Symmetry: A Journey into the Patterns of Nature
Marcus du Sautoy
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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.
shuffles. Interestingly, the cube has the same number of symmetries as the group of symmetries used by Schoenberg and by Dorothy to create their matrix of notes. But the underlying symmetry groups are completely different in structure. The group of symmetries Xenakis is exploiting is called C2 × S4, or the direct product of the cyclic group of order 2 with the symmetric group of degree 4. Xenakis extends his idea of symmetries in music in his composition for 98 instruments, Nomos Gamma, in which
direction, and we never hear a piece played backwards. A mathematical proof too is a very linear work, with its own logical arrow of time. Yet the beauty in a piece of music often reveals itself only when you listen to it again. I often need to hear a piece of music many times before it makes any sense to me. I begin to appreciate how themes later on in the piece are echoing or mirroring ideas I heard earlier. I read mathematical proofs in exactly the same way. My first reading rarely gives me a
the four corners of a square can be described by their coordinates: (0,0), (1,0), (0,1) and (1,1). And similarly in three dimensions: one just adds another coordinate. For example, the eight corners of a cube can be described by eight triples: (0,0,0), (1,0,0), (0,1,0), and so on, up to (1,1,1) (Figure 6). The coordinate (1,0,1) locates or encodes a point on the three-dimensional cube reached by travelling one step east and one step vertically upwards. Fig. 6 Changing geometry into numbers: a
information age. Deep in the heart of this paper, Shannon let slip a description of the simplest of Hamming’s codes, the one that used three check digits to correct errors in messages of length 4. The delay caused by the patent application and the leak in Shannon’s paper allowed another mathematician to slip in and publish his discovery of these codes before Hamming could get there himself. Born in 1902, Marcel Golay was educated in Switzerland before moving to the United States in 1924. His
creative process, doing mathematics can often feel like a theatre improvisation. You set up a tableau with conditions for collisions of ideas and then let the thing run. Very often it goes nowhere, but sometimes there is a dynamic created that clicks. Like the rules of a theatre game, the conditions push you in extraordinary, unexpected directions that too much freedom would stifle. When the producer and director Peter Brook talks about his work in the theatre, he could easily be discussing the