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Mathematics is an indispensable part of modern science, so the beauty of mathematics can be seen everywhere in science. The sense of beauty is related to culture, and people's appreciation of beauty is related to the level of individual education.

Science is also a kind of culture, and the beauty of science is also related to a person's education level and scientific literacy. Even for science and engineering students, not everyone can appreciate the beauty of mathematics in scientific theory. Theoretical physicists often say that the Maxwell equation and the two theories of relativity embody the beauty of mathematics. However, people without certain mathematical accomplishment only see a lot of complicated and boring mathematical formulas, how can there be any "beauty"?

Can mathematical formulas stimulate the "aesthetic feeling" of mathematicians? Scientists use scientific experiments to test and prove this, and also study the relationship between the source of beauty and brain activity. For example, the famous British mathematician M. M.   Atiyah scanned the brain using magnetic resonance imaging in 2014, and the results confirmed that mathematicians' aesthetic perception of mathematics comes from the same area of the brain as people's aesthetic perception of art such as music and painting: the A1 area of the preorbital prefrontal cortex.

Atia chose to provide 60 mathematical formulas covering many fields, asked 16 mathematicians to test them, rated them from ugliness to beauty, and scanned the brain at the same time to measure the area and extent of emotional activity in the brain when producing mathematical beauty. In their paper, they explain the results of experimental analysis, showing that mathematical or abstract formulas not only stimulate beauty and mental excitement, but also share the same emotional areas with artistic beauty in the brain.

Interestingly, among the 60 formulas provided to them, these math professionals chose one of the "ugliest" and one of the "most beautiful" mathematical expressions. They are the following two.

The "ugliest" formula:

The "most beautiful" formula:

The ugliest one has nothing to comment. it is a seemingly complex and puzzling expression that uses an infinite series to calculate 1 / π. Moreover, this is only selected from 60 formulas, if more choices are given, there must be more complex and uglier ones!

The most beautiful formula is called "Euler identity", and of course it only stands out from 60 formulas. However, the Euler identity has always been well received by scientists. For example, American physicist Richard Feynman once called it "the most wonderful formula in mathematics."

What's so wonderful about it? Because it integrates the five most basic and important mathematical constants e, I, π, 1 and 0 into one beautifully. Where e is the base of natural logarithm, I is the unit of imaginary number, π is pi, and the remaining 1 and 0 are self-evident in mathematics. The wonderful thing is, why are these five constants so succinctly linked together? It also includes things like   π = 3.141   592   653... , eBay 2.718   281   828... This strange transcendental number.

This identity first appeared in a book published by the Swiss mathematician Leonhard   Euler in 1748. It is the following complex analysis of the special case of the Euler formula when x = π:

However, people who do not understand mathematics cannot appreciate the beauty of Euler identity and Euler formula. If we don't know the meaning of e, I   and π symbols, complex numbers, power, irrational numbers, trigonometric functions and the geometric meaning they represent, we will not be able to understand the beauty embodied in these two formulas.

And the more you understand the power and connections of these concepts in mathematics, quantum mechanics, and engineering, the more you will admire the beauty of this concise formula!

From the results of the above "most beautiful" and "ugliest" formulas, it can also be found that most mathematicians regard simplicity as an important attribute of the beauty of mathematics. Brevity is also an important attribute of scientific theory. Scientific theory needs to be condensed and condensed, which is the beauty of simplicity, such as the Euler constant equation above.

Image source: pexls simplifies complex things. It is a skill and wisdom. Simplicity is not equal to simplicity, great wisdom as if foolish, the road to simple, with simple to complex, with less than more. Zheng Banqiao, a famous Chinese painter and calligrapher in the Qing Dynasty, showed his calligraphy and literary ideas with "deleted and simplified Sanqiu trees"  , advocating that he should show the richest content with the most concise and clear pen and ink and extraordinary ideas. In essence, it has the same meaning as the so-called "Occam razor" in western logic-the principle of "do not add substance if it is not necessary", and belongs to the "beauty of simplicity".

The purpose of science is to find the simplest and most beautiful description of natural phenomena. Delete all unnecessary superfluous "entities" and leave the least. Simplicity in diversity means harmony between things.

The Occam razor principle in scientific research means that when you are faced with two theories that lead to the same conclusion, choose the simplest and least substantial one!

Figure source: network for example, physicists study a unified theory, the basic laws of physics, various particles and interactions are the entities of the theory. Then, what the unified theory pursues is a kind of succinct beauty. It is to use the least number of physical laws to describe natural phenomena, to use the least number of purpose "indivisible elementary particles" to form all matter, and to use the least kinds of "forces" to describe the interaction between matter. This is in line with the Occam razor principle!

The theory of fractal and chaos describes the seemingly complex phenomena in nature and scientific theory with several simple equations through "self-similar properties". It is also an example of the pursuit of "concise beauty".

James   Clerk   Maxwell,1831-1879 integrates Gauss law, Gauss magnetic law, Faraday induction law and Maxwell-ampere law in electromagnetism, establishes Maxwell equations, describes the properties of electricity and magnetism, and successfully draws the conclusion that light is also a kind of electromagnetic wave.

The Maxwell equation we see now consists of four simple and beautiful equations. However, in the year when Maxwell died of stomach cancer, Maxwell's equations were not in their current concise form. At that time, the system of equations contained 20 equations, which did not look beautiful and had no experimental evidence for the time being, which made people oppose Maxwell's point of view and not accept his theory.

The simplicity and beauty of today's Maxwell equations can be attributed to a self-taught Englishman, Oliver   Heaviside,1850-1925.

When he was a child, Hewitzai had a poor family and scarlet fever, which made him a little deaf. It was such a strange legend who had no formal higher education and taught himself calculus and electromagnetism, the highest theory in the world at that time. Hewitzai is good at discussing and calculating with intuition, and has made many original achievements in mathematics and engineering. But perhaps it has something to do with his self-study background, he does not attach much importance to strict mathematical argumentation, so his operator calculus was opposed by mathematicians at the beginning.

Ignoring the opposition of others, Hewitzai independently founded vector calculus, the method of vector analysis commonly used in physics today. Using the newly invented vector calculus symbol, six years after Maxwell's death, in 1885, Hewitzer rewrote the Maxwell equations into a concise and symmetrical form of the four equations that people know today.

In 1891, Hewitzai became a member of the Royal Society. In 1905, the University of Gottingen in Germany awarded Hewitzai an honorary doctorate, which was recognized and commended by the academic circles to this self-taught scholar.

Source: "what is Science?" author: Zhang Tianrong part of the picture source network copyright belongs to the original author Editor: Zhang Runxin ★ Book introduction ★

The author of "what is Science?" Zhang Tianrong Tsinghua University Press starts from the perspective of the history of science. Science originated from ancient Greece, which likes to study the laws of nature and is willing to explore the relationship between man and nature, which is also the essence of science. The ancient Chinese Mohist school is a school that attaches importance to natural science research and technological discussion. Mozi's research on the imaging of small holes is a rare scientific discovery in ancient China. Modern science was finally born in Europe. There are four essential factors that distinguish science from pseudoscience: questionability, quantification, falsifiability, verifiability and universality. The author uses good and interesting examples to show us how scientists think. Even if all specific knowledge is lost, scientists can still rediscover it using scientific methods. It is also necessary for us ordinary readers to master these scientific thinking and scientific methods. A brief introduction to ★ authors ★

Zhang Tianrong is a physicist and popular science writer. Doctor of theoretical Physics, University of Austin, Texas, USA. He has written popular science books: what is Science, the Mystery of the Butterfly effect: approaching Fractal and chaos, the Ghost of the Century: approaching Quantum entanglement, Eternal temptation: the Mystery of the Universe, from Dice to AlphaGo: interesting talk probability, and so on. This article comes from the official account of Wechat: Origin Reading (ID:tupydread), author: Zhang Tianrong, Editor: Zhang Runxin

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