In mathematics, the magic square contains the richest philosophical thoughts. The Book of Changes is a philosophical book, which has influenced almost all kinds of philosophical thoughts at home and abroad. However, Yi-ology scholars have found through various studies that Yi-ology originated from the Hutu Luoshu, and Luoshu is the third-order magic square. The layout law and construction principle of the magic square contain a living structure that summarizes all things in the world, and it is a mathematical model that explains the emergence and development of the universe. My essays "The Thought of Change of the Fourth-order Perfect Magic Square" and "The Fifth-order Magic Square and the Easy Number System" are further discussions on the philosophical thoughts contained in the higher-order magic square. Interested readers can refer to the No.1 issue of Zhouyi Research in 1999 and No.1 issue in 2.
Second, the magic square is applied to art design
Magic square can be widely used in art design. The western architect Bratton found that the symmetry of magic square is quite rich. It uses magic square to form many beautiful patterns. He called the lines in the square in the pattern "Moline" and applied it to the design of light industrial products and cover packaging. In the work "Melancholia" by the famous German printmaker A. Duller, because one of them can indicate, Regarding the "Moline" map, Japanese magic square expert Abe Yuefang has also done a lot of work. Ji Guangzhong, a teacher in Anyang, Henan Province, has worked out various magic maps and dedicated them to the Central Academy of Arts and Crafts. Beijing Ding Baoxun published 17 "magic line charts" in "Magic Square Album", all of which are very beautiful. The mathematical layout in the magic square is very symmetrical and balanced, and there are rich changes. Therefore, if the numbers are connected in sequence, a strange "Rubik's cube array structure map" can be formed, and a very beautiful art pattern can be obtained after color processing. This pattern shows a variety of symmetrical beauty, but also has the rational law of the magic square principle, so it is intriguing and can be called the work of an axe.
third, the aesthetic value of the magic square.
Mathematics is beautiful, and the magic square is even more beautiful. Magic Square is a system in which mathematics is arranged according to a law. Each magic square is not only an intellectual achievement, but also an artistic masterpiece. With the characteristics of uniformity, balance, symmetry and harmony, generate shines the beauty of mathematics and has high aesthetic value. In mathematical aesthetics, the aesthetic value of magic square is pushed to the top. Because all the contents in mathematics are closely related to numbers, this beautiful structure of magic square can permeate all kinds of mathematical knowledge, showing a variety of fun, so that we can understand many mysteries of mathematical knowledge in the appreciation of magic square.
fourth, the intelligence development function of magic square. The magic square is simple and easy to get started, which will soon arouse teenagers' interest in discussion. It can be said that magic square has played a very important role in intellectual development. Digging into the history of Chinese mathematics, we will see that interesting mathematics, calculation tools and board games are all internally related to the magic square. In the history of algorithms, there were Jiugong calculations first, followed by Taiyi calculations, abacus and electronic computers. In the history of game development, Jiugong was rearranged first, followed by chess, Go and Huarong Road games. Weiqi board is a 19th-order square, and the elephant board is an 8th-order square (its General Palace is a 3rd-order square). Their playing principles are all related to the layout principle of magic squares. The computer game of "digging mines" is closely related to Nine palace map.
In recent years, Chinese magic square researchers have invented many intelligence development games by applying the magic square principle. Liu Zhixiong, Liaoning Province, designed a double-sided magic square with a set of pictures, which won the bronze medal; Wang Zhonghan, Anhui Province, designed an interesting magic square chess; Jiang Yajing, Hunan Province, designed a series of magic square digital game machines; and the author also successfully designed the "Jiugong wonderful chess", which has nine functions and more than 2 game modes, and is an excellent garden for primary school students' mathematical operation training.
5. The influence of magic square in mathematics teaching.
Magic Square has the functions of improving students' learning interest, beautifying teaching materials and enlightening students' thinking in mathematics teaching. The rich variety of numbers in the magic square links all the contents in mathematics textbooks, such as equation magic square, radical magic square, fractional magic square, black hole number magic square, product magic square, difference magic square, square magic square, etc. They can all be used in mathematics teaching to make mathematics content attractive. Figure 1 is a perfect fifth-order magic square. When a student learns the addition and subtraction of rational numbers and gives this figure to the students for discussion, the students will have a strong interest in all aspects of learning activities. They will find that the sum of five numbers in the form of "ten, one, x,/"is all , and the sum of six numbers with "△" in Figure 1 must be equal to the number with "○". The magic square is an important content in the book of Olympic Mathematics.
sixth, the inspiration of magic square to science.
River map can be regarded as a second-order magic square model, and Luoshu is a third-order magic square. Because of their wide spread, they have given people a lot of scientific enlightenment from ancient times to the present. For example, in Einstein's Theory of Relativity, 11 formulas were used to calculate the relative increase and decrease of time and space, and the Heluo number was very enlightening to him. Jiao Weifang, a Chinese-American scholar, once wrote books on Luo Shu matrix, Luo Shu Ji He and Luo Shu space, which promoted the development of mathematics. Fu Xiru of Henan studied Goldbach conjecture by using Luo Shu. We know that the computer is based on automatic control theory, and the inventor of American automatic control theory made a series of control theories by studying China's "three-three maze diagram" (the line diagram of the third-order magic square). As can be seen from the data here, the computers that are now popular all over the world have actually gone to the magic square field. The magic square has a natural attribute, although it is a numerical relationship, but it is often very abstract and general. When people think about it repeatedly, it may inspire a scientific theory and promote its development. In the traditional culture of China, we can see a great deal of information about Luo Shu's application in military, traditional Chinese medicine, astronomy, meteorology, qigong and other fields, which shows that the close relationship between Magic Square and various disciplines can not be ignored.
VII. The magic square is applied to science and technology.
The magic square has been applied to position analysis and combinatorial analysis of "Building Road", "Jordan Curve" and "Seven Bridges". The magic square leads to Laplace's guiding coefficient, Goss Theorem, Gerry Theorem and Henri Storck Theorem, and also leads to the electronic equations of Posen and Brutin. The magic square also led to Sang Nan's theory of automatic control, which led to the birth of electronic computers. Computers have three sources, namely, binary (gossip), abacus and magic square. Electronic science has regarded the arrangement route of magic square as an ideal figure of electronic circuit network. We can see from Taiwan Province Li Kaixuan's A Brief Talk on Yi Shu that the first lesson of Taiwan Province pilots who learn aircraft knowledge from Japan is magic square knowledge, because the construction principle of magic square is closely related to the electronic circuit setting on the plane. Wu Longsheng, an expert in electrical engineering in Taiwan Province, invented the 64th-order square array instrument, which can be used in computers, measuring instruments, communication exchange instruments and control systems of water, electricity, fire and aviation, and has obtained patents. For floating buildings on the sea, the first problem to be solved is to divide the building surface into square grids, and the determination of the building weight of each grid needs to be arranged as skillfully as constructing a magic square, because as long as the weights of all lines and directions are balanced everywhere, it will not tilt. Mr. Tian Jian, a Shaanxi Provincial Political Consultative Conference, wrote a book and is applying magic square to study the theory of traditional Chinese medicine. He studies the digital characteristics of human etiology and the configuration of traditional Chinese medicine from the digital structure of magic square. His research work has attracted the attention of many medical scholars. The author applied the construction principle of the tenth magic square to study the TCM theory of "55 Shengong Yuanqi Bag" and achieved certain results. Liu Jixi in Sichuan once designed magic square cultural products for toy factories, handkerchief factories, ball factories, umbrella factories and porcelain factories, and Xu Zhongyi in Jiangsu had the design of "magic square carpet". Beijing Gaoxuefeng has many patents on "magic square cloth" and "magic array treatment".
VIII. The role of magic square in cutting-edge science.
I would like to focus on the research results of Professor Qi Dongxu, vice president of north china university of technology and doctoral supervisor. In his book Fractal and Its Computer Generation, there is a section "kronecker Multiplication and Magic Square of Matrix", which discusses that the magic square has gradually developed its application from being regarded as just a "strange phenomenon". If M-order magic square A and N-order magic square B are taken as matrices, then Kronecker product A? B is also a magic square. If m×n squares are set on the computer screen, and the gray level of each square corresponds to the element value of m×n matrix A in sequence, and the square corresponding to aij, every time it is divided into P×q small squares, it is colored according to the value of aij*B, and this process continues. As you can imagine, the infinitely nested structure obtained from the magic square has self-similarity (external or internal) and can be regarded as a holographic corresponding structure. Because the magic square is a special numerical matrix, Professor Qi Dongxu found that the Bezier -Bernstein surface generated by using the magic square as the data matrix of the control network has the property of one-way integral invariance, while other well-known approximation methods, such as B-spline interpolation or polishing, lagrange interpolation, etc., do not have this property.
Professor Qi Dongxu and his doctoral student Ding Wei wrote an article entitled Digital Image Transformation and Information Hiding and Camouflaging Technology, which was published in chinese journal of computers. This paper puts forward the technology of "image scrambling transformation according to magic square", which can scramble the image that needs to be kept secret and then restore it according to the principle of magic square. This scrambling transformation can be carried out many times. The author thinks that the classification, counting, construction program and transformation of magic square can be used in information hiding technology, and its application prospect will be very broad.
I recently read about the computer network system. There are five kinds of network topological structures * * *, each of which has its own advantages and disadvantages. But when we think about the structure of the fifth-order perfect magic square, the five kinds of network structures can be integrated into one, which may become the most perfect network architecture, and it is somewhat like the "five-element system" in our human body (Chinese medicine term). Shandong Wu Shuoxin's theory of α (q, A) is very close to the basic principle of computer, and this theory derived from magic square will certainly find its application prospect in computer. Huang Jun Di of Gansu applied the binary theory to study the magic square. It decomposed the magic square into several blocks, all of which are composed of black and white, and they are harmonious and balanced. These black and white blocks can definitely be used in computer technology. I hope everyone can study and develop them.
With the further development of electronic computers, magic squares are more widely used in artificial intelligence, graph theory, game theory, experimental design, arts and crafts, electric sub-circuit principle, position analysis and so on. We can say that the magic square has made great contributions to human civilization in the ancient past, and today, in the information age, it will also have a broad application prospect.