First of all, the beauty of simplicity
Einstein said, "Beauty is simple in essence." He also believes that only with the help of mathematics can we achieve simple aesthetic standards. Einstein's aesthetic theory is also recognized by most people in the field of mathematics. Simplicity and simplicity are its external forms. Only simple, delicate and profound can be called the most beautiful.
The formula given by Euler: V-E+F = 2 is a model of "simple beauty". How many polyhedrons are there in the world? No one can tell. But their vertex number v, edge number e and surface number f must obey the formula given by Euler. Such a simple formula summarizes the identity characteristics of countless polyhedrons. Isn't it amazing? Many equally wonderful things can be derived from her. For example, the number of points V, the number of edges E and the number of regions F of a plane graph satisfy V-E+F = 2, which has become the basic formula of two important branches of modern mathematics-topology and graph theory. Many profound conclusions can be drawn from this formula, which has played a great role in the development of topology and graph theory.
In mathematics, there are many theorems with simple form, profound content and great effect, such as Euler formula. For example:
The circumference formula of a circle: C=2πR
Pythagorean theorem: the sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse.
Average inequality: For any positive number
Sine theorem: the radius r of the circumscribed circle of ABC, then
This concise beauty of mathematics cannot be explained clearly by several theorems. Every progress in the history of mathematics makes the existing theorems more concise. As the great Hilbert once said, "every step of real progress in mathematics is closely related to the discovery of more powerful tools and simpler methods."
Second, the beauty of harmony.
Selberg, a master of number theory, once said that one of his motivations for liking mathematics is the following formula: This formula is really beautiful and can give the combination of odd numbers 1, 3, 5, …. For a mathematician, this formula is like a beautiful picture or landscape.
Euler's formula:, once won the title of "the most beautiful mathematical theorem". Euler established a wonderful and interesting connection between the most important constants in mathematics of his time, which was so harmonious and orderly. The Demefer-Euler formula related to Euler formula is -( 1). This formula closely combines trigonometric function and exponential function, and people think they have no * * * identity. People first marvel at their combination, and then admire it-it is indeed a "match made in heaven", because many beautiful and useful conclusions can be derived from their combination.
For example, from formula (1). From these two formulas, the definition domain of trigonometric function can be extended to complex number domain, that is, radian can be regarded as the angle of complex number. The newly defined cosine function is consistent with the usual cosine function that we are already familiar with.
The beauty of harmony is numerous in mathematics. For example, the famous golden ratio is 0.6 1803398.
In a regular Pentagon, the ratio of side length to diagonal length is the golden ratio.
There is a famous Fibonacci sequence in mathematics {an? }, defined as follows:
a 1？ = 1，a2= 1，
When n≥3, an = an- 1+an-2.
It can be proved that when n tends to ∞, the limit is.
The beauty of Venus is universally acknowledged, and her figure proportion is just the golden ratio.
The golden ratio is widely used in many works of art and architectural design. Leonardo da Vinci called the golden ratio "sacred ratio". He believes that "aesthetic feeling is completely based on the sacred proportional relationship between the parts".
There are still many related problems, and she is worthy of the reputation of "golden section" and "sacred proportion".
Third, the beauty of singularity and mutation.
Two influential magazines around the world jointly invited mathematicians around the world to select "the best mathematical problems in the past 50 years". Among them, there is a fairly simple question: what scores are obtained by B unreasonably, but the results are correct?
After a simple calculation, you can find four scores:. This problem involves the mistake of "incorrect operation and correct result", which not only gives people a surprise, but also shows a strange beauty.
There are also some "crooked equations", such as
A satellite, planet, comet, etc. There may be ellipses, hyperbolas or parabolas due to different motion speeds. These curves are defined as follows:
The ratio of that distance from a fixed point to a fixed straight line is the locus of a point with a constant e,
When E < 1, an ellipse is formed.
When E > 1, a hyperbola is formed.
When E = 1, a parabola is formed.
The constant e changes from 0.999 to 1 and then to 0.00 1. The difference is small, but it forms a completely different curve in shape and nature. And these kinds of curves can be regarded as the sectional lines obtained from different plane truncated cones.
Is there any connection between ellipse and sine curve? Do an experiment, roll the thick paper several times to make a cylinder. Cut the cylinder diagonally into two parts. If the cylinder is not disassembled, the cross section will be elliptical, and if the cylinder is disassembled, the notch will form a sine curve. Isn't the mystery weird and beautiful?
The disordered chaotic state is usually considered to be unable to be studied by mathematics. Random phenomena can be generated by iteration of some phenomena (a quadratic function λx( 1-x)), that is, disordered chaotic states can be generated by iteration of a quadratic equation. This links two completely different types of mathematical problems. This profound discovery makes people lament the magic of natural laws. Moreover, feigenbaum did a lot of calculations on many iterative functions, and got the constant 4.6690 1629 …, which is not a coincidence, although the essence of this number is still unclear. It is this peculiar beauty of mathematics that makes the mysterious, serious and stylized mathematical world full of vitality.
Fourth, the beauty of symmetry.
In ancient times, the word "symmetry" meant "harmony" and "beauty" In fact, the original meaning of this word translated from Greek is "matching and harmony in the arrangement of some things" Pythagoras school believes that the most beautiful of all spatial graphics is spherical; Of all the plane figures, the most beautiful is the circle. A circle is a circle with a symmetrical center-the center of the circle is its symmetrical center, and the circle is also an axisymmetric figure-and any diameter is its symmetrical axis.
The area formula of trapezoid: s =,
Arithmetic progression's first n terms and formulas:,
Where a is the length of the upper bottom edge and b is the length of the lower bottom edge, where a? 1 is the first item, and an is the nth item. In these two equations, a and a 1 are symmetric, and b and an are symmetric.
H and n are symmetrical.
Symmetry is not only beautiful but also useful.
Wave equation of electromagnetic wave;
Where b is the magnetic field strength, e is the electric field strength and c is the speed of light. In this equation, b and e are symmetrical. Maxwell deduced possible electromagnetic waves from these equations by pure mathematical methods, which were later discovered by Hertz, so that the unity of electric field and magnetic field can be obtained.
There are many forms of symmetrical beauty, which is not only appreciated by mathematicians. People's pursuit of symmetrical beauty is natural and simple. Such as lattice symmetry,14th century, Al Chamra Palace in Granada, Spain had all the lattice symmetry, and it was not until 1924 that the type of supernormal symmetry was proved. There is also lattice symmetry, such as logarithmic spiral and snowflake that we like. If we know part of it, we can know all of it. Li Zhengdao and Yang Zhenning also discovered the law of parity non-conservation by studying symmetry. From this, we realized the beauty and success of symmetry.
Fifth, the beauty of innovation.
Euclidean geometry used to be a perfect classical geometry, in which axiom 5: "Only one straight line is parallel to the known straight line at a point outside the straight line" and the conclusion that the sum of the inner angles of a triangle is equal to two right angles seem to be absolute truth. Roman Chevsky adopted the conclusion of different axiom 5: "At least two straight lines are parallel to the known straight lines at a point outside the straight lines", and in this geometry, "the sum of the internal angles of a triangle is less than two right angles", thus creating Roche geometry. Riemannian geometry has no parallel lines. These theories, which are contrary to traditional ideas, are not illusory. When we make long-distance astronomical measurements, it is very convenient to use Roche geometry, and it also has applications in atomic physics and special relativity. In Einstein's general theory of relativity, Riemann geometry is used more to overcome the difficulties in mathematical calculation. Every theory needs constant innovation, every whimsy, every seemingly unreasonable and incredible idea may open a new world. Isn't it difficult to broaden our horizons, broaden our minds and give us completely different feelings? If we boldly imagine again, is there a broader geometry that can include Euclidean geometry and non-Euclidean geometry? In fact, the three kinds of geometry can be unified in the intrinsic geometry of the surface through Gaussian curvature, or through Klein geometry and transformation group. In the process of continuous innovation, mathematics has been developed.
Sixth, unified beauty
The concept of number extends from natural number, fraction, negative number and irrational number to complex number. It has experienced countless ups and downs, its scope has been expanding, and its role in mathematics and other disciplines has been increasing. Then, people will naturally doubt whether the concept of complex number can be further promoted.
Hamilton, a British mathematician, thought hard 15 years and failed. Later, he was "forced to compromise" and sacrificed a property in the complex set, and finally found a quaternion, that is, a1+a2i+a3j+a4k (A1,a2i, a3j and a4k are real numbers), where I, j and k are like imaginary units in complex numbers. If a3 = a4 = 0, quaternion a 1+a2i+a3j+a4k is a general complex number. The study of quaternion promotes the study of linear algebra, and on this basis, the theory of linear associative algebra is formed. Physicist Maxwell established the electromagnetic theory by using quaternion theory.
The development of mathematics is a process of gradual unification. The purpose of unification is as Hilbert said: "Pursuing more powerful tools and simpler methods".
Einstein's lifelong dream is to pursue the theory of universal unity. He revealed the relationship between mass and energy in nature with a concise expression E=mc2, which is a unified work of art. But he still hasn't realized his dream of reunification. Mankind is constantly exploring the complex world and understanding the world from a unified perspective. The universe has no end, and the beauty of unity also needs eternal pursuit.
The beauty of mathematics can be viewed from more angles. Every aspect of beauty is not isolated. They complement each other and are inseparable. She needs people's heart and wisdom to dig deeper and better understand her aesthetic value, her rich and profound connotation and thoughts, and its profound influence on human thinking. If we can explore and discover with mathematicians in the process of learning and get the joy of success and the enjoyment of beauty from it, then we will continue to go deep into it, appreciate and create beauty.