Four colour theorem; number theory; prime numbers; surfaces and knots; higher dimensions; fractals and chaos; cellular automata; Penrose tiling; zero-knowledge proof.
Topology and knots; tilings, sphere packings, and quasi-crystals; fractals, the edge of chaos, and dynamical systems; number theory, polyominoes, pi, acoustics; NP-complete problems; algorithmic complexity; proof, Fermat's Last Theorem
Newton's Laws of Motion, and the workings of our solar system that they describe, are often held up as examples of the clockwork precision of classical physics. Actual brass clockwork models of the solar system, orreries, bolstered this world view. But that never was the entire story...
This excellent book explodes the clockwork myth, with an historical account of the mathematics of celestial mechanics, from the ancient Greeks' insistence on circles within circles, via Tycho, Kepler and Newton's ellipses, through the early chaos results of Poincaré, to current day computational results and Digital Orreries. Newton's elegant laws, as well as having simple clockwork-like solutions of planets moving in ellipses, also have totally chaotic solutions, and our solar system does indeed appear to be chaotic in the long term.
The story is well told, as a complex dance of observation and theory complementing each other -- new theories simplify some observation, but then more precise observations show weaknesses in the theories. This is interspersed with wonderful little morsels -- such as the fact that after Uranus was discovered in 1781, 22 observations of it in the preceding century, including one by Flamsteed in 1690, were discovered in the astronomical records; and that Neptune since its discovery in 1846 has yet to complete one full orbit of the sun.
I found the later chapters -- on the slow discovery of chaos in the solar system -- more interesting, possibly because I am less familiar with this end of the story. We get a slow build-up: our moon's peculiar orbit (because the sun's influence on it is so large); chaos in some asteroid orbits possibly explaining the Kirkwood gaps (Jupiter may make their eccentricity chaotic, but it is then Mars and the Earth which disrupt them completely); the tumbling motion of Hyperion, one of the moons of Saturn; the realisation that all moon systems seem to be chaotic; that Venus and Mars' rotation axes may tumble chaotically, and our moon is all that saves our own from doing the same; finally the realisation that the solar system itself has chaotic behaviour. All this is told in an exciting, but not sensationalist, style. Thankfully, we find no tabloid-style "we're all going to fall into the sun tomorrow!", but rather an interesting discussion of how chaos and stability seem to coexist.
The story is by no means complete. There is obviously still a lot of research to be done: each time more reality is added to the models -- precise details of planetary masses, non-spherical bodies, General Relativistic corrections -- more interesting and subtle behaviour is found. And there's still more to add, such as frictional dissipative forces and the effect of the solar wind.
What is also made clear is the profound effect that computers have had on the discipline. The equations of motion might look simple in the abstract, but there is no closed-form solution even for three bodies, and numerical solutions are amazingly complicated. By using computers, and very cunning algorithms, planetary motions can now be calculated millions of years into the future, enabling long term quasi-periodic and chaotic behaviour to be discovered. In addition, ways of visualising the resulting mass of data have been developed. I would dearly love to see the video that shows how the planetary orbits change, when viewed at 60000 years per second: