Shulou(Shulou.com)11/24 Report--

In the eyes of many people, mathematics may look like this:

In the eyes of mathematicians, mathematics goes like this:

This is the Mandelbrot set (Mandelbrot Set), which is a set of points on a complex plane that correspond to some initial values when iterating over a function, and the iterative results do not diverge to infinity. This function is usually a simple compound function, that is, it is composed of some basic functions, such as square functions.

The Mandelbrot set is a set of all the c that remains bounded. On the complex plane, this set usually presents a very beautiful fractal shape, which is one of the reasons why the Mandelbrot set has attracted wide attention.

The math goes like this:

This is the Julia set (Julia Set), another kind of collection that is closely related to the Mandelbrot set. It is also a set of points on a complex plane, and it is also generated by a function iteration. Like the Mandelbrot set, the Julia set usually presents a fractal shape, and the shape and size depend on the value of c. Therefore, Julia set is also a very beautiful mathematical object, which attracts a lot of people's research and exploration.

It goes like this:

This is the Koch snowflake, a fractal figure, which is constructed by replacing the edges of an equilateral triangle infinitely iteratively. Specifically, we first divide each side of the triangle into three segments, and then replace the middle segment with an equilateral triangle protruding outward, so we get a figure similar to a W shape. Then, we repeat the above curve substitution operation for the three sides of each protruding triangle, and the resulting figure is the first-stage Koch snowflake. Then, we continue the curve substitution operation on the three sides of each small triangle to get the second-level Koch snowflakes, and so on.

After infinite iteration, the shape of Koch snowflakes will be closer and closer to a fractal figure. It has many interesting properties, for example, its perimeter is infinite, but its area is limited and can be calculated. Koch snowflakes also have self-similarity, that is, different parts of Koch snowflakes have a shape and structure similar to that of the whole. These properties make Koch Snowflake an important mathematical object, and it is also widely used in science, art, engineering and other fields.

It goes like this:

This is the Sherpinski triangle (Sierpinski triangle), a fractal figure that is obtained by cutting and removing an equilateral triangle through infinite iterations. Specifically, we first connect three sides in the center of the regular triangle to form an embedded small regular triangle. Then, we cut off the small triangle in the center of the original regular triangle and repeat the above cutting and removal operation for each remaining small triangle to get a smaller Serpinski triangle. After iterating over and over again, the final figure is the Serpinski triangle.

Sherpinsky triangle is the simplest kind of fractal graphics, which has self-similarity and infinite hierarchical structure. For example, any small triangle in the Serpinski triangle can be thought of as a reduced copy of the entire graph. These properties make the Sherbinski triangle a classical mathematical object and are widely used in science, art, engineering and other fields. For example, it can be used to describe the fractal structure in physical systems, and it can also be used as a random number generator in data compression and cryptography.

It goes like this:

This refers to the Serpinski carpet (Sierpinski Carpet) and its three-dimensional counterpart, the Mengel sponge (Menger Sponge), which are fractal figures.

The Serpinski carpet is obtained by cutting and removing a square through infinite iterations. Specifically, we first divide the square into nine small squares, then cut off a small square in the center, and repeat the above cutting and removal operation for each remaining small square to get a smaller Serpinski carpet. After iterating over and over again, the final figure is the Serpinski carpet.

The Mengel sponge is the three-dimensional counterpart of the Serpinski carpet, which is obtained by a similar infinite iterative operation on a cube. Specifically, we first divide the cube into 27 small cubes, then cut off one cube in the center and six small cubes adjacent to it, and repeat the above cutting and removal operations for each remaining small cube. to get a smaller Mengel sponge. After iterating over and over again, the final figure is the Mengel sponge.

These two kinds of fractal graphics also have self-similarity and infinite hierarchical structure, and they are classical objects in fractal geometry. Mengel sponge is also widely used in science, engineering, computer graphics and other fields. For example, in computer graphics, it can be used to construct realistic three-dimensional scenes and simulate textures and shapes in nature.

It goes like this:

This is the "Penrose tiles", also known as the "Penrose diamond" (Penrose diamonds). They are a group of cutting shapes with special geometric properties that can be spliced to fill a plane or space.

Penrose tile was invented by British mathematician Roger Penrose in 1974. It has five symmetries, that is, it can be rotated five times without changing its shape. This special symmetry makes Penrose tiles have a wide range of applications in the fields of mathematics and physics, such as describing crystals and harmonic functions.

Penrose tiles also have another special property, that is, there is no repetitive arrangement. This means that one arrangement cannot be converted into another by translation or rotation. This property makes Penrose tiles one of the representatives of aperiodic crystals, which are also found in nature, such as graphene.

Generally speaking, Penrose tile is a unique and interesting geometric figure, and its unusual symmetrical properties and arrangement make it an important research object in the fields of fractal geometry and crystallography.

It goes like this:

This is Heesch Tiling, which refers to a kind of cutting cover with special properties, which was proposed by Dutch mathematician A. Heesch in 1965. Heesch Tiling is an endless, irregular, aperiodic cutting cover, it can seamlessly cover any area on the plane, and there is no repeated arrangement.

A key feature of Heesch Tiling is that its boundary has special constraints, which is called Heesch number (Heesch number). To put it simply, the Heesch number is the minimum number that can extend the Heesch Tiling indefinitely, that is, for each Heesch Tiling, there is an integer n, so that no matter how extended, it cannot exceed n steps. This constraint not only makes Heesch Tiling regular, but also produces some wonderful geometric and combinatorial properties.

The research of Heesch Tiling is not only to satisfy the curiosity of geometry, but also has certain practical value. For example, Heesch Tiling can be used to design the microstructure of magnetic materials and create seamless wallpapers and patterns. In addition, Heesch Tiling is also used to study topology and self-assembly and other fields.

Generally speaking, Heesch Tiling is an endless cutting cover with special constraints and regular properties, and it has a wide range of applications and research value in the fields of geometry and combinatorial mathematics.

It goes like this:

This is the visualization of Kraz's conjecture. Collatz Conjecture conjecture, also known as 3n+1 conjecture or odd-even normalized conjecture, is a mathematical conjecture proposed by German mathematician Lothar Collatz in 1937.

The conjecture is expressed as: for any positive integer n, if it is odd, multiply it by 3 and add 1, if it is even, divide it by 2, and repeat this operation until n becomes 1. Although this conjecture seems very simple, mathematicians have not yet found a simple proof or counterexample to prove or disprove it.

Although the Kraz conjecture has not been proved, it has been verified many times by computer simulations, including all positive integers with a range of less than 2 to the 68th power. At the same time, Kraz conjecture has also aroused people's thinking and research on the nature of mathematics and algorithm complexity.

It goes like this:

This is the Ulam spiral, a mathematical structure proposed by the Polish mathematician Stanis Stanislaw aw Ulam in 1963. It is a two-dimensional figure formed by natural numbers arranged along a spiral, each of which occupies a grid. Specifically, fill the natural number on a helix in a clockwise direction starting at 1, as shown in the following figure:

Ulan helix has a wide range of applications and research value in the fields of mathematics and computer science. For example, the Wulan helix can be used to study the distribution of prime numbers. By marking the distribution of prime and non-prime numbers on the Wulan helix, it can be found that the distribution of prime numbers has a certain regularity and periodicity. At the same time, the Ulan helix can also be used to generate some interesting mathematical problems and puzzles, such as finding the longest continuous prime sum and so on.

It goes like this:

This is the visualization of Riemann's conjecture. Riemann conjecture (Riemann Hypothesis) is a famous conjecture in the field of mathematics, which was put forward by the German mathematician Bernard Riemann in 1859. This conjecture involves many fields, such as complex function theory, number theory, algebraic geometry and so on. It is one of the most important problems in mathematical research.

The expression of Riemann conjecture is that the zeros of all non-trivial Riemannian zeta functions lie on a straight line on the complex plane, that is, the so-called "critical line". The real part of this borderline is 1 prime 2. Among them, Riemann zeta function is a special complex function, which can be expressed as a series, in which each term involves the power of a natural number.

Although the Riemann conjecture seems very simple, mathematicians have not found a simple proof or counterexample to prove or disprove the conjecture. The proof of Riemann conjecture involves complex mathematical knowledge and technology, which has not been fully proved so far, but many important inferences and conclusions have been obtained, which have a far-reaching impact on the research and application of mathematics.

It goes like this:

This is the visualization of the Pythagorean triple. The Pythagorean triple refers to three positive integers a, b, and c that satisfy the combination of a ²+ b ²= c ². It is named after Pythagoras because he was one of the first ancient Greek mathematicians to discover and study the properties of right triangles. Pythagorean triple is of great significance in mathematics and physics, especially in the study of Pythagorean theorem and trigonometric functions. For example, in computer graphics, Pythagorean triples can be used to generate smooth curves and patterns.

It goes like this:

This is Conway's life game, also known as cellular automata life game, is a simple computer simulation game invented by British mathematician John Conway in 1970. The game is based on a two-dimensional grid, filling each grid with a cell in a "life" or "death" state. These cells evolve and interact according to certain rules, and eventually form a variety of complex patterns and structures.

The rules of the game of life are simple: in each round of evolution, the state of life and death of a cell depends on the number of adjacent cells in the eight grids around it. If there are two or three living cells around a cell, the cell is still living in the next round of evolution; if there are less than two living cells around a cell, the cell dies in the next round of evolution; if there are more than three living cells around a cell, the cell will also die in the next round of evolution. For a dead cell, if there happen to be three living cells around it, the cell will be "reborn" in the next round of evolution, that is, it will become a living cell.

The game of life may seem simple, but it can show a lot of unexpected complex behaviors, including static structure, periodic patterns and mobile starships. In addition, life games also have many applications in mathematics and computer science, such as image processing, data compression, encryption algorithms and so on.

It goes like this:

This is the Rickaman series (Recam á n's Sequence), proposed by the Swiss mathematician Bernardo á n Santos in 1964. The sequence starts at 0, and each successive number is the previous number minus the number in the current sequence, but if the result is negative or has already appeared in the sequence, take the previous number and add the number in the current sequence. For example, the first items of Recamants Sequence are: 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 42, 17, 43, 16, 44, 15, 45, 14, 46, 79, 113, 78, 114, 77, 39, 78, 38, 79, 37, 80, 36, 81, 35, 82, 34, 83, 33, 84.

Rikaman series has attracted much attention because of its unusual way of generation, and has been widely studied and applied in the field of mathematics and computer science. It involves many mathematical fields, such as combinatorial mathematics, number theory, recursion, dynamic programming, etc., as well as polygon segmentation, random walk, algorithm design and so on.

It goes like this:

This is 173 ways to overlap four circles. Four circles are required to be arranged together so that they intersect each other, and the intersection of any two circles is inside or outside the other two circles. There are a total of 173 such arrangements, which were discovered by British mathematician J. B. Wilker in 1969.

The mathematical knowledge involved in this problem includes the geometric properties of circles, combinatorial mathematics, graph theory and so on. It not only has aesthetic value, but also is a mathematical model of some practical problems, such as chip design, molecular structure analysis and so on.

It goes like this:

This is the knight's chessboard tour (Knight's Tour), which refers to the process that the knight starts from a square, moves along the "L" shape, passes through each square exactly once, and finally returns to the starting point on the 8x8 chess board. This is a classical mathematical problem, and there are many solutions, including violence enumeration, heuristic search, backtracking algorithm and so on.

The mathematical knowledge involved in the knight's chessboard journey includes graph theory, Euler circuit, Hamilton circuit and so on. It is not only a classical mathematical problem, but also has practical application value, such as routing planning in computer network, simulation of protein structure in biology and so on.

Last

Mathematics is very intuitive. In order to solve a problem, one must use logical thinking as well as creative thinking and intuition. The mathematical world is dominated by patterns and symmetries, some of which are known, and most of which need to be discovered.

Creativity is exploration, discovery, imagination and innovation. The reason why many people don't see any creativity in mathematics is that mathematics is a boring subject and there are a set of strict rules that people must abide by. When people think of creativity, they think of creating something without the guidance of rules.

But keep in mind that other creative disciplines such as literature, music and art also have their own rules. There are structure, tonality, harmony, rhythm and musical instruments in music, just as algebra, arithmetic, operations and formulas are in mathematics. Once you have learned the language of a subject, any subject can be creative. Science can be as creative as art. Mathematics can be as creative as music.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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