1 the beauty of simplicity of mathematical concepts
There are many concepts in mathematics, but each concept is given in the simplest and most general language. For example, the concept of factorization in algebra: decompose a polynomial into several algebraic expression products. The concept of line segment perpendicular bisector in geometry: "A straight line perpendicular to this line segment and bisecting this line segment, and so on." . For example, how many straight lines can students explore first in the teaching of Basic Drawing Knowledge? Then ask the students to sum up this conclusion in their own words. Finally, the teacher gave "two points determine a straight line", a short sentence, concise and rigorous, rich in connotation, fully let students realize the beauty of simplicity of mathematical theorems; Another example is the definition of a circle in the ninth grade: "A circle is a set of points whose distance to a fixed point is equal to a fixed length". If there is no "set", a point will be formed instead of a circle. The word difference will make the situation different from that of Wan Li, which fully embodies the concise beauty of mathematical concepts.
The beauty of symbol, abstraction and unity.
Most mathematical knowledge consists of numbers and symbols, ranging from four operations to comparative size, as well as large, medium and small brackets in operations. The symbols are all of moderate size and symmetrical up and down. Beautiful figures: first, the beginning of all things, unifying the world and taking the lead; Second, even numbers, double happiness, fly with me; At first glance, it is two or three miles away, and mist hangs over four or five families. There are six or seven pavilions, eighty or ninety flowers (Shao Yong); Seven or eight stars in the sky, two or three points before the rain (Xin Qiji); A sail, an oar, a fishing boat, a fisherman and a hook. A bow and a smile, a bright moon and autumn (Ji Xiaolan). After reading the idioms and poems above, it is obvious to everyone that numbers can show all kinds of thoughts and feelings, whether they are used singly, repeatedly or circularly.
3. The beauty of coordination and symmetry of the structural system.
This symmetry can be seen everywhere in mathematics, such as axial symmetry and central symmetry in geometry; The imaginary root pair of polynomial equation in algebra, the relationship between function and inverse function image (about the symmetry of straight line yzx) and so on all show symmetry. Symmetry gives people beauty and comfort. There are many shapes of quadrilateral, but the most perfect one is square, because its symmetry axis is more than any quadrilateral, and it is also a centrally symmetrical figure. These characteristics make squares popular and widely used. For example, people use the square area with the side length as the unit length as the basic unit to measure the area of other graphics. People also like to beautify the environment with square patterns. For example, laying indoor and outdoor floors with square floor tiles is not only beautiful and elegant, but also simple and easy to construct. Pythagoras said: "The most beautiful three-dimensional figure is spherical, and the most beautiful plane figure is round." Because these two figures are symmetrical in any direction. In fact, things designed according to symmetry can be seen everywhere around us. From erasers and rackets to airplanes and buildings. The famous Great Hall of the People in Beijing; The towering Shanghai Oriental TV Tower; The epitome of the Egyptian pyramids; A fan of realism; Plum blossom petal-shaped combination pattern; Copper coin-shaped round China surface; The beautiful "snowflake" pattern shows the symmetrical beauty and harmonious beauty of geometric figures. 4 universality of formula
There are countless triangles with different shapes and sizes in the world, but the area formula S= 1/2ah is suitable for calculating the area of all triangles, which is also the concrete embodiment of mathematical beauty.
5 universality of application
With the development of science and social progress, mathematics has increasingly penetrated into every field of science and technology and even social life. When you deposit money in the bank, you will encounter interest rate problems; Shot putters must know how to throw in order to achieve ideal results; Football players should also know where to hit the opponent's goal most easily ... In addition, mathematicians give computers intelligence, and computers also make mathematicians smarter. In a word, "Where there is life, there is mathematics". This is also the embodiment of the wide application of mathematics, and it is also an important content of mathematical beauty.
6 strange beauty
Singularity means novelty and development. We take the appearance of "√2" as an example. Before irrational numbers appeared, people thought that the length of any two line segments could be negotiated. However, it was later discovered that the diagonal and side of a square were non-negotiable. And "√2" cannot be expressed as the ratio of two integers. This strange result led to the expansion of the number system, which made people jump out of the narrow circle of rational numbers and produced a new leap in knowledge. Therefore, it is not difficult for us to understand why strangeness is beautiful in mathematics.
In addition, Pythagorean theorem and golden section in mathematics are concrete manifestations of mathematical beauty. Pythagorean theorem is like a bright pearl, with infinite charm, which makes many people fall for it. There are at least 370 existing proofs, which are the theorems with the largest number of exchanges in the world. The golden section is widely used in the fields of architecture, music and art. For example, the side of the five-pointed star is treated according to the golden section; When designing handicrafts or daily necessities, the aspect ratio is often designed to be about 0.6 18 and 0.6 18. This number was discovered by eudoxus in ancient Greece. Interestingly, since then, this number has formed many indissoluble bonds with human beings: the Greek goddess is gentle and attractive. According to experts' research, the ratio of the distance from her feet to her navel to her whole height is exactly 0.6 18. Painters and artists introduce them into painting, sculpture and other artistic fields to make their works more harmonious and beautiful; The announcer on the stage always likes to stand on the stage 0.6 18, which has the best sound effect and makes people look natural and generous. People are most comfortable at around 23℃ and have the best physiological function. These are all derived from the golden section principle.
In addition to the above specific contents, the beauty of mathematics lies in mathematics teaching. The teacher's vivid explanation, incisive analysis, clever guidance, vivid language and reasonable blackboard writing all give students beautiful enjoyment. In teaching, teachers should always consciously explain the history of mathematics development and the wide application of mathematics to students, constantly show the beauty of mathematics and further understand the true meaning of beauty.
The charm of mathematical beauty is attractive, the power of mathematical beauty is enormous, and the idea of mathematical beauty is magical. It can change people's prejudice that mathematics is boring and make people realize that mathematics is also a colorful and beautiful world. If mathematics makes many people feel relaxed and happy, and has devoted their whole lives to it, thus promoting the rapid development of mathematics, then it will certainly inspire more ambitious young people to pursue knowledge and explore the future, because "beauty" exists in mathematics. References [1] (English) Russell's The Development of My Philosophy was published by the Commercial Press 1985: 153 [2] Western aestheticians on Beauty and Aesthetic Feeling edited by Peking University Aesthetics Teaching and Research Section. The Commercial Press1980:/kloc-0.
Follow-up: Are you sure it works? A: Modify the format of some font literature reviews. Baidu changed the font. Q: What if it doesn't work? A: Talking about Beauty in Mathematics Abstract: "In view of the phenomenon that middle school students are tired of learning in mathematics education at present, this paper tries to explore the consistency between aesthetic and intellectual development, teaching principles and aesthetic principles from the perspective of aesthetic characteristics, so as to improve students' interest in learning mathematics and teaching level. Keywords: concise beauty; Symbolic beauty, abstract beauty and unified beauty; The beauty of harmony and symmetry; The universality of the formula; Universality of application; Mathematics, like singular beauty, if viewed correctly, not only has truth, but also has supreme beauty.
Russell
The most beneficial is the most beautiful.
-Socrates
Mathematics can promote people's understanding of the characteristics of beauty: value, proportion, order and so on.
-Aristotle When you wander in the palace of music and listen to beautiful music, you will feel the enjoyment of "beauty" brought by music; When you wander in the world of literature and appreciate the wonderful sentence of "earth-shattering, crying ghosts and gods", you will certainly appreciate the "beauty" brought by literature ... In fact, "where there is mathematics, there is beauty", which is highly praised by ancient philosophers for the beauty of mathematics. There is also "beauty" in mathematics that can enlighten wisdom and cultivate sentiment. The content of mathematical beauty is rich, such as the simple unification of mathematical concepts and the coordination and symmetry of structural relations; The universality of formula, the universality of application and the singularity of formula are the concrete contents of mathematical beauty. Let's talk about my understanding of the beauty of mathematics in combination with elementary mathematics.
The concise beauty of mathematical concepts in 1 simplifies the thinking process and is more reliable.
-Fried food (fried food)
The so-called beauty problem in arithmetic refers to a problem that is difficult to solve; The so-called beautiful answer refers to a simple answer to a difficult and complicated question.
-Diderot
The size of the universe, tiny particles, the speed of rockets, the ingenuity of painters, the qualitative change of the earth, and the mystery of biology. The complexity of daily life, ... can be expressed by mathematics.
-Hua
Mathematics is the language that God uses to write the universe.
Galileo
There are many concepts in mathematics, but each concept is given in the simplest and most general language. For example, the concept of factorization in algebra: decompose a polynomial into several algebraic expression products. The concept of line segment perpendicular bisector in geometry: "A straight line perpendicular to this line segment and bisecting this line segment, and so on." . For example, how many straight lines can students explore first in the teaching of Basic Drawing Knowledge? Then ask the students to sum up this conclusion in their own words. Finally, the teacher gave "two points determine a straight line", a short sentence, concise and rigorous, rich in connotation, fully let students realize the beauty of simplicity of mathematical theorems; Another example is the definition of a circle in the ninth grade: "A circle is a set of points whose distance to a fixed point is equal to a fixed length". If there is no "set", a point will be formed instead of a circle. The word difference will make the situation different from that of Wan Li, which fully embodies the concise beauty of mathematical concepts.
2 symbolic beauty, abstract beauty and unified beauty mathematics is also a language, and it is the most perfect language in the existing structure and content ... It can be said that it is natural to speak in this language; The creator talks to it, and the protector of the world continues to talk to it.
-C. Dielmann In terms of its essence, mathematics is abstract; In the last century, his abstraction was higher than logic.
-Crystal
It is almost impossible for nature not to have a preference for the beauty of mathematical reasoning.
-Yang Zhenning
Most mathematical knowledge consists of numbers and symbols, ranging from four operations to comparative size, as well as large, medium and small brackets in operations. The symbols are all of moderate size and symmetrical up and down. Beautiful figures: first, the beginning of all things, unifying the world and taking the lead; Second, even numbers, double happiness, fly with me; At first glance, it is two or three miles away, and mist hangs over four or five families. There are six or seven pavilions, eighty or ninety flowers (Shao Yong); Seven or eight stars in the sky, two or three points before the rain (Xin Qiji); A sail, an oar, a fishing boat, a fisherman and a hook. A bow and a smile, a bright moon and autumn (Ji Xiaolan). After reading the idioms and poems above, it is obvious to everyone that numbers can show all kinds of thoughts and feelings, whether they are used singly, repeatedly or circularly.
3. The beauty of coordination and symmetry of the structural system.
Symmetry is a broad theme, which is of great significance in both art and nature. Mathematics is his foundation.
-H. Weyl This symmetry can be seen everywhere in mathematics, such as axial symmetry and central symmetry in geometry; There are imaginary root pairs of polynomial equations in algebra, and the relationship between function and inverse function image (about the symmetry of straight line yzx) is symmetrical. Symmetry gives people beauty and comfort. There are many shapes of quadrilateral, but the most perfect one is square, because it has more symmetry axes than any quadrilateral, and it is also a figure with central symmetry. These characteristics make squares popular and widely used. For example, people use the square area with the side length as the unit length as the basic unit to measure the area of other graphics. People also like to beautify the environment with square patterns. For example, laying indoor and outdoor floors with square floor tiles is not only beautiful and elegant, but also simple and easy to construct. Pythagoras said: "The most beautiful three-dimensional figure is spherical, and the most beautiful plane figure is round." Because these two figures are symmetrical in any direction. In fact, things designed according to symmetry can be seen everywhere around us. From erasers and rackets to airplanes and buildings. The famous Great Hall of the People in Beijing; The towering Shanghai Oriental TV Tower; The epitome of the Egyptian pyramids; A fan of realism; Plum blossom petal-shaped combination pattern; Copper coin-shaped round China surface; The beautiful "snowflake" pattern shows the symmetrical beauty and harmonious beauty of geometric figures. 4 universality of formula
There are countless triangles with different shapes and sizes in the world, but the area formula S= 1/2ah is suitable for calculating the area of all triangles, which is also the concrete embodiment of mathematical beauty.
5 universality of application
With the development of science and social progress, mathematics has increasingly penetrated into every field of science and technology and even social life. When you deposit money in the bank, you will encounter interest rate problems; Shot putters must know how to throw in order to achieve ideal results; Football players should also know where to hit the opponent's goal most easily ... In addition, mathematicians give computers intelligence, and computers also make mathematicians smarter. In a word, "Where there is life, there is mathematics". This is also the embodiment of the wide application of mathematics, and it is also an important content of mathematical beauty.
6 strange beauty
Singularity means novelty and development. We take the appearance of "√2" as an example. Before irrational numbers appeared, people thought that the length of any two line segments could be negotiated. However, it was later discovered that the diagonal and side of a square were non-negotiable. And "√2" cannot be expressed as the ratio of two integers. This strange result led to the expansion of the number system, which made people jump out of the narrow circle of rational numbers and produced a new leap in knowledge. Therefore, it is not difficult for us to understand why strangeness is beautiful in mathematics.
Characteristics of mathematical aesthetic methods
1, intuition, aesthetic intuition is an important type of mathematical intuition, and mathematical aesthetic method is mainly a method driven by aesthetic intuition and made aesthetic consideration. Because of this, the successful application of mathematical aesthetic methods has a great relationship with the intuitive ability of the subject. This feature also shows that the conclusions drawn from it can only be established after being tested by logical methods.
2. Emotion
The application of mathematical aesthetic method is based on the mathematical aesthetic feeling of aesthetic subject. Like any aesthetic feeling, people have a strong emotional color for mathematical aesthetic feeling. Pleasure, calmness, liveliness, confusion, interest, satisfaction and even excitement and surprise ... Mathematical aesthetic methods are always accompanied by various emotional experiences, which is in sharp contrast with the pure rationality of logical methods.
Step 3 be selective
The method of mathematical aesthetics is a method of consciously making choices based on aesthetic considerations. It is "very self-sufficient, aesthetic and unaffected by experience (almost unaffected)." This selectivity makes the aesthetic method not a concrete method to solve mathematical problems or obtain mathematical discoveries, but a strategic method to determine the direction and principles. This selectivity is the guiding light that leads to mathematical discovery and invention, so it makes the mathematical aesthetic method creative.
Step 4 evaluate
The aesthetic method of mathematics is often expressed as appreciation and evaluation of the obtained mathematical achievements. Generally speaking, the application of logical methods ends with the solution of problems, while aesthetic methods not only pay attention to whether the problems are solved, but also mainly consider the elegant solution of the problems? The former pays attention to the truth of mathematical problems, while the latter pays attention to the unity of truth, goodness and beauty. Poincare pointed out: "This is not a flashy style", and the history of mathematical development shows that the evaluation of aesthetic methods is indispensable for "fruitful mathematical theory".
Basic approaches to the application of mathematical aesthetic methods
1, enhance aesthetic self-awareness and be good at discovering mathematical beauty.
In mathematical activities, participants' aesthetic consciousness is the dynamic reflection of objective aesthetic objects in participants' minds, which is also commonly called aesthetic feeling. It includes aesthetic taste, aesthetic tendency, aesthetic ability, aesthetic ideal, aesthetic emotion and so on. Although aesthetic feeling is subjective, it ultimately comes from the practice of mathematical activities. Rich forms and factors of beauty in mathematics (aesthetic feeling for short) are the objective basis of aesthetic feeling. Only under the condition that beauty causes the subject's aesthetic feeling can the subject make aesthetic consideration. Therefore, being good at discovering the causes of mathematical beauty and "recognizing the true face of Lushan Mountain" is the premise of applying mathematical aesthetics methods.
2. Mathematical aesthetic activities should pay attention to the combination of logical methods and intuitive methods.
Generally speaking, the generation of aesthetic feeling is intuitive, but this does not mean that rational thinking has nothing to do with aesthetics. Aesthetic research shows that rational thinking plays an important role in aesthetics (especially mathematical aesthetics). In mathematics activities, in order to gain real aesthetic importance, we must combine logical thinking method with intuitive method. Logical thinking can play a tendentious role in standardizing perception and imagination in mathematical aesthetics. The former permeates the latter, so that aesthetic feeling is not a primary perceptual perception or a bunch of illusory subjective imagination, but a dynamic reflection of the essence of mathematical objects.
3. In the process of mathematical cognition, evaluation and creation, we should consciously take mathematical aesthetic standards as guidance.
In addition to the above specific contents, the beauty of mathematics lies in mathematics teaching. The teacher's vivid explanation, incisive analysis, clever guidance, vivid language and reasonable blackboard writing all give students beautiful enjoyment. In teaching, teachers should always consciously explain the history of mathematics development and the wide application of mathematics to students, constantly show the beauty of mathematics and further understand the true meaning of beauty.
The charm of mathematical beauty is attractive, the power of mathematical beauty is enormous, and the idea of mathematical beauty is magical. It can change people's prejudice that mathematics is boring and make people realize that mathematics is also a colorful and beautiful world. If mathematics makes many people feel relaxed and happy, and has devoted their whole lives to it, thus promoting the rapid development of mathematics, then it will certainly inspire more ambitious young people to pursue knowledge and explore the future, because "beauty" exists in mathematics. References [1] (English) Russell's The Development of My Philosophy was published by the Commercial Press 1985: 153 [2] Western aestheticians on Beauty and Aesthetic Feeling edited by Peking University Aesthetics Teaching and Research Section. The Commercial Press1980:/kloc-0. Hehe, others can't reprint it