The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry
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What do Bach's compositions, Rubik's Cube, the way we choose our mates, and the physics of subatomic particles have in common? All are governed by the laws of symmetry, which elegantly unify scientific and artistic principles. Yet the mathematical language of symmetry-known as group theory-did not emerge from the study of symmetry at all, but from an equation that couldn't be solved.
For thousands of years mathematicians solved progressively more difficult algebraic equations, until they encountered the quintic equation, which resisted solution for three centuries. Working independently, two great prodigies ultimately proved that the quintic cannot be solved by a simple formula. These geniuses, a Norwegian named Niels Henrik Abel and a romantic Frenchman named Évariste Galois, both died tragically young. Their incredible labor, however, produced the origins of group theory.
The first extensive, popular account of the mathematics of symmetry and order, The Equation That Couldn't Be Solved is told not through abstract formulas but in a beautifully written and dramatic account of the lives and work of some of the greatest and most intriguing mathematicians in history.
celebrated novel Love Story read, “What can you say about a twenty-five-year-old girl who died? That she was beautiful, and brilliant, that she loved Mozart and Bach. And the Beatles and me.” One can easily paraphrase this sad summary for Évariste Galois (1811–32) and Niels Henrik Abel (1802–29). For Galois it would probably read something like, “What can you say about a twenty-year-old boy who died? That he was a romantic, and a genius, that he loved mathematics. And he succumbed to
thus set Galois on the road to formulating group theory. Groups and Permutations Permutations and groups are intimately related. In fact, the group concept was born out of the study of permutations. For Galois, this was only the first step in a series of ingenious inventions and ideas that paved the way to his brilliant proof. Let me provide a brief reminder of the precise definition of a group introduced in chapter 2. A group consists of members that have to obey four rules with respect
Klein proved that the icosahedral group and the permutations group are isomorphic. But, recall that Galois’s proof on the solvability of equations relied entirely on the classification of equations according to their symmetry properties under permutations of the solutions. The unexpected link between permutations and icosahedral rotations allowed Klein to weave a magnificent tapestry in which the quintic equation, rotation groups, and elliptic functions were all interwoven. Just as the completion
observers at rest) as the muons’ speed approaches the speed of light. Light itself always travels through three-dimensional space at precisely the speed of light. Special relativity tells us that nowhere can light travel at any other speed, nor is it ever possible to catch up with light—light can never be at rest. In this sense, perceiving light is a bit like the perception of motion in a movie. Each frame in the film captures a slightly different scene, and when these frames are flashed rapidly
number of frieze patterns that exist. Princeton mathematician John Horton Conway has given amusing names to the seven different types of strip patterns. The names correspond to the pattern of footprints obtained when each of the actions is repeated: hop, step, jump, sidle, spinning hop, spinning sidle, spinning jump. Figure89 Figure 90 Figure 91 The symmetries of the laws of physics under translations, rotations, and uniform motion (including the invariance of the speed of light)