Professor Yang Shimo and Hong Daoshi went around to investigate the new site of the laboratory, came to Magpie Village, and found a place on the north side of a hillside and the south side of the weir pond, which is what Uncle Liu thinks is a vegetable greenhouse. We all know that the south of the mountain and the north of the water are yang, and the north of the mountain and the south of the water are yin. For example, the general city center is relatively densely populated, and such places have better feng shui. Because there are many people, the yang energy will be more vigorous, and living in a place where the yang energy is strong, the feng shui fortune will naturally be strengthened. In order to shield himself from electromagnetic fields and the influence of human activities, Professor Yang's laboratory was built in a "cloudy" place in the north of the mountain and in the south of the mountain, and the roof and surroundings were blackened. In order to keep it secret, it was also claimed that this was a vegetable greenhouse.

Hu Kehao asked Professor Yang Shimo about the relocation of the laboratory, and Professor Yang said: "The greenhouse has not been built for a long time, and the experimental equipment stored in the warehouse in U City has not yet been moved. ”

"All the experiments have stopped?" Hu Kehao asked.

"While the experimental equipment has not been moved over, the venue is spacious, and I have recently been doing a new experiment. How about coming in and having a look? Coach the work. Professor Yang said.

Professor Yang opened the curtain of the laboratory, and Hu Kehao and Uncle Liu entered the laboratory. Professor Yang often does some strange experiments, and those who don't know him think that he is not doing experiments, but doing some tricks. Sure enough, Hu Kehao saw where Professor Yang was doing experiments, and he was clearly processing products.

There were more than a dozen people in the room, divided into two groups, one group was weaving carpets, and the other group was making mats! This Professor Yang is also true, in the underground laboratory of the abandoned factory area of U City, his experiments are either bullet fighting experiments, or cat breeding experiments, or ball playing experiments, but he said that these experiments are electronic double-slit experiments, Schrödinger cat experiments, beamsplitters and delay experiments, which are really inexplicable and insulting.

Uncle Liu saw the carpet that Professor Yang hired someone to weave and smiled: "Professor Yang, what kind of carpet did you hire someone to weave?" Can this be called a carpet? There are holes everywhere, who are you selling to? ”

"Haha, the drunkard doesn't mean to drink, he cares about the mountains and rivers. My carpet looks like a carpet, but it's not actually a carpet, and I use it for other purposes, not to make money. Professor Yang said.

Hu Kehao took a closer look and saw that these carpets were a square, dividing the square into nine equal parts. Become a nine-square grid and dig out the middle grid. The remaining eight grids, each grid is divided into nine equal parts, and there are a total of seventy-two small grids in the eight large grids, and the small grid in the center of the eight large grids is dug out. Sixty-four cells remained, and each cell was divided into nine equal parts, making a total of five hundred and seventy-six cells out of the sixty-four grids, and the small grid in the center of the sixty-four large cells was dug out. There are five hundred and twelve squares left...... And so on, and so on......

"Professor Yang, when will you achieve this ruined carpet? Keep digging holes in the carpet, and finally the carpet will be gone? Hu Kehao asked.

"My carpet is no ordinary carpet, it has a masculine name - Sierpinski carpet, pronounced Sherpinski carpet." Professor Yang said.

"Professor Yang, I didn't expect you to fall into clichés. Originally, your carpet was a domestic product, did you give it a foreign name, and it became an imported high-end product? Fourth Uncle Liu teased.

Professor Yang smiled but did not speak. After the laboratory greenhouse was built, Professor Yang wanted to seize the time and find a moving company to move the experimental equipment in as soon as possible. One day, he saw the women of Magpie Village knitting carpets under the leadership of village magistrate He Shujun, and he was inspired to make the most of the space to make Sierpinski carpets in an open laboratory.

Hu Kehao and Uncle Liu continued to visit and saw several craftsmen playing mats. In rural areas, tanning mats are common. Skilled craftsmen cut the bamboo into very thin strips of one inch wide, with green slivers with skin and yellow slivers without skin. Strips can be used to weave a lot of agricultural tools, such as back baskets, such as rice sieves, such as baskets, such as dustpans (the word dustpan in this author's dustpan is derived from this), such as drying mats. The drying mat is like weaving cloth, which is nothing more than weaving together the strips that are the warp and weft threads, including the green and yellow strips.

Cotton yarn is woven together to form a piece of cloth, and the strips are woven together to form a "bamboo cloth", which becomes a mat, and farmers can use the mat to dry wheat, millet, corn, peanuts, and cotton. However, most of the time the cotton is dried with a basket, which is woven together with bamboo pieces and bamboo strips, which has the lowest technical content and can be woven by children. Ordinary people can't make up the mat, so you need to hire a special craftsman, and this craftsman is the "craftsman". With the development of society, many craftsmen are about to disappear, such as carpenters, stonemasons, and masons. If these craftsmen disappear, how can the "craftsman" spirit promoted by the government be implemented?

Uncle Liu Si is a farmer, and in his eyes, the mat is a piece of "bamboo cloth", that is, an enlarged version of the bamboo mat he sleeps on in summer. Fourth Uncle Liu couldn't help but smile dumbly when he saw Professor Yang's drying mat: "Why is the boundary of your drying mat crooked, and how do you let people dry things?" ”

Hu Kehao saw that the craftsman first made an equilateral triangular mat, and after a while, this mat had three sides. The craftsman divided each side into three equal parts, and with the middle one as the base, he made an equilateral triangle, and now there were twelve sides of the mat. The craftsman divided each of the twelve sides into three equal parts, and made an equilateral triangle at the bottom of the middle one, so that there were now forty-eight sides of the mat...... And so on and so forth......

"Professor Yang, what is the origin of your snowflake-shaped mat?" Hu Kehao asked.

"Professor Yang will definitely give this mat a foreign name again, so that it will also become an imported product." Fourth Uncle Liu teased.

"Yes, the name of this mat is Koch, which is pronounced as Koch. The curves on the side of the mat are called Koch curves, and these curves make up the snowflake curve. ”

"Professor Yang, I feel that you should name it the harsh mat, you hired these craftsmen to make the mat for you, it is really too harsh for them!" Fourth Uncle Liu said openly.

Professor Yang was not annoyed and smiled without saying a word. According to Professor Yang Shimo, these things belong to the category of fractal geometry. In the traditional sense, we all know that the point is zero-dimensional, the line is one-dimensional, the plane is two-dimensional, the cube is three-dimensional, and the space where ghosts, dreams, and the underworld are located is four-dimensional. They all have a feature that the dimension of the space is an integer, whereas the dimensionality of a fractal space can be not an integer, but a fraction. According to Professor Yang Shimo's experiments on carpet weaving and matting, we know that the area of the Sierpinski carpet is 0 and the length is infinite, and the unenclosed Koch curve is also 0 in area and infinite in length. What kind of existence is a thing with no area but infinite length, and what is its spatial dimension?

We can also look at the dimensionality from another angle: after magnifying the line segment by two times, the resulting line segment can be seen as the superposition of two original line segments. When the square is doubled, the resulting square can be seen as the superposition of the original squares to the second power of 4=2. When the cube is magnified by two times, the resulting cube can be seen as the superposition of the original cubes to the third power of 8=2. Therefore, the spatial dimensions of line segments, squares, and cubes are 1 dimension, 2 dimensions, and 3 dimensions, respectively.

In general, if we magnify the set E a fold, and the resulting new set can be superimposed by the set of a to the power d, then the fractal dimension of the set E is said to be d. According to this algorithm, it is not difficult to get that the dimensionality of the cube is 3 dimensions. Or it can be calculated that if we magnify the set E a times, and the resulting new set can be superimposed by the sets b, then the fractal dimension of the set E is said to be the quotient of the logarithm of b divided by the logarithm of a. The set obtained by magnifying the Sierpinski carpet by 3 times can be seen as the superposition of 8 Sierpinski carpets, so its dimensionality is log8 divided by log3 equals 1.8928. The set obtained by magnifying the Koch curve by 3 times can be seen as the superposition of 4 Koch curves, so its dimension is log4 divided by log3 equals 1.2619.

In this way, many things in our world are no longer integer dimensions, such as trees and flowers, mountains and rivers, smoke clouds, crystal growth, molecular movements, and so on. In order to study this kind of irregular curves or shapes with self-similarity, fractal geometry was born. In 1982, Mandelbrot created and developed fractal geometry. Fractal objects have a self-similar hierarchy, and the part and the whole have statistical similarities in form, function, information, time, space, etc. For example, each part of a magnet has north and south poles like the whole, and as it is divided, each part has the same magnetic field as the whole magnet. The self-similar hierarchical structure of the Scherpinsky carpet and the Cork curve, with the appropriate enlargement or reduction of the geometric size, and the whole structure remains the same, which is the magic of fractals.