The Beauty Of Time And Space In Mathematical Context

Zheng Lei / Wen

The four-dimensional space of human life can be imagined as a moving picture. The most complex shape we can see is three-dimensional. As time changes, the object will take on different shapes. The happiness we see in our eyes and in our hearts comes from the beauty in life. Music, mathematics and language are the tools and carriers of beauty. The tools for mathematics to express the beauty of time and space lie in formulas and graphs.

The beauty of mathematics is not easy to be appreciated by people. One of the important reasons is the lack of popular science works to explain and show the beauty. The color printed "charming graphics" and "beauty of formula" cleverly show the beautiful forms of nature with geometric figures and algebraic formulas that are easy to understand by the public, short and interesting anecdotes and high-definition photos. The two books not only improve the public aesthetic, popularize mathematical knowledge, but also help to enhance readers' interest in mathematics.

The beauty of figures lies in symmetry, while the beauty of formulas lies in simplicity. Square and circle are probably the earliest known figures, while points, lines, surfaces and integers, fractions, rational numbers and irrational numbers are highly abstract mathematical concepts. When the ancients measured the shadow of the sun with a straight rod or a huge stone to determine the time change of the day, the Pythagorean theorem of geometry (known as Pythagoras theorem in the West) gradually took shape. This special shape of three sides is expressed as a ^ 2 + B ^ 2 = C ^ 2. The square operation corresponds to the size of the square, which naturally leads to a question: what is the side length of a square with an area of 2? I remember the first time I met this question, the teacher gave the answer is [~ formula ~]. That's true, but it doesn't make any sense, because it's just a sign. It was not until we knew that the number was a decimal with infinite digits and could not be expressed as the ratio of two integers. There are several irrational numbers that are particularly important. They contain the secrets of the universe. They are PI and E in Euler's formula.

A square can be seen as two right triangles put together, and a circle can actually extend from a right triangle. In the 3rd century B.C., Archimedes discovered the relationship between a right triangle and a circle: if a right angle side is equal to the radius of a circle, and the length of another right angle side is equal to the circumference of a circle, then the area of a right triangle is as large as that of a circle. This knowledge comes from the measurement of the circumference of a circle. Imagine how we measure a curved line. A simple method is to approximately replace the length of a short straight line connecting the two ends of the curve. When the line segment is short enough, the measurement error will be within the acceptable range. So we make a lot of lines on the circumference that connect the head and tail, and then connect them with the center of the circle to get a lot of isosceles triangles (each can be divided into two right triangles). Interestingly, the ancient Chinese mathematics was also very advanced at that time. Nine chapters arithmetic, written before Archimedes' time, gave an example to calculate the circle area.

From the measurement of circumference, human beings have found a new idea -- infinite approach, which is actually an application of the concept of "limit" in modern mathematics. Calculus is the basis of advanced mathematics, but many people have found it difficult to understand the concept of limit. Taking circumference measurement as an example, when the triangle with the center of the circle as the vertex becomes infinite, the bottom edge of the triangle will completely coincide with the arc on the circumference. This process is infinite subdivision, which is an ideological experiment. Cutting each other into infinitely many parts, and then adding up, it will rise from roughly equal to exact equality. Differential and integral operations can be derived from the limit, and velocity, acceleration and force can be defined by differential calculation. Starting from Newtonian mechanics, with the help of mathematics, physics has embarked on a scientific road.

E ^ I π + 1 = 0, which is called "the most beautiful equation", can best reflect the beauty of mathematical simplicity. The five most important constants in mathematics are all embodied in a formula without any redundant numbers or letters. It's like a line of perfect and concise verse. Trigonometric function, Taylor series, probability theory, group theory, complex function, electromagnetism, relativity, quantum mechanics and so on all originate from these five numbers. The ultimate secret of the universe seems to be hidden in the number 3 between E and π. This number is related to the golden section ratio and Fibonacci series. From mountains, rivers and coastlines to the thread patterns on crystals and shells, we can see that they are all fractal geometric figures, which contain the objective law of the birth and fall of everything on the earth, which needs to be solved thoroughly Bottom.

Mathematics is actually very interesting, the key is to have a good learning method, which is the weakest point of conventional mathematics textbooks. It is a process of cognitive gradual development for human beings to express nature with mathematics. There are many interesting stories. There are ingenious connections between geometric mathematics and natural phenomena. When all these things are strung together, you will get a dazzling and beautiful "pearl necklace". This process has become an aesthetic enjoyment instead of a boring mixture of letters, picture elements and numbers. No matter how complicated the mathematical concept is, the simple concept of its source can be deduced. The number is no more than 0 and 1, and the algorithm is nothing but addition and subtraction. For example, multiplication and division are intermediate addition and subtraction, while calculus is only advanced addition and subtraction. Learning mathematics needs to master the thinking of "dimension reduction". Solid geometry is transformed into plane geometry by projection. One and two integrals are nothing more than calculating area or volume. Understanding these, mathematics is not only not difficult, but also contains the beauty of all things in the universe.

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