(A) beautiful language
Mathematics has its own unique language-mathematical language, which includes:
The Language of Number 1-Symbolic Language
As for "∏", "Nine Chapters of Arithmetic" says: "Cut it carefully, lose less, cut it again, and blend with the circle without losing anything"; Facing the irrational number "√2", we will never forget that hippasus (a member of Pythagoras School in the 5th century BC) was thrown into the sea because of the "mathematical paradox" that a square with a side length of 1 cannot be expressed as the ratio of integers. And sin? , ∞, etc. The language of one number after another shows the perfection and exquisiteness of numbers to the fullest.
Two-form language-perspective language
From the perspective of formal symmetry (how many touching stories echo from a distance with "central symmetry" and "axial symmetry"); Proportion (the beautiful "golden section" divides the wonderful configuration of the figure? ); Harmony (such as logarithmic notation, base number and true number) is actually the best combination of classics! ); Being different ("the biggest" and "the smallest") reminds us of "the power of the mountain" and "the gentleness of the water". We deeply realize that where there are mountains and water, why is it always the inner charm of outstanding people? ) and novelty (the appearance of one mathematical paradox after another keeps the freshness and vitality of mathematics and even all natural sciences) and so on.
(B) concise beauty
Einstein said, "Beauty is simple in essence." He also believes that only with the help of mathematics can we achieve simple aesthetic standards. 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? !
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 = 2π r.
Pythagorean theorem: the sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse+=.
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."
Poincare pointed out: "What gives us beauty in solving and proving?" It is the harmony of all parts, their symmetry, their ingenuity and balance. "
(4) Harmonious beauty
Beauty is harmonious. Harmony is also one of the characteristics of mathematical beauty. Harmony means that the formal structure is elegant, rigorous or not contradictory.
No subject can clarify the harmony of nature more clearly than mathematics.
-Paul carus
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. At first, people were surprised at their combination, but later they admired it ―― it was really a "match made in heaven".
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. The aspect ratio of building windows is generally: the kneecap is the golden point of the thigh and calf, the elbow joint is the golden point of the arm, and the navel is the golden point of the height. When the temperature is 23 degrees Celsius, people feel most comfortable. At this time, the body temperature is about 0.6 18 at 23:37. The theme of famous paintings is mostly painted at 0.6 18 of the picture, and the sound code of stringed instruments is placed at 0.6 18 of the strings, which will make the sound sweeter. The exquisiteness of architectural design, the mystery of human science, the elegant style of artistic works and the beautiful rhythm of musical works are all integrated into the symmetrical beauty and harmonious beauty of numbers.
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".
(4) Strange beauty
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
The orbit of a satellite, planet, comet, etc. Because of the different speed of motion, it may be ellipse, hyperbola or parabola. These curves are defined as follows: the locus of a point whose ratio of the distance from a point to a fixed point to its distance from a fixed line is 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?
(5) Symmetrical beauty
In ancient times, the word "symmetry" meant "harmony" and "beauty" 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.
There are many forms of symmetrical beauty, which is not only appreciated by mathematicians. People's pursuit of symmetrical beauty is natural and simple. For example, we like logarithmic spiral and snowflake. If we know some of them, we can know all of them. 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.
(6) 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.
(7) 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 number set, and finally found a quaternion, that is, a1+a2i+a3j+a4k (A1,a2, a3 and a4 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 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.
(8) Analogical beauty
Algebraic language in analytic geometry has played an unexpected role because it does not need to be considered geometrically. Considering the equation, we know that it is a circle. Where does the perfect shape, symmetry and endless end of a circle exist? In the equation! For example, it has symmetry and so on. Algebra replaced geometry, thought replaced eyes! In the properties of this algebraic equation, we can find out all the properties of the circle in geometry. This fact enables mathematicians to explore deeper concepts through algebraic representation of geometric figures. That is four-dimensional geometry. Why can't we consider the following equation? What about the shape equation? This is a great progress. By analogy alone, we can move from three-dimensional space to high-dimensional space, from tangible to intangible, and from real world to virtual world. What a wonderful thing it is! This process can be accurately described by the poem of Cheng Hao, a famous philosopher in the Song Dynasty: Tao leads the world to the outside, but thinking into the abnormal environment.
(9) abstract beauty and freedom beauty
From the basic concepts of elementary mathematics to the principles of modern mathematics, there is a general abstraction and generalization. As Kepler said: "The main purpose of studying the external world is to discover the reasonable order and harmony endowed by God, which is revealed to us by God in mathematical language".
The first characteristic of mathematics is that she has the ability of abstract thinking. What she deals with in mathematics is abstract quantity, which is expressed by symbols divorced from the content of specific things. It can be an algebra of any specific number, but it is not equal to any specific number. For example, "n" stands for a natural number, not n jobs, n chickens, nor n photos ... nor a specific number. I wonder if it is 0? Is it 1? Or 100? ..... "Know" includes "don't know" and "concrete" is full of "not concrete", which is such an abstract number!
Leonardo da Vinci was a master of art and science from 15 to 16 century. He summed up the idea of his art monograph in one sentence: "No one appreciates my work except mathematicians."
Many famous figures in history are obsessed with music, such as the great mathematician Kranak. Why is a math prince so obsessed with music? There may be many reasons. In my opinion, the most important point is that mathematics and music are abstract languages, full of abstract beauty and free beauty. Moreover, mathematics and music are two artificial bright worlds. The former created an infinite and absolutely true world with only ten Arabic numerals and a few symbols, while the latter created an infinite and absolutely beautiful world with only five lines and some tadpole-shaped notes. If music is the best expression of human emotional activities, then mathematics is the most magical product of human rational activities.
(10) dialectical beauty
Anyone who is familiar with mathematics realizes that mathematics is full of dialectics. If every science contains rich dialectical thoughts, then mathematics has its own special expression, that is, it can clearly express various dialectical relations and transformations with mathematical symbolic language and concise mathematical formulas.
For example: in elementary mathematics: the corresponding relationship between points and coordinates; The relationship between curve and equation; The inherent relationship between inevitability and contingency revealed by probability theory and mathematical statistics. And the contents involved in senior three mathematics: the concept of limit, especially modern limit language, well embodies the dialectical relationship between finite and infinite, approximation and accuracy; Newton-Leibniz formula describes the connection and mutual transformation between differential and integral operations, and so on.
Such examples abound in mathematics. Of course, to truly master the beauty of mathematics, it is not enough to just know some mathematical knowledge. We should also be good at discovering the relationship between various mathematical structures and mathematical operations, and establishing and applying the relationship and transformation between them. Only in this way can we exert the power of dialectical thinking contained in mathematics. The dexterity of many calculation methods in mathematics proves the beauty of methods, and the way of thinking is often formed by comprehensive utilization of various relationships and appropriate transformation.
Mastering the dialectical comparative thought of "the two advantages are the most important, and the two disadvantages are the least" will master the key to understanding this kind of topic. In fact, mathematics in all parts of the country implements the principle of "choosing the most important of the two, and choosing the least of the two". Mathematics embodies dialectics everywhere, and mathematicians look at problems with dialectical eyes all the time. Professor Chen Shengshen said in a lecture in Peking University in the 1980s: "People often say that the sum of the internal angles of a triangle is equal to 180, but this is wrong!" ..... "It is wrong to say that the sum of the angles in a triangle is 180, not that this fact is wrong, but that this way of looking at the problem is wrong. It should be said that the sum of the outer angles of the triangle is 360! Looking at the internal angles, you can only see that the sum of the internal angles of the triangle is180; The sum of the internal angles of the quadrilateral is 360; The sum of pentagonal interior angles is 540...n The sum of pentagonal interior angles is (n-2) * 180. Although the formula for calculating the sum of inner angles is found, it contains the number of sides n ... What if you look at the outer corner? The sum of the external angles of a triangle is 360, the sum of the external angles of a quadrilateral is 360, the sum of the external angles of a pentagon is 360, …, and the sum of the external angles of an n-polygon is 360.
This summarizes many situations with a very simple conclusion, and replaces the formula related to n with a constant unrelated to n, and finds a more general law. "In fact, mathematics is not aesthetics?
The power of mathematics is infinite. The beauty of mathematics is like the poem in Dante's Divine Comedy. Beautiful and harmonious music, unique paintings and magnificent buildings will also make mathematics learners passionate and interested! 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 with mathematicians and teachers 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. I believe that our math study will definitely achieve better academic results.
Personal introduction: A high school math teacher, who has been teaching for ten years, published a paper "Analogy Triangle Formula, Finding the Entrance to Solving Problems" and "A Stone Arouses a Thousand Waves".
Adopt me! ! !