Symmetry: A Journey into the Patterns of Nature (Anglais) Relié – 11 mars 2008
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Descriptions du produit
Présentation de l'éditeur
Symmetry is all around us. Our eyes and minds are drawn to symmetrical objects, from the pyramid to the pentagon. Of fundamental significance to the way we interpret the world, this unique, pervasive phenomenon indicates a dynamic relationship between objects. In chemistry and physics, the concept of symmetry explains the structure of crystals or the theory of fundamental particles; in evolutionary biology, the natural world exploits symmetry in the fight for survival; and symmetry—and the breaking of it—is central to ideas in art, architecture, and music.
Combining a rich historical narrative with his own personal journey as a mathematician, Marcus du Sautoy takes a unique look into the mathematical mind as he explores deep conjectures about symmetry and brings us face-to-face with the oddball mathematicians, both past and present, who have battled to understand symmetry's elusive qualities. He explores what is perhaps the most exciting discovery to date—the summit of mathematicians' mastery in the field—the Monster, a huge snowflake that exists in 196,883-dimensional space with more symmetries than there are atoms in the sun.
What is it like to solve an ancient mathematical problem in a flash of inspiration? What is it like to be shown, ten minutes later, that you've made a mistake? What is it like to see the world in mathematical terms, and what can that tell us about life itself? In Symmetry, Marcus du Sautoy investigates these questions and shows mathematical novices what it feels like to grapple with some of the most complex ideas the human mind can comprehend.
Biographie de l'auteur
Marcus du Sautoy is a professor of mathematics and the Simonyi Professor for the Public Understanding of Science at Oxford University. He is a frequent contributor on mathematics to The Times, The Guardian, and the BBC, and he lives in London.
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1) Quintic Equation: Both Abel and Galois proved the quintic equations have no radical solutions. Abel proved 'No solution' by reductio ad absurdum; while Galois proved 'Why No?' with the beautiful Group Theory. How could a 19-year-old French boy thought of such grand math theory? It was a shame he was not recognised by the grand mathematicians like Cauchy, Gauss, Fourier, etc. He wrote the Group Theory down the night before his deadly duel and scribbled "Je n'ai pas le temps" (I have no time)... it took another 10 years for Group Theory to be rediscoverd by Prof Liouville of the Ecole Polytechniques (whose ignorant examiners ironically failed Galois twice in Entrance Concours Exams).
2) Moonshine: Monster Group dimensions (dj) & relationship with Fourier expansion of coefficients (cj) in Modular Function (page 333):
x^-1 + 744+196,884x + 21,493,760 x^2 + 864,229,970x^3 +...
cn= c1+c2+...cn-1 + dn
where d1 = 196,883
d2 = 21,296,876
d3 = 842,609,326
and c1 = 1+ d1 = 196,884
c2 = c1+d2 = 21,493,760
c3 = c1 + c2 + d3 = 864,229,970
What a coincidence! no wonder Conway said this discovery was the most exciting event in his life.
3) 'Atlas of Finite Group': the book covered the insider story of the 5 Cambridge mathematicians led by Conway, in an attempt to create the 'Periodic Table' of Group's building blocks (Monster Group is the last one).
3) Icosahedron symmetry (20-sided polygon of triangular faces): this is the way viruses 'trick' our body cells to reproduce for them, by this deadly icosaherdon beauty. In nature, bees are tricked by flowers' symmetry. In human, we are 'tricked' by opposite sex's body symmetry:)
4) Arche de la Defense @ Paris: a Hypercube architecture (cube of 4-dimensions), shows us we can visualize 4-dimension objects in our 3-dimension world.
5) Chap 7 (Revolution) compared the Anglo-Saxon and French Math culture:
"Anglo-saxon temperament tend towards the nitty-gritty, revelling in strange examples and anomalis. The French, in contrast, love grand abstract theories and are masters at inventing language to articulate new and difficult structures." I agreed, having been taught in anglo-saxon (UK, USA) math before entering into French Grande Ecole (Engineering University), I found great difficulty to compete with French classmates in abstract math, but beat them in applied math by my high-school 'anglo-saxon' math training. You notice France has never won IMO Math Olympiad Championship like USA, China do, but France invented most of the modern algebra and modern analysis.
This book is a grand-tour of the most exciting modern math - Group Theory. For all math students who hate reading the boring abstract modern math textbooks, you will be 'hooked' by the underlying beauty of modern math after reading du Sautoy's Symmetry.
Imagine an equilateral triangle. You can leave it where it is, or you can rotate it around by a third, or by two thirds, and it looks just the same. You can flip it around three different axes, and it looks the same. Those are its six symmetries. The Greeks were fascinated with the symmetry of solid figures, the Muslims with that of tiles and plane figures. But shapes and tilings are not all there is to symmetry; different ways of shuffling a pack of cards have symmetry, as does the number lock on a piece of luggage. The change ringers who team up to ring five bells in the exactly 120 different orders in which five bells can be rung are performing a symmetrical operation. There are symmetries of many properties of matter and physics, and it might be that they will help explain string theory. Indeed, the great story here is that like prime numbers, symmetric groups are at the heart of the mathematics of our universe, but unlike prime numbers, there is only a finite number of symmetries, and they have all now been found. It was thought in the 1920s that group theory had reached a dead end and there were no further families of symmetry to be found. But in 1965, a new group was found that didn't fit into any of the previous families, and like the athletes who found themselves able to do a four minute mile once one person had done so, mathematicians were inspired by this discovery into finding further such groups. The largest of the groups is called the Monster. "The Monster is like some huge, great symmetrical snowflake that you can see only when you get to 196,883-dimensional space," says John Horton Conway, one of the heroes of the quest described here. Forget the six symmetries of that equilateral triangle; there are more symmetries in the Monster than there are atoms in the Sun. Even more wonderful is that there are mathematicians like Conway who somehow can picture such an object in their minds, and have included it and all the other possible symmetries in the massive, inclusive _Atlas of Finite Groups_. If the math (even at the level of this popularization) gets dense, there are always funny stories about mathematicians, who are often just as strange as you would expect such eggheads to be.
Symmetries is not a textbook on symmetry, and the concepts here require not only textbooks but mathematical textbooks, textbooks only a few people, even a few mathematicians, are ever going to read. What du Sautoy has done is to give a feeling of the importance of the hunt and the excitement of it. He has also done readers the service of explaining just what a mathematician does. "Many of my friends have the impression that I'm sitting in my office doing long division to a lot of decimal places, and wonder why a computer hasn't put me out of a job by now." It is hunting for patterns and discovering them that keeps him going. The twelve chapters here not only go through the history of group theory but each tells of a month in his life, the joys and frustrations of mathematical research, of mathematical teamwork and competition, or of the push to publish one's ideas. Maybe you will never master group theory, but if you want to know what mathematicians do, and why it is important, this is a wonderful guide.
The book could have used a little tighter editing--some glaring typos in the math and imprecise language were a distraction at times, but only because of the high quality of the book as a whole. For me it was a pleasure to read!