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.
to form exactly four triangles, in which all the sides of all the four triangles are equal. Try this for a few minutes, but be aware that the solution requires an unconventional approach. In case you have not succeeded, don’t despair; most people have trouble with this problem. The solution is shown in appendix 10. Galois’s proof concerning which equations are solvable by a formula (chapter 6) is the embodiment of thinking outside the box—to answer a question about algebraic equations he invented
ideas of the people around you. The younger you are, the more likely you are to be truly original. Psychologist Howard Gardner makes a similar distinction between mathematicians and scientists on one hand and artists on the other: It is important to note here a decisive difference from creation in the sciences or mathematics. Individuals in these latter areas begin to be productive at an early age and certainly have the option of making numerous innovations during their early years.
interpretation was later questioned: Hines 1998. 278 Anderson and Harvey showed that while Einstein’s brain: Anderson and Harvey 1996. 279 McMaster University neuropsychologist: Witelson, Kigar, and Harvey 1999. 280 The autopsy report on Galois’s brain: Dupuy 1896. 281 “the weight of the brain”: The value of the Parisian pound at the time was somewhat larger than the current value; 489.75 grams compared to 453.59 grams. 282 Mathematician and author Ian Stewart expressed: Stewart 2004. 283
but unless you turn it as in figure 27b, so that the axis of symmetry is vertical, you may not perceive the symmetry. Cognitive scientist Irvin Rock of Rutgers University and collaborators conducted a series of experiments designed to test the dependence of perception of form on orientation. In particular, they wanted to test if the perception of bilateral symmetry depends on whether the axis of symmetry is truly vertical in the retinal image or whether it is only perceived as being vertical. The
particles, such as electrons and quarks. The same applies to the force carriers as well. Messenger particles such as gluons or the W and the Z owe their existence to yet other harmonics. Put simply, all the matter and force particles of the standard model are part of the repertoire that strings can play. Most impressively, however, a particular configuration of vibrating string was found to have properties that match precisely the graviton—the anticipated messenger of the gravitational force.