The Three-Body Problem

23 Translator’s Note: Mozi was the founder of the Mohist school of philosophy during the Warring States Period. Mozi himself emphasized experience and logic, and was known as an accomplished engineer and geometer.

24 Translator’s Note: This was a Chinese 16-bit minicomputer modeled on the American Data General Nova.

25 Translator’s Note: A maser is like a laser, but for electromagnetic radiation, typically microwaves, not in the visible light range.

26 Translator’s Note: The Third Front program was a secret, military-led industrialization effort during the Cultural Revolution that built factories in China’s interior, where they would be less vulnerable to American and Soviet attacks.

27 Translator’s Note: Hand-written posters using large Chinese characters have become associated with the Cultural Revolution in the popular imagination. However, they have a long history in China as tools of propaganda as well as protest both before and after the Cultural Revolution.

28 Author’s Note: How three bodies would move under the influence of their mutual gravitational attractions is a traditional problem in classical mechanics that arises naturally in the study of celestial mechanics. Many have worked on it since the sixteenth century. Euler, Lagrange, and more recent researchers (aided by computers) have all found solutions for special cases of the three-body problem. Karl F. Sundman later proved the existence of a general solution to the three-body problem in the form of a convergent infinite series, but the series converges so slowly that it is practically useless.

29 Translator’s Note: Poincaré showed that the three-body problem exhibited sensitive dependence on initial conditions, which we would now understand as characteristic of chaotic behavior.

30 Translator’s Note: For details, please see Alain Chenciner and Richard Montgomery, “A remarkable periodic solution of the three-body problem in the case of equal masses,” Annals of Mathematics, 152 (2000), 881–901.

31 Author’s Note: Roche limit: Édouard Roche, French astronomer, was the first to calculate the theoretical distance between two celestial bodies such that the smaller body will be torn apart by tidal forces from the larger body. The Roche limit is usually expressed as a function of the densities of the bodies and the equatorial radius of the larger body.

32 Translator’s Note: Chairman Mao was often compared to the “red sun,” especially during the years of the Cultural Revolution.

33 Translator’s Note: The Chinese term for “sunspot” () literally means “solar black spots.” Black, of course, was the color of counter-revolutionaries.

34 Translator’s Note: Alpha Centauri, though appearing to the naked eye as a single star, is actually a double-star system (Alpha Centauri A and Alpha Centauri B). A third star, called Proxima Centauri and invisible to the naked eye, is probably gravitationally associated with the double-star system. The Chinese name for the objects () makes it clear that the “star” is really a system of three stars.

35 Translator’s Note: The Five Black Categories, the targets of the Cultural Revolution, were five political identities used during the revolution: landlords, rich farmers, counter-revolutionaries, “bad elements,” and right-wingers.

36 Translator’s Note: During the initial phase of the Cultural Revolution, all classes ceased at colleges and elementary, junior high, and high schools as older students became Red Guards. The resulting chaos finally caused the leadership in Beijing to ask students to return to class in late 1967 and continue the revolution in a more controlled manner.

37 Translator’s Note: “Cowsheds” were locations set up by work units (factories, schools, towns, etc.) during the early phases of the Cultural Revolution to detain the counter-revolutionary “Monsters and Demons” (reactionary academic authorities, rightists, the Five Black Categories, etc.) at the work unit.

38 Translator’s Note: This meeting marked the beginning of the “Reform and Opening Up” policy and was seen as the moment when Deng Xiaoping became the leader of China.

39 Translator’s Note: In the later years of the Cultural Revolution, privileged, educated urban youths were sent down to the poor, mountainous countryside to live with and learn from the farmers there. Many of these so-called “Rusticated Youths” were former Red Guards, and some commentators believe that the policy was instituted by Chairman Mao to restore order by removing the rebels, who had gotten out of control, from the cities.