves that cannot be clearly seen in the air will make the coastline longer, L2 > L1; For example, if we change the track gauge with the length of 1m, we can count all the neglected bends, and the results will continue to increase, but there are still several centimeters and dozens of centimeters of bends that are ignored, and the obtained length is L3 > L2 > L1; By analogy, the more accurate the measurement, the more details the coastline reveals, and the longer the coastline you get (Figure 19). It is conceivable that the measured length will be an astronomical figure in terms of molecular and atomic scales. Although this has no practical significance, it shows that as the measurement unit becomes infinitely small, the coastline length will also become infinitely long, so it is uncertain. So the length is.
Of course, as far as manpower is concerned, you may stop measuring with the 1m gauge, and physicists may think that this measurement process must reach the theoretical limit at the atomic level, but from the idealized point of view of mathematicians, this increasingly detailed measurement process can continue indefinitely, which means that the corresponding measurement results will increase indefinitely, that is to say, there is no exact mathematical definition of the so-called coastline length. Usually, the coastline length we say is only at a certain scale. Benoit Mandelbrot said, in fact, the length of any coastline is infinite in a sense, or the length of the coastline depends on the length of the ruler.
Mandelbrot first encountered the problem of coastline length in a little-known obscure paper in the final manuscript of British mathematician Lewis Fry Richardson. This problem aroused his great interest, and he devoted himself to studying it. Among them, many controversial topics he explored later became part of chaos theory. At first, Lewis Fry Richardson looked through the encyclopedias of Spain, Portugal, Belgium and the Netherlands in order to understand the tortuous coastline length of some countries. He found that there was a 20% error in the estimation of the coastline length of the same country in the book. Lewis Fry Richardson pointed out that this error was caused by their use of different length scales. At the same time, he found that the relationship between coastline length L and measurement scale S is as follows. It is worth noting that the relationship between log( 1/s) and log(L) is linear, and its slope is a certain value d:, that is, where lgk≈3.7 and d≈0.24. Obviously, if we draw lgL, the slope of the straight line is D.
Mandelbrot found that Richardson's empirical formula of boundary length L (r)= Kr 1-a in1year can be used as such a parameter to describe the characteristics of coastline, which he called "canonical dimension", which is one of the famous fractal dimensions. The study of this problem has become a turning point in Mandelbrot's thought, and the concept of fractal has sprouted. He finally unified Cantor's three diversities, Koch curve and other mathematical objects regarded as "morbid" and "monster" by traditional mathematics for a century into a brand-new geometric system, and made a new branch of mathematics-fractal geometry rank among the modern mathematics.
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A simple metaphor for measuring the length of coastline with different gauges
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Obviously, there will be a world of difference between measuring with people's footsteps and measuring with an ant, because ants will climb many more bends than people, so the measurement result will be far greater than that of people. Suppose there is an infinitely small creature, then the measurement result will be infinite. Don't forget that in the eyes of ants, we are bigger than whales. Gulliver's Travels has a wonderful description of the scale of observation. When Gulliver arrived in the land of giants, he found that no woman was beautiful, because in his small eyes, he could clearly see every ferocious pore of a woman. The text to be read can be compared to the British coastline. To say that there is no solution to a text is not to say that there is no solution. It is the British coastline, but how long it is, different readers have different measurement results. But who will win? Don't the results of ants count?
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Please help the special envoy of the United Nations.
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Problem: A and B have the same land boundary line, which bends in the direction of B (Figure 20). There is a strategic highland opposite the border line, which was originally owned by both countries. In 1980s, country A re-measured the boundary, and the measured boundary length was greater than the original record length. According to the newly measured length, this highland completely falls within the territory of country A, so country A asks country B to put it in.
Scheme: It is pointed out to the two countries that the boundary line is a fractal curve, and the determined length cannot be obtained by traditional measurement methods. With the decrease of measurement units, the measurement length will increase. The length measured by a new country is larger than the original record, precisely because she used a smaller yardstick when measuring. Therefore, on the one hand, the fractal geometry theory can be used to explain the two countries, on the other hand, the two countries can be used to demonstrate the border.
Thinking:
1. Why is length no longer a characteristic quantity of coastline?
2. Why does the coastline become longer and longer with the decrease of measurement units when measuring the length of coastline?
3. Understand and explore the generation process of Koch Snow Curve, and understand why Mandelbrot chose Koch Snow Curve as the mathematical model of coastline.