The origin of natural logarithm e?

The origin of natural logarithm e?

The aesthetic significance of spiral, especially logarithmic spiral, can be expressed in the form of index
φkρ=αe
Where α and K are constants, φ is the polar angle, ρ is the polar diameter, and E is the base of natural logarithm. For the convenience of discussion, we define e or the form of transformation and composition by E as "natural law". Therefore, the core of "natural law" is e, whose value is 2.71828 Is an infinite loop number
Count, beautiful?
1. The beauty of numbers
Among them, the Pythagorean School, which prevailed in ancient Greece in the sixth century B.C., had a more profound insight. They first studied the harmony of musical rhythm from the perspective of mathematics and acoustics, and found that the qualitative differences of sound (such as length, height, weight, etc.) were determined by the differences in the number of pronunciations, such as the length of pronunciations (such as strings), the length of pronunciations (such as strings), the number of pronunciations (such as the length of strings), the number of pronunciations (such as the length of strings), the number of pronunciations (, Therefore, the basic principle of music lies in the quantitative relationship
The Pythagorean School extended the principle of harmony in music to architecture, sculpture and other arts, explored what kind of proportion would produce the effect of beauty, and came up with some empirical norms. For example, it is said that they discovered the "golden law" which has a long-standing influence in Europe (some people say that the golden law "has a long history in Europe), It is Cai Mi who 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 equals the short part multiplied by the whole line segment. "If the length and width of something are composed according to this ratio, then it is more beautiful than the rectangle composed of other ratios.")
This group of scholars also applied the principle of mathematics and harmony to the study of astronomy, thus forming the concept of so-called "music of the heavens" or "harmony of the universe". They believed that the stars in the sky also produce a kind of harmonious music when they move in accordance with a certain orbit. They also believed that the function of the human body is harmonious, just like a "small universe", The beauty of the universe is due to the coordination of the speed, distance and turnover time of various material units and stars. All these are the harmony of numbers
The ancient Chinese thinkers also had similar views. Lao Tzu of Taoism and the biography of Xici of Zhouyi both tried to explain the formation of the universe by mathematics, and later developed into the school of image and number of Zhouyi. In Zhouyi, the hexagram of Ben represents the beauty of simplicity, the hexagram of Li represents the beauty of magnificence, and the so-called "extreme number determines the image of the world", Xunqing, a Confucianist, also said: "all things are different from the universe. It is not appropriate but useful to be human, and it is also difficult to count." Chuang Tzu treated the "small self" and "big self" equally, and "small year" and "big year" were the same as Pythagorean School, which confirmed the "small universe" and "big universe" mutually, The idea of seeing beauty from the harmony of numbers deeply influenced the Chinese aesthetics of later generations
2. The beauty of golden rule
The golden rule has always been colored with magnificent and mysterious colors, and it is known as the most beautiful form of "natural and reasonable". We know that the golden rule is not only the principle of composition, but also the best state of natural things. The Italian mathematician fabnesher found that many plant leaves, petals and pineal shell petals are arranged in an approximate value of 0.618:1 from small to large, This is the famous "fibnescher sequence": 1, 2, 3, 5, 8, 13, 21, 34 The color patterns on the animals are also in line with the golden ratio. The ratio of shoulder width to waist, and the ratio of above waist to below waist are also in line with the golden ratio
Modern scientists have also found that when the ratio of high frequency to low frequency of "beta" brain waves presented by the brain is an approximate value of 1:0.618 (12.9 Hz to 8 Hz), people's mind and body are most happy. Even when the ratio of natural temperature (23 ℃) to human body temperature (37 ℃) is 0.618:1, it is most suitable for people's physical and mental health and most comfortable, Mathematicians for industrial and agricultural production system optimization method, put forward the best proportion of ingredients, the best proportion of organizational structure, etc., are also generally in line with the golden law
However, this does not mean that the golden law is more aesthetically significant than the law of nature
φkρ=αe
When k = 0.0765872 and P1 / P2 = 0.618, the ratio of adjacent polar radii on the same radius line in the spiral has a golden section. In fact, when f (x) is equal to the x power of E, take x as 0.4812, then f (x) = 0.618
In other words, the law of nature includes the golden law
The golden law shows the relative static state of things, while the "natural law" shows the general state of things' movement and development. Therefore, in a sense, the golden law is the solidified "natural law" and the "natural law" is the moving golden law
3. The beauty of "natural law"
"Natural law" is the form of E and its transformation and composition
1（1+——）
The x power of X, the limit when x approaches infinity
When people study some practical problems, such as the cooling of objects, the reproduction of cells, and the decay of radioactive elements, they have to study them
1（1+——）
The x power of X is the limit when x approaches infinity. It is this kind of finite value obtained from infinite change that develops from two opposite directions (when x approaches positive infinity, the limit of the above formula is equal to e = 2.71828 When x tends to negative infinity, the result of the above formula is also equal to e = 2.71828...) The common form fully embodies the essence of the formation, development and decline of the universe
Modern cosmology shows that the universe originated from the "Big Bang" and is still expanding. This description is consistent with the second law of thermodynamics, one of the two great discoveries in the second half of the 19th century. The entropy law points out that the evolution of matter is always in the direction of destroying information and disintegrating order, The limit of degradation is the disordered equilibrium, that is, the state of maximum entropy, a state of inaction and stillness. What does this process look like? It's not difficult to understand as long as we look at the pictures of spiral galaxies in astrophotography. If we have to find the kind of dynamic cause that Aristotle said, then, We can think of the universe as being organized by each pre tightened spring, or we can think of the whole universe as a huge spring. History is just a process in which the spring constantly strives for freedom and releases energy
The evolution of life body has the opposite characteristics. It tends to be very different from the entropy described by the second law of thermodynamics. It enables the living matter to avoid the tendency to decline with the environment. Any life is a dissipative structure system, so it can avoid the death state of approaching the maximum entropy, The essence of metabolism is to make the organism successfully eliminate all the entropy that it has to produce when it is alive
On the one hand, the "law of nature" reflects the collapse process of the natural system in the direction of chaos (such as the decay of elements), On the other hand, it shows the essence of life system that only through an orderly process can it maintain its own stability and promote its own development (such as cell reproduction). It is precisely with the characteristics of order and disorder, vitality and death in the same form that "natural law" has important aesthetic value
If the barren and boundless desert is the entropy increasing state of disorder and stillness of "natural law", then the vast and vigorous grassland is the dynamic and stable structure of "natural law". Therefore, the desert makes people feel solemn, boundless, meditative, and reminiscent of all kinds of difficulties and frustrations in the course of life, while the grassland makes people excited and excited, Let people feel the joy and happiness of life
e=2.71828…… It is a quantitative expression of "natural law". The image expression of "natural law" is spiral. The mathematical expression of spiral usually has the following five kinds: (1) logarithmic spiral; (2) Archimedes spiral; (3) chain spiral; (4) hyperbolic spiral; (5) cyclotron spiral. Logarithmic spiral is the most common in nature, and other spiral also has a certain relationship with logarithmic spiral, The logarithmic spiral was introduced by Descartes in 1638. Later, the Swiss mathematician Jacob Bernoulli studied it in detail and found that the involute and involute of the logarithmic spiral are still logarithmic spiral, and the tangents of the poles at each point of the logarithmic spiral are still logarithmic spiral. Bernoulli marveled at these interesting properties, He left a will to draw a logarithmic spiral on his tombstone
The famous British painter and art theorist Hogarth deeply felt that the spiral shape or spiral shape gradually shrinks to their center, which is the shape of beauty, We can easily find spirals in the works of ancient and modern artists. Why do our senses and our "spiritual" eyes often instinctively and intuitively get satisfaction from such a form of spirals? Doesn't this mean that there is a more primitive isomorphic relationship between our spirit, our "inner" world and the external world than history?
As we know, protein, as the basic material of life phenomenon, participates in the whole work of life process in living objects. Its function is so complex, efficient and mysterious, which is closely related to its structure. Chemists have found that the multi titanium chain of protein is mainly spiral, and the structure of nucleic acid, the material determining heredity, is also spiral
The ancient Greeks had a musical instrument called the wind sonata. When its strings vibrated in the wind, it could produce beautiful and pleasant tones. This kind of tone is called "vortex wake effect", The inner ear structure of the human auditory organ evolved over a long period of time also has a vortex shape. Is this for the convenience of appreciating the ancient Greek wind chime? As well as our fingerprints, hair spirals and so on, this physiological structure of the aesthetic subject corresponds to the isomorphism of the external world, that is, the natural basis for the harmony between "internal" and "external"
Some people say that mathematical beauty is the brilliance of "one", which has invariance under the action of as many transformation groups as possible. That is to say, it has the expression of common laws of nature, and is the unity of "many" and "one". So the "natural law" is also shining with the brilliance of "one". Who can say that E = 2.71828 How much convenience and success has it brought to mathematicians? People praise the rigidity, clarity and frankness of straight lines, appreciate the beauty, change and implication of curves, but they don't know that any straight line and curve can be made up of enough parts from 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 the "law of nature" 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 law of development. What gives this form vivid expression? Spiral!
Some people say that beauty lies in the rhythm of things, and "natural law" also has this rhythm; some people say that beauty is a dynamic balance and an eternity in change, so "natural law" is also a dynamic balance and an eternity in change; some people say that beauty lies in the dynamic structure of things, so it is“