## Myself and Another Me

Life, should be as perfect and harmony as a sphere which has the philosophical closeness and symmetry. Naturally, I keep this obsession into science, then I found I was wrong, on both.

The family condition of mine is not quite lovely, that is not very far from the “typical family problem” in many shows, and because of that I tend to absorb information more than expressing. Back to the time, the most frequently used body parts of mine to interact with the world are brain, eyes, and ears, the nose is not included because of my rhinitis. Sit at the back left of a car, I staring at the front seat to create extraordinary connections between leather patterns, hear all the things people around discussing, maybe shocking them by pointing out a better solution, then close my eyes go into my inside mind palace.

Me as such did not have the willingness to build interaction with others. People around talking always carried with emotions which hide their real original ideas, also words from my mouth maybe more compact but still lose its accuracy during transportation. Such encoding and decoding are low efficient, I am not a nerd, but people get tired when they know they need to do something not that useful every day. However, human being have an original impulse to communicate with people, I did not find other people as a friend but create another me instead. Since that, I got a step into a loop.

The way people to think is talk or ask the questions toward someone inside his mind, but that is more like to be talking to self, just using the language to help thinking. But I let the one who am I talking to become a real entity, he is no longer an intermediary but a real personality. I use “you” to call him, he could immediately understand my point sometimes even before I really say something. He is a good talker, also a great listener. I was amazed by such a high efficiency of communication. Not only “him” in current, but an infinite number of “him” in the past. One day I went out from my own apartment and found it was raining. When I was in the mood of manic, surprisingly I found there was an umbrella in my bag, what I did first? Saying to myself: “Thanks for your considerate, me in the past.” Every time I cannot control my emotion, he would bring me to think out of the box and do the best logical decision. They are both fathers and brothers, also, they are me. That’s really an extraordinary relationship.

That’s a boring black raining night, quite a silence, only the noise of my 3D printer. I was waiting for the time I set to go to the bed by browsing random stuff. Suddenly, There was something called “Parity violation” shocking to me. Basically, if we look at a spinning particle in a mirror, it should spin in a inverse direction, just like a person shaking his left hand, but in the mirror, it looks the right hand is moving. However, some particles in some condition, even are mirror reflected, still, spin in the same direction. That’s just like saying a mirrored particle has different properties. The Person in the mirror is not that person, but a different one.

I saw a slim bright light beam outside the window, then heard a powerful noise came over then“Let’s take a shower, shall we?” I said. “Yeah, why not?”He follows back.

The white noise produced by the water drops hitting the ground is perfect for helping people think deeply. “You always know what I gonna say, you are my best brother, my firmest umbrella”  “But, where are you?” Only the silence but the echo of my voice. “You! Locked me into a loop! Whatever I say, whatever you agree or not, your answer is what I want, my answer is what you want. We are one particle, but you in the mirror. The night still black, raining, and quite, but I am sure I was renewed with the water fog reflecting the shining golden bathroom heater light around.

The key to social communication is not the exchange of information, but makes everyone happy. I began to say something meaningless just let people around smile, I do need to show my ability but my classmates would spontaneously invite me to be the team leader. Sometimes, people come to me is not for finding a solution but just for spreading their emotion. It is human being. He is still around me, help to keep myself, think outside the emotional box, but we are just not in the loop anymore.

“Diversity is essential to happiness” It’s my new favorite motto. People always tried to set Romanesque symmetric definitions. Just like lock the sex to male and female, but now we realize there are LGBT or even more than 68 different sex, in the strict sense, the symmetry of male and female is broken. Or Do the physics really describe the world beautifully and accurately? The universe is running in its own way, maybe we called “ grateful formulas” are just try to approach the world this function. Thus, whatever life or science, people should get rid of the paranoid idea of beauty in form, but respect the fact of chaotic, diversity, and unknown.

## [Serial] The Automorphism Rocket

*2019/7/8

Before ALL:

Hello folks, since the summer began, I have to say a project I have planed for a long time should get into its path now. And it is a serial, so I will update the lastest processes as I do.

Basically, the goal is to build a fully fictional rocket which has a Thrust Vector Control(TVC) which allows the rocket could control the thrust direction stay toward the earth, and a panel with velocity sensors, direction sensors on so the SCM could compute.

Then, I would add more functions to the rocket, such as muti-thrusts. So finally allow the rocket to get into space(stratosphere?).

OK, there are few steps I need to do in order to make this goal comes true.

Stage One, build a TVC

Stage Two, build a control panel and write software

Stage Three, build a rocket as a whole

Then, test, test, test…

If I can make it, then add more functions and let it go the space.

Now, everything starts here.

Stage One: Build a TVC

Prototype 1:

The video was made a long time ago, so there are many things not been added compared to now I, but the idea is the same. Basically, I 3D-printed a model to physically exam my idea, and know I find there are two main problems need me to fix.

1. I need grooves and rooms to fit my two servo motors
2. The outside ring needs to control the inner ring but not directly connect the central, otherwise, the move of one axis would affect the other one, simply to say, they will be stuck.

*2019/7/28

The first “usable” TVC is done for building!

But there are still a lot of problems

1. the size of the inner ring is too big
2. the total design is not compact enough

*2019/8/1

After Serval models failed, the usable prototype is coming!

*testing platform is done!

## 手把手教你怎么搭建基岩版Minecraft我的世界服务器

What？

1.基岩版（XBOX，WIN10 UWP， 手机）

2.JAVA版（也就是之前电脑上玩的那种）

3.网易版（辣鸡）

Why？

How？

1.去阿里云（https://chuangke.aliyun.com/invite?userCode=z66uogjh）(吃我邀请码得优惠）买一台ECS服务器，最低配置就可以了，位置选靠近你地理位置的地方，系统注意了：要选Ubuntu 18.04 及以上的，第一次建议用按量付费试试水

2.这样你的服务器算是开好了，点管理，左上角找到“重置实例密码”来设定密码

3.因为我们需要19132端口来连接服务器，所以必须在安全组中添加

4.得到 public key:还记得之前我们设置过密码嘛？因为新版本的Xshell需要密匙文件才让你登录，所以我们这边再下载一下这个文件（妥善保管，不会让你下第二次的）在网络与安全-密匙对 中创建密匙，名字随意，把.pem文件保存. 然后点绑定密匙，将你的服务器绑定此密匙 然后点重启

5. 现在我们需要一个工具来连接这台服务器，别人喜欢用Putty，但我更爱Xshell

6.下面来配置服务器

1.输入apt install unzip回车下载解压命令

2.输入mkdir mc回车来创建名为”mc”的文件夹

3.输入cd /root/mc回车 进入此文件夹

4.输入wget https://minecraft.azureedge.net/bin-linux/bedrock-server-目前版本.zip回车, 下载服务器安装包

5.输入unzip bedrock-server-目前版本.zip回车, 解压

6.输入LD_LIBRARY_PATH=. ./bedrock_server回车 开启服务器 并进入服务器后台

(在后台输入 stop回车 来关闭后台)

screen -r [那几个数字] 恢复窗口

*先写到这里, 之后写怎么把地图迁移进服务器

Deemo

## The Chemistry Behind The Forming of Different Colors of M42 Nebula

PS:IT’S NOT A RESEARCH PAPER

Abstract

This paper begins with the Messier 42 nebula, talking about how different color lights form due to the electron transport in Bohr’s model. How these lights are absorbed by gas and form the absorption spectrum, then how these lights are captured by sensors with the explanation between CCD and CMOS sensors. The history of spectrometer also includes plus the motivation and the reason why people are keeping doing that.

Key Words: Messier 42, Spectrum, Bohr Model, Sensor, Photoelectric effect, Spectrometer, Wavelength, Energy transform, Frequency, Coulomb’s equation

Introduction

When we were just children, sometimes laying on a green field at night the light pollution is not that bad. People lie in the starlight, and the huge Milky Way fills all of the eyesight. Surely, it is night, but when your eyes adapt to the dim, the elegant shining beauty would amaze you, indeed. Sometimes people would notice there are something different lie on the sky, something bigger with different colors. After that people who are curious about that seek for answers, Finally, they found that is called “nebula”.

The “M42”

M42 or the Orion Nebula, mostly the most popular object in the Messier objects table. It is 1,344 ± 20 light years away(Bringmann, 2017). M42 is a diffuse nebula, which means it is formed ionized gases and diffuse lights with different wavelengths. Every day, there are stars forming, M42 is the closest factory of star from the planet Earth

This paper going introduce and explain the theory of how these colors were formed, how people transform light to digit, and the way people identify chemicals from light, plus its history, and why we do it with the foreword of M42.

Excitation and emission spectra

Light is an electromagnetic wave, it could transport energy, and because of its wave-particle duality, we can say photons have energy inside. In the Bohr Model, it could be easier to explain the whole thing because it is very clear without talking about the probability wave function or The Heisenberg Uncertainty.

As the diagram shows left, when an electron absorbs energy it would be excited, and goes outside from its ground state orbit, and it goes back to inside orbits when they release energy. The different colors of the light release directly connect to the difference of energy because the difference of colors is the result of diverse of the wavelengths, and energy relates to the frequency of the wave follows:$E=hf$

Put these two into the common equation:$v=f\lambda$  since the velocity just equal to the speed of light, which is a constant, higher frequency result in low wavelength, vice versa. So when there is an inside orbit which contain more energy because of the Coulomb’s equation: $\int{F=\int\frac{1}{4\pi\varepsilon_0}\cdot\frac{q_1q_2}{r^2}dr}=U=-W=KE=\frac{1}{4\pi\varepsilon_0}\cdot\frac{q_1q_2}{r}$

As r gets smaller, the energy gets higher, so the inner orbit emits light with higher energy, which means it is in higher frequency and low wavelength result in more red light.

The forming of three main colors

There are three main colors the M42 nebula has: Red, Blue-violet, and Green.

Red light is the result of H- radiation, mainly, it talks about Hydrogen atom in Bohr’s model, the electron jump from n=3 orbit to n=2 orbit, and result in long wavelength of 656.28 nm(H-alpha Emission, 2013) which is red.

Blue-violet light is the result of massive O-Class stars radiation at the core of the nebula. By Harvard spectral classification, stars are classified to O B A F G K M from high temperature to low temperature. There is a pithy formula: “Oh Boy! A Fine Girl Kiss Me” Since O-class is hot, which means it has high energy, so O-class has high frequency and short wavelength, result in Blue-violet light.

Green used to be mysterious in the early 20th century, because there is no known spectral lines could explain that, people even think they discovered a new element. However, by the developing of modern physics, people found the green hue is caused by the low probability transition in doubly ionized oxygen, that only could happen in the lab, but still possible happen in deep space wherein extreme condition.

Photoelectric effect: How to get a photo of a deep space object

We all know the camera could take photos, but how? Why light could be transformed to digit version, that is the contribution of the Photoelectric effect. Einstein gave the mathematical explanation for it. Mainly, if a photon carries enough energy, it could let the metal emit an electron carries with same amount kinetic energy: $KE=\frac{1}{2}mv^2$

Sensor Mechanism: CCD and CMOS

The sensor is where the camera accepts lights, and the place photoelectric happens. There are two main kinds of sensor exist in the market: CCD(Charge-coupled Device) and CMOS(Complementary Metal Oxide Semiconductor). The difference between these two main in the way they collecting voltage signals.

From Figure 6, each block is a pixel that the photoelectric effect happens, when there is a photo hit the pad, one electron would go out and create a voltage signal, the difference between CCD and CMOS is shows up on their signal processing mechanism. Each pixel of CCD only transform photo to electron signal then process line by line, but each pixel of CMOS could process independently.

In Astrophotography, people more like to use CCD because it would produce less noise and record faster. Heat is the main cause of useless data pixel because the heat would mess up the voltage signal, since CMOS process data on every pixel it would cause huge heat during a long time capturing which is very common in astrophotography, but it is much better on CCD because of it process data line by line. However, to minimize the noise, some people would cool down their sensor, so there is something called cooled CCD, people cool their sensors (normally 253K) to prevent noise as possible as they can.

Spectrometer: Way to identify the chemical by its emitted light

Since the different color of light it the result of different elements jump orbitals, we could inversely use this property to identify the element combination of one object. Then, the spectrometer is invented. The grating spectrometer has the clearest mechanism with the cheap prize. It could separate one combined light source to spectrum distribution just like how Figure 7 shows.

When a continuous light goes through a cloud, most of the spectrum would not be absorbed, but if the cloud meets the light with a certain spectrum which has the energy just enough to let its atom jump excited, the cloud would absorb it. So when people on earth observe it, they would find there are some black gaps in the continuous spectrum.

Take our star as an example, it is a G-class star with yellow looking. The black lines called Fraunhofer lines. Here in figure 9, for example, D-line represent element sodium, G-line is iron and calcium, C-line is $H\\alpha$…etc.

History of Spectrometer

Newton seems to be the first one who builds a spectrometer, but he did not use that to build some vital theory theories.

Bunsen invented a better flame source called Bunsen’s burner, now is common in labs and Kirchhoff designed the rest of it. Later, when Kirchhoff used this spectrometer to observe the sun, he found there are several black lines with different brightness, he guesses that might cause by different elements, then he uses some flames made by different sources to prove this hypothesis, and he finds he was right. After proven his hypothesis, he graphed 570 lines to indicate different elements.

The reason why we doing this

Using spectrometer could help us discover what made the early universe, and how the universe evolved because the nebula is the birth room of stars, it has a similar condition like the early universe

To get a better image of deep space target, know their spectrum is very important because people could choose a more suitable filter to cut off other disturbing spectra to improve the image quality.

And the mountain is there, a human being must climb it or disappear in inner dispute history.

Conclusion

M42 is a very beautiful nebula which is the closest star birth room, the light goes through the gas cloud be to absorb some with a certain wavelength, comes to earth with absorption spectrum. The light is captured by sensors(mostly CCDs), because of the photoelectric effect, it is transformed to a voltage signal, finally become digit data, processed by computer become a gorgeous image. Knowing that helps people know more about the universe and how it formed, people use this information to take a better photo by selecting more suitable filters. Human being knows the world around it a little more, exited their fundamental curiosity, which finally becomes the reason people come out of the earth but not root on it.

Reference

Bringmann, T., Conrad, J., Cornell, J. M., Dal, L. A., Edsjö, J., Farmer, B., … & Scott, P. (2017). DarkBit: a GAMBIT module for computing dark matter observables and likelihoods. The European Physical Journal C, 77(12), 831.

. A stellar spectral flux library: 1150–25000 Å. Publications of the Astronomical Society of the Pacific, 110(749), 863.

(n.d.). Retrieved June 2, 2019, from, https://wps.prenhall.com/wps/media/objects/476/488316/ch09.html

H-alpha Emission. (2013, May 08). Retrieved June 5, 2019, from https://ismlandmarks.wordpress.com/h-alpha-emission/

Stefano, M. (n.d.). . Retrieved June 5, 2019, from http://meroli.web.cern.ch/lecture_cmos_vs_ccd_pixel_sensor.html

Anthony, S. (2013, October 02). Every color of the Sun’s rainbow: Why are there so many missing? Retrieved June 5, 2019, from https://www.extremetech.com/extreme/167878-every-color-of-the-suns-rainbow-why-are-there-so-many-missing

Spectrometer. (n.d.). Retrieved June 5, 2019, from https://zh.wikipedia.org/wiki/Spectrometer#/media/File:Spectrometer_schematic.gif

History of spectroscopy. (2019, March 04). Retrieved June 5, 2019, from https://en.wikipedia.org/wiki/History_of_spectroscopy#/media/File:Kirchhoffs_first_spectroscope.jpg

## “The Ring”

At the wide, white house, the “master” dressed in black, stands upon the platform. “Let’s celebrate the coming of our friends, shall we?” he says to the children before him. He is their father, but not by blood. The children, dressed in white, reply “yes” one by one, with planned perfect meticulous smiles.

“Sorry, sir, but what are we celebrating?”, the youngest little “stander,” said.

The “master” askance at him with unknown mean. Saying: “Oh, you people do not hear the history yet” “Surely, you must familiar with the story before the Cold War, because that’s the thing people want to tell”

“Why?”

“Because that is the period of good.”

“How bad after that? Are we live in good?”

“People work in Caltech found if they using quantum entanglement technique, they could transform one matter to gas in sudden, and affect matters near it, transform them to the same phase as it, they called it “Phase Assimilation” .”

“That’s powerful, isn’t it?”

“That’s why the Soviet Union let it become a weapon, and create a world we now live in”

The master points outside, the child’s line of insight follows that, ends at the giant ring.

“Do you know the Rings of Saturn?”

“Yes, the rings of Saturn are the most extensive ring system of any planet in the Solar System, that orbit about Saturn. The ring particles are made almost entirely of water ice, with a trace component of rocky material. And that may be formed by objects which come inside the Roche limit and split to pieces…”

“They all taught you in the class, right? But have you ever pay attention to this giant plate around our planet earth?”

The little child stays for silence for a few seconds, looking at the ring. Whispering by his heart: “how couldn’t I notice that elegant thing?”

It is a huge ring, spreads 300 kilometers wide on the ground on average, around the whole planet. At the start of its formation, the heavier matter of earth first be phase-changed, projectile to the most outside the Roche limit zone of the earth which are the edge of matters comes together or split to pieces, people call it “Heavy Hand”. At that time, the self-rotation axis changed by the unbalanced losing of self-mass, divide the world to two half by the ring, 20% of USSR left the to the side which located with the USA, people later called it the “United hemisphere”, and the rest of the USSR body locate at the another, called the “Communized hemisphere”. The mass of earth still losing due to the gravity force from the “Heavy Hand” and the Centrifugal net force created by the self-rotating, the gravity of the earth is not that strong anymore.

“Yes, Sir, No man-made things could ever come close to the ring…yet, so we disconnect with another hemisphere for 80 years”

“That’s why!” The “master” suddenly goes rage, his eyes extended to large with a crazy laugh, “We live in such buildings, white, immutable. And train you to workers, works for me! To recover the freaking space industry!” He keeps himself in silence for a second, then “I know you would not understand me. I born at the period of the Civil War, really great time. I chose to be a rocket scientist, and rocket chose me. I am 87 years old now, at that time, I am only 17, 17! I am old now, getting old in such a pathetic world. And you, you are all pathetic.

“Sorry, sir… I do not understand, what you mean “pathetic”?”

“You would not understand…forever” The master goes back to cold immediately, no one knows what he exactly thinking about.”

“Should they be called children?” The master stand by, think in his head “These standers are puppets, they could only be free in their “mother’s” wombs” “At the day I become master from rocket scientist, I visited the “Red House” for producing standers, all the things happened still embarked in my head.” “Children originally tend to play and do things they like, people using conditioning to control their future behaviors, if babies tried to play, they would play a very loud shocking noise, I was shocked by that, to be honest. Then these babies would not cry, because…I do not know what they experienced, but they just do cry, after that, they just stop play and stand up for the next instruction. They become the standers. What a “beautiful” world”

“I let you go outside for 35 minutes,” The master said with unexpected

“OK”

“Why? Why? Why? Why does he let me go out? Am I just said anything wrong, would I be killed? What would happen?” beta thinks in his head.

34 minutes, 37 seconds

Beta open the heavy white door, it is the first time he gets out of the house after he got five years ago. He was 2 at that time, now he is seven. The door open with the sound of old, apparently the master himself does not like to walk out.

The world is shockingly the same, Beta still remember there are some men dress in grey walking slowing when he was 2 on the street, but now, things seem to be just an older version of them, just more feeling of fade. People do not talk to each other. There is no brightness in their eyes. They are still walking dead.

“Why the master wants to go out? There are no interesting things, nothing new, just like five years ago”

17 minutes, 14 seconds

The ring still big, strong, even getting larger than before, would people from two sides meet? Do they want to meet?

“It is a beautiful, elegant”

5 minutes, 32 seconds before ship across the “giant hand”

All the people suddenly stand up, do not move, freeze, like old trees but with an unbelievable strong spirit, they are waiting for something, they look at the sky, the ring.

“What’s wrong? What are they doing?”

3…

2…

1…

Beta feel there is something shined a little. Then something he would never forget in his life happens.

People cry, with a smile on their face, old men, young people, grey, white, black…

They are all celebrating. They run to the grassland stand down, smile and cry in silence…

The world becomes different… become like 80 years ago

Beta go back to the “home”, he finds the master is smiling, saying: “Our ship met the ship from another side, and we fight…”

“Our friend is back, the long depression period ends, it is the new beautiful world”

Beta look up, look at the wall, there is a maxim on there:

THE WORLD KEEPS RUNNING, NOT BECAUSE OF PEACE AND LOVE, BUT ENDLESS DISPUTE AND COUNTERMINE

## Using Conditioning to Improve Teaching Efficiency

When a little young baby just born, learning occur at every moment the person lives on the earth, study let people become completed, also helps them to live in this society. One’s future is to decide by learning.
Teachers teach students? Right? But how many of these relationships could be in a highly efficient way? When you in school, how many times you see there are some students set away from the speaker, and do their things but not actually hear what does the person who is currently talking saying? When the people talking accompany with other people’s voice, he could be very tired because he needs to higher the frequency and the intensity of the voice.

So, it is vital to get all the students’ attention at one moment with low cause, but how? Even many strict teachers set roles to his students, after few times class, there always be some students not pay their attention to the teacher at the time they need.

I want to mention my AP Biology teacher, Ms. Rosales, she is the one who demonstrates the psychology tool is such a strong tool to apply in real life.
I still remember the first thing Ms. Rosales ask us to do is something called if…then. It really simple, if she hit the table or things in a certain rhyme, the students should response that with another rhyme for quite.

It looks very simple, and the reality also, but it really let the order of class improve in a massive way. I still remember when Ms. Rosales out of the school for some personal issue, there is a teacher called Ms. Romasa, well, she is kind of nice, but she do not have the what we now called “pa-pa” skill as Ms. Rosales has, mostly, when the class getting disorder and loud, she tries to take the control by yelling in a loud way, that really takes some time and spirit, and only when students think she is angry then they become quite a little. The class still goes on, but not that orderly.

There actually are some very deep and integrated mechanisms behind that. In psychology, what Ms. Rosales apply is called Classical conditioning. There is a very famous experiment did by Palov involved with his dog.

A dog naturally has the willingness to eat meat, so they saliva when they see meat, it is called unconditioned response because it is naturally happening. However, how could we let dog saliva without using meat, but something else, something like…the sound of the ring? The sound of the ring called natural stimuli because the sound of the ring itself has nothing to do with letting dong saliva. What Palov did is ring the ring when he shows the dog with meat, and repeated it for several times, then he finds when he only rings the bell, the dog continues to saliva without meat, the natural stimuli become conditioned stimuli.

What Ms. Rosales do to the students is a similar thing from the perspective of mechanism. At the first days Ms. Rosales began to teach us, she could be very serious because she keeps let students train that. Students naturally response as quiet and tractable when their teacher gets serious. So here teacher get serious is an unconditioned stimulus, and students become quite is an unconditioned response. And the “pa-pa” hits are just natural stimuli, it is true that students know what they should do when they hear it, but their body is not be conditioned yet which means there are not that many people would respond in a quick way. However, after one week’s training, the quantity change cause a quality change, students could response the “pa-pa” sign immediately without any delay, and they would be quiet and pay attention to the teacher.

However, it is not the end, there is something called generalization, which just means the subject would response similar stimuli with one conditioned response or response the stimuli with a similar response. In Palov’s dog experiment, the dog not only responds to the sound of the ring, but other things make a similar sound, like whistle or computer “bi-bi” sound.
Just like in the class, sometimes Ms. Rosales would make different rhyme by mistake, but student response that with same high speed. Also, students would not only be quiet and pay attention to the teacher but also turn down the laptop, iPad or other devices.

The key component here is repeated, if not, extinction happens, or we can say “forget”. If Palov does not train his dog in a certain curve, finally the dog would no saliva to the sound of the ring.
Ms. Romsas had taught us for a long time period, on the first day Ms. Rosales’s class, many students “forget” to respond to the “pa-pa” hit. They got extinction.

To hold the best curve of conditioning? Try to do it in random time gap and random frequency in densely to prevent the appearing of extinction.
Hope you can apply to your life successfully.

## Why The Interval to Graph a Trig Equation in Polar Differ?

Boys, set your calculator to polar mode, and draw a graph of
r=4sin(3θ), then set the first interval of [0,π] and the second one: [0, 2π]

They looks exactly the same, isn’t it?

And if we take of the integral of these two to find the area, for the graph with interval [0,π], area = 4π, but for the graph with [0,2π] the area is 8π, thus, which one is the “true” one we want to find?

I. The Form of Trig Polar Equation

For a trig polar equation, the form is looks like: r=asin(bθ) or cos etc…

a decide the “size of the graph”

b decide the number of “blades”

II. Odd number b and Even number b

The thing is vital :

*The number of blades = 2b

-For even number of b, the blades created are separated(here b=4, so there are 8 “blades”):

-For odd number b, the “blades” are covered with each other, take example of r=4sin(3θ) with [0, 2π], it actually graphed 6 blades, but each two are in the same position covered the another one, so we could only see three. If we do the integral to calculate its area, we get the area of 6 blades.

Thus, if we want to find the area of 3 blades, you can set the interval of [0, π], or just simply divide the area with interval [0, 2π] by two

It’s a very interesting trick dealing with polar equation.

P.S. : hpbsd for me~

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