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The mystery of the Sierpiński carpet: how to create infinite complexity from a square?
In the field of mathematics, the Sierpiński carpet is a fascinating flat fractal first described by Wacław Sierpiński in 1916. This carpet is not only a mathematical example, but also a model of infinite complexity. The graphics generation process behind it is amazing, constantly revealing the subtle relationship between mathematics and aesthetics.
Construction process
The construction of a Sierpiński rug begins with a square. The initial square is cut into nine identical small squares to form a 3x3 grid, and then the middle small square is removed. This process is then recursively applied to the remaining eight small squares, and so on until an infinite level is reached. This process of recursively removing squares reveals an exquisite structure that makes one marvel at the logic behind it.
Attribute analysis
The area of this rug is zero.
Under the standard Lebesgue metric, the area of a Sierpiński rug approaches zero. This result can be proved by the recursive process: assuming that the area of the i-th layer is ai, then the area of the i+1th layer will be (8/9)ai. As i increases, ai will tend to zero.
The interior of this rug is empty.
This can be proven by legislation. If it is assumed that there is a point P located inside the carpet, then there is a square completely contained in the carpet, which will result in at least one small square being hollowed out in the k+1 iteration, so this assumption cannot be satisfied. , causing conflicts.
Random process and its applications
In recent years, Brownian motion on the Sierpiński carpet has attracted considerable attention from mathematicians. The study found that a random walk on this carpet would spread slower than an unrestricted random walk on a flat surface. This means that a random walker walking on a carpet would need more steps to cover the same distance.
Wallis sieve method
The Wallis screen is a variation of the Sierpiński carpet. Its construction process is similar to that of the Sierpiński carpet, starting with dividing the unit square into nine smaller squares and removing the middle one. Subsequently, in each layer of segmentation, each small square is divided into 25 smaller squares, and the middle one is also removed. This process continues.
Application scope
Sierpiński carpets have important applications in modern technology, especially in the design of mobile phones and Wi-Fi fractal antennas. Due to their self-similarity and scale invariance, these antennas are able to easily support multiple frequencies and outperform conventional antennas, making them an excellent choice for portable devices.
Conclusion
The world of mathematics is so mysterious and fascinating. Sierpiński carpet challenges our traditional understanding of space and dimension with its unique structure and beauty. When we witness this infinite complexity evolving from a square, we can't help but think: How much unknown beauty is hidden on the other side of mathematics?
