What about the base e of natural logarithm?

1. the laws of nature-the wealth of aestheticians and artists

Vortex or spiral is a very common form of natural things, such as: a wisp of smoke curling up, ripples swaying gently on the blue lake, several snails slowly climbing the fence and countless stars dancing in the quiet night sky. ...

The aesthetic significance of spiral, especially logarithmic spiral, can be expressed in exponential form:

φkρ=αe

Where α and k are constants, φ is polar angle, ρ is polar diameter, and E is the base of natural logarithm. For the convenience of discussion, we define e or the form composed of e as "natural law" Therefore, the core of "natural law" is E, and its value is 2.7 1828 ... This is an infinite cyclic number.

Count, is it beautiful?

1, the beauty of numbers

For a long time, people have a deep understanding of the beauty of numbers. Among them, the Pythagorean school, which prevailed in ancient Greece in the sixth century BC, had profound insights. They first studied the harmony of music rhythm from the viewpoint of mathematics and acoustics, and found the qualitative differences of sound (such as length, height, weight, etc.). ) is determined by the number difference of pronunciation methods. For example, if the pronunciation body (such as strings) is long, the sound will be long; When the vibration speed is fast, the sound is high; When the vibration speed is slow, the sound is low. So the basic principle of music lies in the quantitative relationship.

The Pythagorean school extended the principle of harmony in music to other arts such as architecture and sculpture, explored what proportion would produce beautiful effects, and obtained some empirical norms. For example, Huang Jinlv, which has a long-term influence in Europe, is said to have been discovered by them (some people say that Bicai put forward the so-called "golden section law" in 1854. The so-called golden section law is to take a line and divide it into two parts, so that the square of the long part is equal to the short part multiplied by the whole line segment. ""If the length and width of a thing are all made according to this ratio, then it is' more beautiful' than a rectangle composed of other ratios. " )。

Scholars of this school also apply the principles of mathematics and harmony to the study of astronomy, thus forming the so-called concept of "heavenly music" or "cosmic harmony", and believe that the stars in the sky also produce a harmonious music when they follow a certain orbit. They also believe that the function of human body is harmonious, just like a "small universe". The human body is beautiful because of its parts-head, hands, feet, five senses and so on. -Appropriate proportion and coordinated action; The beauty of the universe is due to the coordination of speed, distance and turnaround time between various material units and various stars. These are the harmony of numbers.

Ancient thinkers in China had similar views. Laozi, a Taoist, and The Book of Changes, The Legend of karma, both tried to explain the formation of the universe with mathematics, and later evolved into the School of Images and Numbers in the Book of Changes. In the Book of Changes, Ben's divination means simple beauty, deviation from divination means gorgeous beauty, and the so-called "too many, which determines the image of the world" are all similar to the conclusions of mathematical reasoning. Xun Qing of Confucianism also said: "Everything is different from the universe. It is not appropriate but useful to be a person. " Zhuangzi treated the "ego" and "me" equally, so did "off-year" and "New Year", similar to the Pythagorean school's verification of "small universe" and "big universe". The so-called "from the hand to the heart, the mouth can't speak, there are several things in the middle." This idea of seeing beauty from the harmony of numbers deeply influenced China's aesthetics in later generations.

2. The beauty of Huang Jinlv

Huang Jinlv has always been dyed with magnificent and mysterious colors, and is called the most wonderful form proportion of "natural rationality". We know that Huang Jinlv is not only the principle of composition, but also the best state of natural things. Fibonacci, an Italian mathematician in the Middle Ages, found that the arrangement order of leaves, petals and pine nut shells of many plants from small to large is an approximate value of 0.6 18: 1, which is the famous Fibonacci sequence: 1, 2,3,5,8,1. The comparison dimensions set by dance coaches and gymnastics experts, such as the ratio of shoulder width to waist, and the comparison between waist and lower waist, are also generally in line with the golden ratio.

Modern scientists have also found that when the ratio of high frequency to low frequency of "β" brain waves presented by the brain is the approximate value of 1: 0.6 18 (the ratio of 12.9 Hz to 8 Hz), people's body and mind are happiest. Even when the ratio of natural temperature (23 degrees Celsius) to human body temperature (37 degrees Celsius) is 0.6 18: 1, it is most suitable for people's physical and mental health and comfortable. In addition, mathematicians put forward the best proportion of ingredients and the best proportion of organizational structure. For the optimization method of industrial and agricultural production system, this is also generally in line with Huang Jinlv.

However, this does not mean that Huang Jinlv has more aesthetic significance than "natural law". We can prove that when the logarithmic spiral:

φkρ=αe

When Huang Jinlv takes the equal ratio, that is, k=0.0765872, and the equal ratio P 1/P2=0.6 18, the radius ratio of adjacent poles on the same radius line in the helix has the golden section relationship. Actually, when the function f(X) is equal to the x power of e, let x be 0.48 12, then f (x) = 0.6 18. ...

Therefore, the hidden logic of gold law firms is "natural law". In other words, "natural law" includes Huang Jinlv.

Huang Jinlv showed the relative static state of things, while "natural law" showed the general state of things' movement and development. Therefore, in a sense, Huang Jinlv is a solidified "natural law" and "natural law" is a moving Huang Jinlv.

3. The beauty of "natural law"

"natural law" is the form of e and e after certain transformation and compound. E is the essence of "natural law", which is a function in mathematics;

1( 1+——)

The x power of x, the limit when x approaches infinity.

People should study some practical problems, such as the cooling of objects, the reproduction of cells and the decay of radioactive elements.

1( 1+——)

The x power of x, the limit when x approaches infinity. It is this finiteness obtained from infinite change that develops from two opposite directions (when X tends to positive infinity, the limit of the above formula is equal to E = 2.7 1828 ..., and when X tends to negative infinity, the result of the above formula is also equal to E = 2.7 1828 ...), which fully embodies the formation of the universe.

Modern cosmology shows that the universe originated from the Big Bang and is still expanding. This description is consistent with the entropy law, that is, the second law of thermodynamics, which is one of the two great discoveries in the second half of19th century. The law of entropy points out that the evolution of matter is always in the direction of destroying information and disintegrating order, and gradually degenerates from complex to simple and from high to low. The limit of degradation is disorderly balance, that is, the state of maximum entropy, a state of inaction and death. What is this process like? It is not difficult to understand just by looking at the photos of spiral galaxies in astrophotography. If we must find the motive force that Aristotle said, then we can regard the universe as a clockwork mechanism that is wound up in advance, or simply regard the whole universe as a huge clockwork. History is just the process of this clockwork constantly striving for freedom and releasing energy.

The evolution of organisms has opposite characteristics, which is very different from the entropy described by the second law of thermodynamics. It enables organisms to avoid the decline of trends and environment. Any life is a dissipative structure system, and the reason why it can avoid the death state approaching the maximum entropy is because life can continuously absorb negative entropy from the environment through metabolic processes such as eating, drinking and breathing. The essence of metabolism is to let the organism successfully eliminate all the entropy it has to produce when it is alive.

On the one hand, the "law of nature" embodies the collapse process of natural systems in the direction of chaos (such as the decay of elements), on the other hand, it shows the essence that life systems can maintain their stability and promote their development (such as cell reproduction) only through an orderly process. It is precisely because of this feature that order and disorder, life and death are in the same form, and "natural law" has important aesthetic value.

If the barren desert is the entropy increasing state of "natural law" in disorder and death, then the vast and vibrant grassland is the dynamic and stable structure of "natural law" in order and prosperity. Therefore, the desert makes people feel solemn, boundless and thought-provoking, reminding people of all kinds of difficulties and bumps in the course of life; Grassland, on the other hand, makes people excited, happy and feel the joy and happiness of life.

E = 2.7 1828 ... is a quantitative expression of "natural law". The image expression of "natural law" is a spiral. There are five mathematical expressions of spiral: (1) logarithmic spiral; (2) Archimedes spiral; (3) interlocking helix; (4) Hyperbolic spiral; (5) spiral. Logarithmic spiral is the most common spiral in nature, and other spirals are also related to logarithmic spiral, but at present we have not found the general formula of spiral. The logarithmic spiral was introduced by Descartes in 1638. Later, Swiss mathematician Jacob Bernoulli made a detailed study on it, and found that the involute and extension line of logarithmic spiral are still logarithmic spiral, and the tangent of each point and pole of logarithmic spiral is still logarithmic spiral, and so on. Bernoulli was surprised by these interesting properties and left a will on his tombstone to draw a logarithmic spiral.

Willianm Hogarth, a famous British painter and art theorist, deeply felt that spirals or spirals gradually narrowed towards their centers, which were all beautiful shapes. In fact, we can easily find the spiral in the works of ancient and modern artists. Why can our feelings, our "spiritual" eyes, often be satisfied instinctively and intuitively from such a spiral form? Doesn't this mean that there is a more primitive isomorphic correspondence between our spirit, our "inner" world and the external world than history?

As we know, protein, as the basic substance of life phenomenon, participates in all the work of life process in life, and its function is so complex, efficient and mysterious that it is closely related to its structure. Chemists found that protein's titanium chain is mainly spiral, and the nucleic acid structure that determines heredity is also spiral.

The ancient Greeks had a musical instrument called Feng Qin Ming. When its strings vibrate in the wind, it can make a beautiful tone. This tone is called "vortex tail effect". It is thought-provoking that the inner ear structure, an auditory organ formed by human beings through long-term evolution, also has a vortex shape. Is this for the convenience of enjoying the ancient Greek wind chimes? And our fingerprints, hair clips, etc. The physiological structure of this aesthetic subject corresponds to the isomorphism of the external world and is the natural basis for the harmony between "inside" and "outside".

Some people say that the beauty of mathematics is the brilliance of "one", which is invariant under the action of as many transformation groups as possible, that is, it is the expression of the universal law of nature and the unity of "many" and "one", so "natural law" also flashes the brilliance of "one". Who can say how much convenience and success E = 2.7 1828 ... have brought to mathematicians? People praise the rigidity, clarity and truthfulness of straight lines and appreciate the beauty, change and implication of curves, but they don't know that any straight line and curve can be composed of enough parts on the spiral. Some people say that beauty is the unity of subject and object, the unity of inner spiritual world and outer material world, so "natural law" also has this unity. Human understanding develops according to the law of negation of negation, and the history of society and nature also follows this dialectical development law. What gives this form a vivid expression? Spiral!

Some people say that beauty lies in the rhythm of things, and the "laws of nature" also have this rhythm; Some people say that beauty is a dynamic balance and eternal change, so "natural law" is also a dynamic balance and eternal change; Some people say that beauty lies in the dynamic structure of things, so the "laws of nature" also have this structure-such as hairspring in watches, clockwork in machines and so on.

"Natural law" is the unity of formal reasons and dynamic reasons, the image expression of things, and the same expression of concrete and abstract. Limited life is rooted in infinite nature, and the pulse of life consciously adjusts its movement and rhythm according to the melody of the universe ... organic and inorganic, internal and external, social and natural, and all things are one. Is this all the aesthetic mystery revealed by the "laws of nature"? Don't! "Natural law" always has endless aesthetic connotation, because it symbolizes the vast and profound nature. Because of this, it attracts people worthy of unremitting exploration, thus showing the essential power of human evolution. (Originally published in the 4th issue of Science Spring (1984), with the original title: laws of nature-the treasure of aestheticians and artists)

2. Euler's natural logarithm base formula

(approximately equal to the base of natural logarithm 2.7 1828-E)

Euler is known as Shakespeare in the digital world. He is the most prolific mathematician in history and a scholar who has written the most books in various fields (including all branches of mathematics and mechanics, optics, acoustics, water conservancy, astronomy, chemistry, medicine and other theories and applications). ). In the history of mathematics,18th century is called "Euler era".

Euler was born in Switzerland. At the age of 365,438+0, he lost his right eye vision and was blind at the age of 59. But he is optimistic, and his memory and concentration are amazing, which enables him to think complex problems accurately in the noisy environment of 13 children.

Euler was humble all his life and never named his discovery after himself. Only the base of natural logarithm, which is about 2.7 1828, was named E by him. However, due to his extensive contributions to mathematics, important constants, formulas and theorems named after him can often be seen in many branches of mathematics.

Many mathematical symbols that we are used to now were invented and introduced by Euler, such as function symbols f(x), π, e, ∑, logx, sinx, cosx, imaginary number I and so on. High school teachers often use a joke about the base e of natural logarithm to help students remember a very special differential formula: in a mental hospital, a patient says to others all day, "I differentiate you, I differentiate you." I don't know why, these patients all have simple calculus concepts. They always think that one day they will be differentiated into zero and disappear like ordinary polynomial functions, so they avoid him. One day, however, he met an unmoved person. He was surprised. This person simply said to him, "I am the x power of E."

This differential formula is: No matter how many times E differentiates X, the result is still E! No wonder math students use e to describe unswerving love!

Compared with π being the first letter of a circle in Greek, the origin of E is less known. Some people even think that Euler takes the first letter of his name as the natural logarithm.

Euler chose e for two reasons: first, after the four letters A, B, C and D that are commonly used, the first letter that is not often used is e, so he naturally chose this symbol, which represents the base of natural logarithm; One is that e is the first letter of the index. Although you may suspect that the native language of Swiss Euler is not English, it is actually an index in French and German.