by Liu Cixin
The ground glowed red like a piece of iron in a blacksmith’s furnace. Bright rivulets of lava snaked across the dim red earth, forming a net of fire that stretched to the horizon. Countless thin pillars of flame erupted toward the sky: The dehydratories were burning. The dehydrated bodies inside gave the fire a strange bluish glow.
Not far from him, Wang saw a dozen or so small pillars of flame of the same color. These were the people who had just run out of the pyramid: the pope, Galileo, Aristotle, and Leonardo. The fiery pillars around them were translucent blue, and he could see their faces and bodies slowly deforming in the flame. They focused their gazes on Wang, who had just emerged. Holding the same pose and lifting their arms toward the sky, they chanted in unison, “Tri-solar day—”
Wang looked up and saw three gigantic suns slowly spinning around an invisible origin, like an immense three-bladed fan blowing a deadly wind toward the world below. The three suns took up almost the entire sky, and as they drifted toward the west, half of the formation sank below the horizon. The giant fan continued to spin, a bright blade occasionally shooting above the horizon to give the dying world another brief sunrise and sunset. After a sunset, the ground glowed dim red, and the sunrise a moment later flooded everything with its glaring, parallel rays.
Once the three suns had completely set, the thick clouds that had formed from all the evaporated water still reflected their glow. The sky burned, displaying a hellish, maddening beauty.
After the last light of destruction finally disappeared and the clouds only glowed with a faint red luminescence reflected from the hellish fire on the ground, a few lines of giant text appeared:
Civilization Number 183 was destroyed by a tri-solar day. This civilization had advanced to the Middle Ages.
After a long time, life and civilization will begin again, and progress once more through the unpredictable world of Three Body.
But in this civilization, Copernicus successfully revealed the basic structure of the universe. The civilization of Three Body will take its first leap. The game has now entered the second level.
We invite you to log on to the second level of Three Body.
16
The Three-Body Problem
As soon as Wang logged out of the game, the phone rang.
It was Shi Qiang, who said it was urgent that he come down to Shi’s office at the Criminal Division. Wang glanced at his watch: It was three in the morning.
Wang arrived at Da Shi’s chaotic office and saw that it was already filled with a dense cloud of cigarette smoke. A young woman police officer who shared the office fanned the smoke away from her nose with a notebook. Da Shi introduced her as Xu Bingbing, a computer specialist from the Information Security Division.
The third person in the office surprised Wang. It was Wei Cheng, the reclusive, mysterious husband of Shen Yufei from the Frontiers of Science. Wei’s hair was a mess. He looked up at Wang, but seemed to have forgotten they had met.
“I’m sorry to bother you, but at least it looks like you weren’t asleep,” Da Shi said. “I have to deal with something that I haven’t told the Battle Command Center yet, and I need your advice.” He turned to Wei Cheng. “Tell him what you told me.”
“My life is in danger,” Wei said, his face wooden.
“Why don’t you start from the beginning?”
“Fine. I will. Don’t complain about me being long-winded. Actually, I’ve often thought about talking to someone lately.…” Wei turned to look at Xu Bingbing. “Don’t you need to take notes or something?”
“Not right now,” Da Shi said, not missing a beat. “You didn’t have anyone to talk to before?”
“No, that’s not it. I was too lazy to talk. I’ve always been lazy.”
WEI CHENG’S STORY
I’ve been lackadaisical since I was a kid. When I lived at boarding school, I never washed the dishes or made the bed. I never got excited about anything. Too lazy to study, too lazy to even play, I dawdled my way through the days without any clear goals.
But I knew that I had some special talents others lacked. For example, if you drew a line, I could always draw another line that would divide it into the golden ratio: 1.618. My classmates told me that I should be a carpenter, but I thought it was more than that, a kind of intuition about numbers and shapes. But my math grades were just as bad as my grades in other classes. I was too lazy to bother showing my work. On tests, I just wrote out my guesses as answers. I got them right about eighty to ninety percent of the time, but I still got mediocre scores.
When I was a second-year student in high school, a math teacher noticed me. Back then, many high school teachers had impressive academic credentials, because during the Cultural Revolution many talented scholars ended up teaching in high schools. My teacher was like that.
One day, he kept me after class. He wrote out a dozen or so numerical sequences on the blackboard and asked me to write out the summation formula for each. I wrote out the formulas for some of them almost instantaneously and could tell at a glance that the rest of them were divergent.
My teacher took out a book, The Collected Cases of Sherlock Holmes. He turned to one story— “A Study in Scarlet,” I think. There’s a scene in it where Watson sees a plainly dressed messenger downstairs and points him out to Holmes. Holmes says, “Oh, you mean the retired sergeant of marines?” Watson is amazed by how Holmes could deduce the man’s history, but Holmes can’t articulate his reasoning and has to think for a while to figure out his chain of deductions. It was based on the man’s hand, his movements, and so on. He tells Watson that there is nothing strange about this: Most people would have difficulty explaining how they know two and two make four.
My teacher closed the book and said to me, “You’re just like that. Your derivation is so fast and instinctive that you can’t even tell how you got the answer.” Then he asked me, “When you see a string of numbers, what do you feel? I’m talking about feelings.”
I said, “Any combination of numbers appears to me as a three-dimensional shape. Of course I can’t describe the shapes of numbers, but they really do appear as shapes.”
“Then what about when you see geometric figures?” The teacher asked.
I said, “It’s just the opposite. In my mind there are no geometric figures. Everything turns into numbers. It’s just like if you get really close to a picture in the newspaper and everything turns into little dots.”
The teacher said, “You really have a natural gift for math, but … but…” He added a few more “but”s, pacing back and forth as though I was a difficult problem that he didn’t know how to handle. “But people like you don’t cherish your gift.” After thinking for a while, he seemed to give up, saying, “Why don’t you sign up for the district math competition next month? I’m not going to tutor you. I’d just be wasting my time with your sort. But when you give your answers, make sure to write out your derivations.”
So I went to the competition. From the district level up through the International Mathematics Olympiad in Budapest, I won first place each time. After I got back, I was accepted by a top college’s math program without having to go through the entrance examination.…
You’re not bored by my talking all this time? Ah, good. Well, to make sense of what happened later, I have to tell you all this. That high school math teacher was right. I didn’t cherish my talent. Bachelor’s, master’s, Ph.D.—I never put much effort into any of them, but I did manage to get through them all. However, once I graduated and went back to the real world, I realized that I was completely useless. Other than math, I knew nothing. I was half asleep when it came to the complexities of relationships between people. The longer I worked, the worse my career. Eventually I became a lecturer at a college, but I couldn’t survive there either. I just couldn’t take teaching seriously. I’d write on the blackboard, “easy to prove,” and my students would still struggle for a long while. Later, when they began to eliminate the worst teachers, I was fired.
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sp; By then I was sick of everything. I packed a bag and went to a Buddhist temple deep in the mountains somewhere in southern China.
Oh, I didn’t go to become a monk. Too lazy for that. I just wanted to find a truly peaceful place to live for a while. The abbot there was my father’s old friend—very intellectual, but became a monk in his old age. The way my father told it, at his level, this was about the only way out. The abbot asked me to stay. I told him, “I want to find a peaceful, easy way to just muddle through the rest of my life.” The abbot said, “This place isn’t really peaceful. There are lots of tourists, and many pilgrims too. The truly peaceful can find peace in a bustling city. And to attain that state, you need to empty yourself.” I said, “I’m empty enough. Fame and fortune are nothing to me. Many of the monks in this temple are worldlier than me.” The abbot shook his head and said, “No, emptiness is not nothingness. Emptiness is a type of existence. You must use this existential emptiness to fill yourself.”
His words were very enlightening to me. Later, after I thought about it a bit, I realized that it wasn’t Buddhist philosophy at all, but was more akin to some modern physics theories. The abbot also told me he wasn’t going to discuss Buddhism with me. His reason was the same as my high school teacher’s: With my sort, he’d just be wasting his time.
That first night, I couldn’t sleep in the tiny room in the temple. I didn’t realize that this refuge from the world would be so uncomfortable. My blanket and sheet both became damp in the mountain fog, and the bed was so hard. In order to make myself sleep, I tried to follow the abbot’s advice and fill myself with “emptiness.”
In my mind, the first “emptiness” I created was the infinity of space. There was nothing in it, not even light. But soon I knew that this empty universe could not make me feel peace. Instead, it filled me with a nameless anxiety, like a drowning man wanting to grab on to anything at hand.
So I created a sphere in this infinite space for myself: not too big, though possessing mass. My mental state didn’t improve, however. The sphere floated in the middle of “emptiness”—in infinite space, anywhere could be the middle. The universe had nothing that could act on it, and it could act on nothing. It hung there, never moving, never changing, like a perfect interpretation for death.
I created a second sphere whose mass was equal to the first one’s. Both had perfectly reflective surfaces. They reflected each other’s images, displaying the only existence in the universe other than itself. But the situation didn’t improve much. If the spheres had no initial movement—that is, if I didn’t push them at first—they would be quickly pulled together by their own gravitational attraction. Then the two spheres would stay together and hang there without moving, a symbol for death. If they did have initial movement and didn’t collide, then they would revolve around each other under the influence of gravity. No matter what the initial conditions, the revolutions would eventually stabilize and become unchanging: the dance of death.
I then introduced a third sphere, and to my astonishment, the situation changed completely. Like I said, any geometric figure turns into numbers in the depths of my mind. The sphereless, one-sphere, and two-sphere universes all showed up as a single equation or a few equations, like a few lonesome leaves in late fall. But this third sphere gave “emptiness” life. The three spheres, given initial movements, went through complex, seemingly never-repeating movements. The descriptive equations rained down in a thunderstorm without end.
Just like that, I fell asleep. The three spheres continued to dance in my dream, a patternless, never-repeating dance. Yet, in the depths of my mind, the dance did possess a rhythm; it was just that its period of repetition was infinitely long. This mesmerized me. I wanted to describe the whole period, or at least a part of it.
The next day I kept on thinking about the three spheres dancing in “emptiness.” My attention had never been so completely engaged. It got to the point where one of the monks asked the abbot whether I was having mental health issues. The abbot laughed and said, “Don’t worry. He has found emptiness.” Yes, I had found emptiness. Now I could be at peace in a bustling city. Even in the midst of a noisy crowd, my heart would be completely tranquil. For the first time, I enjoyed math. I felt like a libertine who has always fluttered carelessly from one woman to another suddenly finding himself in love.
The physics principles behind the three-body problem28 are very simple. It’s mainly a math problem.
“Didn’t you know about Henri Poincaré?” Wang Miao interrupted Wei to ask.29
At the time, I didn’t. Yes, I know that someone studying math should know about a master like Poincaré, but I didn’t worship masters and I didn’t want to become one, so I didn’t know his work. But even if I had, I would have continued to pursue the three-body problem.
Everyone seems to believe that Poincaré proved that the three-body problem couldn’t be solved, but I think they’re mistaken. He only proved sensitive dependence on initial conditions, and that the three-body system couldn’t be solved by integrals. But sensitivity is not the same as being completely indeterminable. It’s just that the solution contains a greater number of different forms. What’s needed is a new algorithm.
Back then, I thought of one thing: Have you heard of the Monte Carlo method? Ah, it’s a computer algorithm often used for calculating the area of irregular shapes. Specifically, the software puts the figure of interest in a figure of known area, such as a circle, and randomly strikes it with many tiny balls, never targeting the same spot twice. After a large number of balls, the proportion of balls that fall within the irregular shape compared to the total number of balls used to hit the circle will yield the area of the shape. Of course, the smaller the balls used, the more accurate the result.
Although the method is simple, it shows how, mathematically, random brute force can overcome precise logic. It’s a numerical approach that uses quantity to derive quality. This is my strategy for solving the three-body problem. I study the system moment by moment. At each moment, the spheres’ motion vectors can combine in infinite ways. I treat each combination like a life form. The key is to set up some rules: which combinations of motion vectors are “healthy” and “beneficial,” and which combinations are “detrimental” and “harmful.” The former receive a survival advantage while the latter are disfavored. The computation proceeds by eliminating the disadvantaged and preserving the advantaged. The final combination that survives is the correct prediction for the system’s next configuration, the next moment in time.
“It’s an evolutionary algorithm,” Wang said.
“It’s a good thing I invited you along.” Shi Qiang nodded at Wang.
Yes. Only much later did I learn that term. The distinguishing feature of this algorithm is that it requires ultralarge amounts of computing power. For the three-body problem, the computers we have now aren’t enough.
Back then, in the temple, I didn’t even have a calculator. I had to go to the accounting office to get a blank ledger and a pencil. I began to build the math model on paper. This required a lot of work, and in no time at all I went through more than a dozen ledgers. The monks in charge of accounts were angry with me, but because the abbot wished it, they found me more paper and pen. I hid the completed calculations under my pillow, and threw the scratch paper into the incense burner in the yard.
One evening, a young woman suddenly dashed into my room. This was the first time a woman had shown up at my place. She clutched a few pieces of paper with burnt edges, the scratch paper I had thrown out.
“They tell me these are yours. Are you studying the three-body problem?” Behind her wide glasses, her eyes seemed to be on fire.
The woman surprised me. The math I used was unconventional, and my derivations took large leaps. But the fact that she could tell the subject of my study from a few pieces of scratch paper showed that she had unusual math talent and that she, like me, was very devoted to the three-body problem.
I didn’t have a good impres
sion of the tourists and pilgrims. The tourists had no idea what they were looking at, only running around to snap pictures. As for the pilgrims, they looked much poorer than the tourists, and all seemed to be in a state of numbness, their intellect inhibited. But this woman was different. She looked like an academic. Later I found out that she had come with a group of Japanese tourists.
Without waiting for my answer, she added, “Your approach is brilliant. We’ve been searching for a method like this that could turn the difficulty of the three-body problem into a matter of massive computation. Of course, it would require a very powerful computer.”
I told her the truth. “Even if we were to use all the computers in the world, it wouldn’t be enough.”
“But you must have an adequate research environment, and there’s nothing like that here. I can give you the use of a supercomputer. I can also give you a minicomputer. Let’s leave together tomorrow morning.”
The woman, of course, was Shen Yufei. Like now, she was concise and authoritarian, but she was more attractive then. I’m naturally a cold person. I had less interest in women than the monks around me. This woman who didn’t adhere to conventional ideas about femininity was different, though. She attracted me. Since I had nothing to do anyway, I agreed right away.
That night, I couldn’t sleep. I draped a shirt over my shoulders and walked out into the yard. In the distance, I saw Shen in the dim temple hall. She knelt before the Buddha with lit joss sticks, and all her movements seemed full of piety. I approached noiselessly, and as I came by the door to the temple hall, I heard her whisper a prayer: “Buddha, please help my Lord break away from the sea of misery.”
I thought I must have heard wrong, but she chanted the prayer again.