2. Photon is the basic particle of matter.
When it comes to everything in the universe, elementary particles need to start with the elementary particles of matter. There are several reasons to think that photons are the basic particles of matter: 1, photons are very common particles around human beings, and matter can emit photons under any conditions; 2. The birth of life, the existence of human beings and the influence of the universe on human beings are all completed by photon information; 3. The photon model proposed by Einstein; One of the foundations is that energy is related to the speed of light of photons; 4. A positron with a certain energy collides with a negative electron with a certain energy, and the electron pair disappears and a pair of photons is released at the same time; 5. The interaction of photon clusters can synthesize ordinary particles.
Einstein put forward the photon theory for decades, and the research on photons continues. In particular, the theory that photons are basic particles of matter has not been established. Why? The reason is that people are unwilling to give up the current material model: the quality of matter is inherent in the object itself and has nothing to do with the environment; Material world: According to Einstein's understanding of photons, if photons are the basic particles of matter, then any object has no mass in a relatively static state, and the meaning of matter without static mass is terrible. We can't touch objects, we can't see each other, and we can't use anything. This is an impossible physical model in our real life. Because it is an impossible physical model, we must change the way of thinking about nature, otherwise there is no way to understand nature. This is how nature plays tricks on human beings, and it is not allowed for human beings to know elementary particles so easily. If photons are really basic particles of matter, then the matter around us does not have static mass, because matter is composed of basic particles such as photons, and according to Einstein's understanding of photons, photons should not have static mass, photons have no static mass, and substances composed of photons will not have static mass. In fact, matter has no static mass. What we usually see is the static mass of matter. The moving mass of matter is produced in this way: because matter constantly interacts with the surrounding matter, this function is to absorb and emit photons. The greater the photon energy per unit time, the more its mass will be displayed to the outside world. The smaller the photon energy absorbed and emitted per unit time, the smaller the mass of matter. If a substance stops interacting with photons in the environment, it will have no mass. This phenomenon is almost impossible. If possible, it will appear in later events. First, matter is at absolute zero, that is, in an area without photon energy. Second, the photon information of the substance itself is too far from the photon information of the environment, and the substance will not absorb it at all. If this event happens suddenly, then the object seems to suddenly disappear. In layman's terms, objects have entered another time and space.
Matter constantly interacts with the environment to make photons (absorption and emission), thus showing the quality of matter. In a certain period of time, when the number of photons interacting with the environment is large and the energy is large, it shows more mass. This can be expressed by a mathematical formula: where is the highest frequency at which a substance absorbs and emits photons. When the photon energy acts within a certain period of time, the greater it is, the greater the mass of the substance we feel; or vice versa, Dallas to the auditorium
In this sense, all the substances around us, as long as there is rest mass, are not the basic particles of matter, only photons, without rest mass, and photons themselves are the basic particles of matter. All electrons, protons and even smaller particles of matter, as long as they have static mass, can not be regarded as basic particles of matter, but only as photon information groups composed of photons. Due to the different composition of photon information, the corresponding lifetime of photon information is also different. If the corresponding photon information in the environment is particularly large, then after it absorbs and emits photon information, its own photon information changes very little per unit time, that is, the corresponding particles exist in nature for a long time. Generally speaking, this kind of particle has a long life. If the photon information of the corresponding particle exists very little in nature, then there are two ways for such particles to exist. First, there is less photon information of the absorbing substance. My static mass shows less and my life is longer, and the other is that I absorb more photon information. My static mass is greater, I change more in unit time, and my life span is shorter. Matter actually adopts the former method, which can be explained by the frequency of the dark line of the atomic absorption spectrum of matter corresponding to the bright line of the atomic emission spectrum of matter, indicating that matter only absorbs the photon information with the frequency corresponding to its own photon information.
Because there are so many photons in nature, the lifetime of photons is infinite, and that is exactly the case. Photons exist for the longest time and cannot disappear, because there are laws of energy transformation and conservation in nature. If a photon disappears and its lifetime has a certain value, this law will not hold. Photons can only be absorbed by an object and emit one photon at the same time, but the combination of emitted photon information is different. This different combination of photon information expresses certain photon information, especially the photon information emitted by an object after it absorbs photon information. This is how objects convey information. There is a maximum speed for information transmission in nature, which is the speed of photons in nature. Human vision uses the highest speed in nature.
In addition, in the process of evolution, the human body makes full use of the photon information in nature, and the five senses of human beings are all evolved on the basis of the original feeling of photon information. Photon information is a form of information expression that is constantly communicated between substances. As long as a substance exists, it must constantly absorb and emit photon information, and at the same time emit photon information with its own characteristics to express its material identity to nature. This is the characteristic of the existence of matter. In nature, human beings constantly absorb photon information on the earth's surface. Because the photon information on the earth's surface is constantly changing, in order to adapt to the changes in nature, human beings have evolved the visual organ-eyes. In particular, the most sensitive light wave of human eyes is the photon with the central frequency of the sun, and the most sensitive light wave of other animals' vision should be the central frequency of the light wave absorbed by this animal during its life. All other organs of human beings have evolved under the same information language of matter, because the existence of matter is to constantly absorb photon information while sending it, so the photon information between matters is the same language of matter and matter, people, animals, plants and plants, and the communication between these matters, so photon information is not only a type of matter that must be absorbed and sent when matter exists, but also a language for exchanging information between matters. This language is the same language of all substances, and it is a language that all substances can understand. Whether it is inanimate matter or living matter, whether it is between plants and animals, or between people and animals, it is a pity that human beings can communicate with each other. Due to the expansion of material desires, various organs develop perfectly without being used. I don't understand the content of this photon information. When human beings encounter the change and existence of this photon information, they only know that they have a feeling, but they don't know what this feeling represents and what the information content of this feeling is. This is a pity for human beings in the evolution of nature, giving up the natural perception of nature, giving up the fastest and most accurate feeling of photon information in nature, and using the five senses of the human body, especially the human vision only uses a small part of light waves. However, due to the development of science and nuclear technology, human beings have created mobile phones, which have replaced part of human natural perception and supplemented part of the functions that human beings have lost.
Because photon information is a natural perception in nature, it is called natural language, which is used by all substances and can be understood by all substances. So not only people have spirits, but all things have spirits, so it is not surprising that animals can understand some human behaviors.
Because the task of photons is too big, one is to act as a spokesman for displaying matter. Without the absorption and emission of photons, matter cannot exist, its quality cannot be displayed, there is no inertia, and it cannot serve and utilize other matter. The second is the composition of photons-photon information, which is the language for all substances in nature to exchange information. It is not enough to say that photons are the basic particles of matter.
.
3. It is a mixture of iron hydroxide and ferrous hydroxide because:
2 sodium hydroxide+ferric chloride = iron hydroxide +2 sodium chloride
While Fe(OH)2 will be rapidly oxidized to Fe(OH)3.
Fe(OH)2 is a white precipitate, while Fe(OH)3 is a reddish-brown precipitate.
Because only part of Fe(OH)2 is oxidized to Fe(OH)3, the gray-green precipitate is a mixture of two kinds of precipitates, which looks gray-green.
4。 I don't know.
5。 . Mathematics is the mother of all sciences ","Mathematics is the gymnastics of thinking ". It is a science that studies numbers and shapes, and it is everywhere. To master technology, we must first learn mathematics well, and to climb the peak of science, we must learn mathematics well. What are the characteristics of mathematics compared with other disciplines? What is its corresponding way of thinking? What kind of subjective conditions and learning methods does it require us to have? This lecture will briefly explain the characteristics, ideas and learning methods of mathematics.
First, the characteristics of mathematics
Three characteristics of mathematics: preciseness, abstraction and wide application.
The so-called rigor of mathematics refers to the strong logic and high proficiency of mathematics, which is generally reflected by axiomatic system.
What is the axiomatic system? It refers to selecting a few undefined concepts and propositions without logical proof, and deducing some theorems to make them a mathematical system. In this respect, the ancient Greek mathematician Euclid is a model, and his Elements of Geometry studies most problems in plane geometry on the basis of several axioms. Here, even the most basic and commonly used original concepts cannot be described intuitively, but must be confirmed or proved by axioms.
There are still some differences in rigor between middle school mathematics and mathematics science. For example, the continuous expansion of several sets in middle school mathematics, the expansion operation law of several sets is not strictly deduced, but obtained by default. From this point of view, middle school mathematics is still far from rigorous, but to learn mathematics well, we must not relax the requirements for rigor and ensure the scientific content.
For example, arithmetic progression's general term is summed up through the recursion of the previous items, but it needs to be strictly proved by mathematical induction to be confirmed.
The abstraction of mathematics is manifested in the abstraction of spatial form and quantitative relationship. In the process of abstraction, it abandons the specific characteristics of more things, so it has a very abstract form. It shows a high degree of generality and symbolizes the concrete process. Of course, abstraction must be based on concreteness.
As for the wide application of mathematics, it is well known. Only in the past teaching and learning, we often paid too much attention to the abstract meaning of theorems and concepts, but sometimes gave up their wide application. If abstract concepts and theorems are compared to bones, then the extensive application of mathematics is like flesh and blood, and the lack of any one will affect the integrity of mathematics. The purpose of increasing the application space of mathematics knowledge and research-based learning in the new high school mathematics textbook is to cultivate students' ability to solve practical problems by applying mathematics.
Let's look at an interesting question in life.
In any meeting, the number of people shaking hands with an odd number must be even. Try to prove it.
If we grasp two keys: first, the total number of handshakes must be even,
Second, the characteristics of high school mathematics
There are often students who do not adapt to mathematics learning after entering high school, which in turn affects their enthusiasm for learning and even their grades plummet. Why is this happening? Let's take a look at the changes in high school mathematics and junior high school mathematics.
1. reinforcement theory 2. Add courses. Increase the difficulty by 4. Ask for improvement.
Third, master mathematical thinking.
High school mathematics is closer to advanced mathematics in learning methods and thinking methods. Learning it well requires us to master it from the height of methodology. When we study mathematical problems, we should always use materialist dialectical thinking to solve mathematical problems. Mathematical thought is essentially a reflection of the application of materialist dialectics in mathematics. The mathematical thoughts that should be mastered in middle school mathematics learning are: set and correspondence, initial axiom, combination of numbers and shapes, movement, transformation and transformation.
For example, the concepts of sequence, linear function and straight line in analytic geometry can be unified by the concept of function (special correspondence). For another example, the concepts of number, equation, inequality and sequence can also be unified into the concept of function.
Let's take a look at the following example of solving problems from a "contradictory" point of view.
Given that the moving point Q moves on the circle x2+y2= 1, fix the point P (2 2,0) and find the locus of the midpoint of the straight line PQ.
Analyzing this problem, P, Q and M are mutually restricted, and the movement of Q drives the movement of M; The main contradiction is the movement of point Q, whose trajectory follows the equation x02+y02 =1①; Secondary contradiction: m is the midpoint of the straight line PQ, and the coordinates (x, y) of m can be expressed by the midpoint formula with the coordinates of point Q.
x=(x0+2)/2 ②
y=y0/2 ③
Obviously, by substitution, x0 and y0 in the problem can be eliminated and the desired trajectory can be obtained.
Mathematical thinking method is different from problem-solving skills. In proving or solving, it can be said that solving problems by induction, deduction and method of substitution is a technical problem, and mathematical thinking is a guiding general thinking method. When solving a problem, from the overall consideration, how to start, what are the methods? It is a common problem under the guidance of mathematical thinking methods.
With mathematical ideas, we should master specific methods, such as method of substitution, undetermined coefficient method, mathematical induction, analysis, synthesis and induction. Only under the guidance of problem-solving thought and flexible use of specific problem-solving methods can we really learn mathematics well. It is often difficult to make mathematics learning enter a higher level by mastering specific operation methods without considering problems from the perspective of problem-solving thinking, which will bring great trouble for further study in universities in the future.
In terms of specific methods, commonly used are: observation and experiment, association and analogy, comparison and classification, analysis and synthesis, induction and deduction, general and special, finite and infinite, abstraction and generalization.
If you want to win the battle, you can't just fight bravely, not afraid of death or suffering. You must formulate tactics and strategies that have a bearing on the overall situation. When solving mathematical problems, we should also pay attention to solving the problem of thinking strategy, and often think about what angle to choose and what principles to follow. Generally speaking, the general idea adopted to solve problems is a principled thinking method, a macro guidance and a general solution.
Mathematical thinking strategies commonly used in middle school mathematics are:
Simple control of complexity, combination of numbers and shapes, mutual advancement and retreat, turning adversity into familiarity, overcoming difficulties, harmony, dynamic and static transformation, separation and integration complement each other.
If you have correct mathematical thinking methods, appropriate mathematical thinking strategies, rich experience and solid basic skills, you will certainly learn high school mathematics well.
Fourth, the improvement of learning methods.
In the strange circle of exam-oriented education, every teacher and student can't help falling into the sea of questions. The teacher doesn't talk about some questions, and neither will the college entrance examination. Students are afraid to do one less question. In case the exam loss is too heavy, in such an atmosphere, the cultivation of learning methods is often ignored. Every student has his own method, but what kind of learning method is correct? Is it necessary to "read a lot of questions" to improve your level?
Reality tells us that it is a very important issue to boldly improve learning methods.
Learn to listen and read.
We listen to teachers and read textbooks or materials at school every day, but are we listening and reading correctly?
Let's talk about listening (listening, classroom learning) and reading (reading textbooks and related materials). What students learn is often indirect knowledge, abstract and formal knowledge, which is refined on the basis of previous exploration and practice, and generally does not include the process of exploration and thinking. So be sure to listen to the teacher, concentrate and think positively. Find out what the content is. How to analyze it? What is the reason? In what way? Is there a problem? Only in this way can we understand the teaching content.
The process of attending classes is not a process of passive participation. On the premise of attending class, we need to analyze: what thinking method is used here, and what is the purpose of doing so? Why can the teacher think of the shortest way? Is there a more direct way to solve this problem?
"Learning without thinking is useless, thinking without learning is dangerous", so we must have positive thinking and participation in the process of listening to classes, so as to achieve the highest learning efficiency.
Reading mathematics textbooks is also a very important way to master mathematics knowledge. Only by reading and reading mathematics textbooks can we master mathematics language well and improve our self-study ability. We must change the bad tendency of using textbooks as dictionaries to look up formulas without reading books. When reading textbooks, we should also strive for the guidance of teachers. When reading the content of the day or the content of a unit or a chapter, we should consider comprehensively and have a goal.
For example, learning the arcsine function, in terms of knowledge, through reading, should ask the following questions:
(1) Does every function have an inverse function? If not, when does a function have an inverse function?
(2) Under what circumstances does a sine function have an inverse function? If so, how to express its inverse function?
(3) What is the relationship between the image of sine function and the image of arcsine function?
(4) What are the properties of the arcsine function?
(5) How to find the value of the arcsine function?
(2) learn to think.
Einstein once said: "The development of the general ability of independent thinking and independent judgment should always be in the first place." Being diligent and good at thinking is the most basic requirement for us to learn mathematics. Generally speaking, we should try our best to do the following two things.
1, good at finding and asking questions.
2. Be good at reflection and reverse seeking.
6. Menelaus theorem
Menelaus, an ancient Greek mathematician, first proved Menelaus theorem. It is pointed out that if a straight line intersects with three sides AB, BC, CA of △ABC or its extension lines at points F, D and E, then AF/FB×BD/DC×CE/EA= 1.
Prove:
The passing point A is the extension line where AG‖BC intersects DF at G.
AF/FB=AG/BD，BD/DC=BD/DC，CE/EA=DC/AG
Multiply by three formulas:
AF/FB×BD/DC×CE/EA = AG/BD×BD/DC×DC/AG = 1
The inverse theorem also holds: if there are three points F, D and E on the sides of AB, BC and CA or their extension lines, and AF/FB×BD/DC×CE/EA= 1, then the three points F, D and E are * * * lines. Using this inverse theorem, we can judge the trisection line.
8 ... an important role in matter. Quantum physics believes that this force is generated by exchanging some kind of energy packet. This energy envelope is called quantum. I suggest reading: Beyond Time and Space-A Scientific Journey through Parallel Universes, Curvature of Time and the Tenth Dimension, in which the collapse of superstring theory by European geometry is expounded in very popular language. Covers the content of modern high-energy physics. The author of this book, Kakumichio, is a Japanese-American physicist and a professor of physics at City College of City University of new york. In addition to this book, he also wrote Beyond Einstein, Quantum Field Theory and Introduction to Superstrings. Kakumichio believes that discovering the beauty and mystery of the universe is the meaning of life. In this book, he wrote: "Simplicity and elegance are the qualities that inspire great artists to create masterpieces handed down from generation to generation, and also inspire scientists to explore the laws of nature. Just like a work of art or a moving poem, the equation itself has some beauty and rhythm. "
quantum theory
& lt& lt quantization >>
What is quantum? Some people often talk about what quantum mechanics means, but they don't really understand what quantum means. In fact, the concept of quantum is for integer matter (not decimal), and there is no so-called continuous separability. For example, some people think that half of 10cm is 5 cm, and half of 5 cm is 2.5cm According to the truth, you can divide it infinitely, but the concept of quantum tells us that this division is infinite.
Let me give you a very simple example. At the ATM, you can withdraw 65,438+000 yuan, 200 yuan, 300 yuan, etc. These are multiples of 65,438+000. You can't withdraw 105 yuan or 105.5, because the ATM only issues paper money, not coins, 105 or 65438. ! It's just that ATMs can't handle change.
This is how the quantum world is quantized. Above, the money paid by ATM is a multiple of 100, which is similar to photon wavelength. We use 4000? ,400 1? ..... wait, it's all 1? Does that mean there is no 4000.5? What about wavelength light? No, it doesn't make sense in quantum mechanics. The wavelength can only be changed by quantum jump. Must be a basic unit (such as 1? Integer multiples of).
I will introduce these basic units one by one:
Planck constant
Planck constant (h)
Planck constant comes from an important formula E=hf (energy =h wavelength), where h is the relationship between the energy of electromagnetic wave and its wavelength. H is 6.626x 10-24J per second, such as 4543? The wavelength of light, 4543 per second? More energy.
In many formulas, h/2p is better, the code name is H plus a picture, and reading h-cross.h-cross is also called Planck constant. In fact, it doesn't matter whether there is 2p or not, because there is little difference between them.
Planck length
Planck length
Although gravity acts infinitely far, it fails in a short distance (Planck length), which is about 10-33cm. This is the shortest length.
planck time
planck time
The time required for light to travel through Planck's length is 10-43 seconds. It is the smallest unit of time, and any less time is meaningless. So the universe only makes sense after the big bang.
Off-topic, if it is not Planck, it must be the smallest unit, such as:
planck mass
planck mass
Suppose that the mass of an imaginary particle with Planck length is about 10-5g, which is 9 times that of the proton. Planck mass is not the smallest unit of mass.
Planck energy
Planck energy
The energy required to make a particle with Planck mass (E=mc2) is about 10 19GeV.
Planck temperature
Planck temperature
The temperature corresponding to Planck energy is1032 K.
Planck density
Planck density
The intrinsic density of an imaginary particle with Planck length and Planck mass is 1094g/cm3, which is 1060 times higher than the density of the nucleus! This assumes that particles are so dense that they can only exist in the Big Bang.
& lt& lt secondary quantization >:>
Quantization has also been introduced into the field, and some people call it the second quantization. For example, gravity is no longer a sphere of influence, and the concept of "scope" is not too unscientific ... Quantization tells us that gravity is transmitted by some particles called gravitons, which are virtual particles. We can't see them when they transmit power. For example, when two magnets are close together, you won't see photons passing through the middle. ), but when they have enough energy, they will be born into real particles and will no longer transmit force. For example, the photons we see every day are not force-transmitting.
The particle exchange in the field is usually marked by Feynman diagram. Don't think of it as just input and output. The meaning behind it can be very complicated. For example, the intersection of two straight lines actually involves multiple numbers (such as Maxwell equation and Schrodinger equation). Of course, this textbook is for beginners, so I try not to involve numbers (it takes me several times to sort out the materials), so I don't plan to do it in the future.
(Photo: lkastro-group)
& lt& lt non-Apels >:& gt；;
Regarding the calculus of quantum mechanics, we have to say non-Abelian groups, because it often involves matrices. In the usual addend or multiplier calculus, the inversion of terms will not affect the answer, such as axb=bxa or a+b=b+a, which is called Abel group. Obviously, neither subtraction nor division belongs to Abel group. ! ! Axb won't wait for bxa! ! !
Of course, if you don't learn pure mathematics, it is difficult to understand. Let me give you a simple example. There are four kinds of trees, namely flowers, shrubs, shrubs and trees. We want to know how much water each plant needs to grow (suppose one liter, two liters, three liters and four liters of water), and we will test them one by one, but we don't want to put all sixteen trees in the same row, because a long and narrow field is rare.
Matrix one
(Photo: lkastro-group)
But later we found that this arrangement was not beautiful enough, because all the trees would cover the flowers from the back, so we rearranged it so that no flowers and trees were in a straight line:
Matrix 2
(Photo: lkastro-group)
When we express the position of a tree, we can use the row and column method, such as the tree in the second row and the third column. We can write BoC=A, and the tree in the third row and the third column is naturally CoC=C (the function of circle O is similar to the multiplication sign X). We can see that AoA=A, BoB=B, CoC=C and DoD=D in the matrix, which is called idempotent, and it is easy to see one.
However, the above two matrices do not meet another characteristic: switching law, AoB=C, but BoA=D, AB inversion will make the answer different. This number is called non-Abelian. In the 4th-order (4x4) matrix, there is no matrix that is both idempotent and commutative, but only an inempotent commutative matrix, as shown below:
Matrix three
(Photo: lkastro-group)
In fact, all orders have power matrices, in addition, all orders have commutative matrices, but only odd idempotent matrices have commutative solutions, and even idempotent matrices can never be commutative (non-Abelian). As for why, you don't need to know, because it involves the reduction to absurdity of high number theory (probably, if there is an even idempotent commutative matrix, then this matrix will be both odd and even, which is contradictory.
< < scattering >; & gt
Scattering was first introduced into quantum theory, and there are three main types of scattering:
Elastic scattering: incident particles have no kinetic energy conversion after scattering (electrons are separated from charges when they hit electrons).
Inelastic scattering: incident particles have kinetic energy after scattering (part of kinetic energy is converted into heat energy when billiards hit billiards).
Deep inelastic scattering: all kinetic energy of incident particles is converted into heat energy after scattering (electrons collide with electrons to become photons).
Scattering is a profound knowledge. In the so-called Raman effect, it is found that using a single band (for example, 4 140? When photons enter the particle pile, the wavelength of some photons will become longer (such as 4 150? ), some wavelengths will become shorter (such as 4 130? ), because the motion of particles is random (Brownian motion), the change of length and shortening can only be calculated by statistical principle.
Raman believes that the blue of the sea is not caused by the scattering of sunlight by suspended particles in the sea, but by the scattering of sunlight by water molecules. Just as the blue sky is caused by molecules in the air rather than suspended particles, short waves are the easiest to scatter, making the sea and sky blue. The same can happen when X-rays, X-rays or gamma rays collide with electrons, and photons will lose energy and increase wavelength, which is the so-called Compton effect.
(Photo: lkastro-group)
Some of them use the Compton effect to criticize BIGBANG for no purpose. They think that quasars are all Moncha, and that the light emitted by quasars will produce a strong Compton effect through free electrons, so the cosmic redshift implies that galaxies are not far away from us (Doppler effect), but because the light of galaxies is constantly scattered by interstellar electrons before coming to us (there is no denying that there are electron and positron pairs in interstellar space), if that is true.
In addition, I read an article < < BIGBANG is under attack & gt& gt, in which the reasons for challenging the Big Bang include finding that many stars are older than the age of the Big Bang universe (12 billion years), and the giant wall organization also thinks that it will take more than10 billion years to form. If our galaxy is only12 billion years old, it will only rotate 45-60 times at most, which is not enough to form a vortex. The biggest problem is that the Big Bang has a singularity, which makes them ask what was before the Big Bang.
& lt& lt Chaos Theory >>
For example, quantum theory is not only used in physics, but also used to explain another theory-chaos theory. Chaos theory is a very rapid theory in recent years, and the department of mathematics of the university is also reading it. A central concept of chaos theory is the butterfly effect, which means that a butterfly flapping its wings in Beijing will slightly change the air pressure, and these disturbances will change the air pressure in nearby places, so they will spread widely.
Isn't that ridiculous? But in the nonlinear system, any small disturbance will increase exponentially at the beginning, that is, after 1 s, the difference will be 4 times, then 8 times, 16 times, and so on. This will cause the system to become more and more unpredictable over time.
This prediction error stems from our inaccurate understanding of the beginning of the system and too many variables.
Linear systems can be represented by such images. Even if a piece of lead falls straight, even if there are external factors such as wind direction, even if we change its position at first, we can probably guess its landing point. The nonlinear system can start from a point on the circle, and its landing point can be square or octagonal, so the result depends entirely on external interference.
(Photo: lkastro-group)
The unpredictability of complex systems seems inevitable, such as weather, because there are too many initial variables, and our understanding of them is not comprehensive enough.