Oxford's Very Short Introductions are concise introductions to a wide gamut of topics. Currently available titles cover advertising, Anglo-Saxons, British politics, the Dead Sea Scrolls, folk music, and “stars”. This entry into that intriguing list is a basic, even cursory, overview of the group-theoretical underpinnings of symmetry. I imagine it breaks the stride of voracious VSI readers. A reading of online reviews proves that to be the case: “…not a beginner's book”, “…concepts difficult to follow”, “tough slogging”, and even “abandon all hope” are all typical comments from readers.
In fact, Stewart supposes very little mathematical sophistication of the reader. Giving “the mathematical skeleton of the argument”, he reduces game-theoretic analysis of rock-paper-scissors to a grocery list of logical facts. When matrix algebra is about to make a cameo in defining symmetry, Stewart casually remarks “we won’t go into that.” Earlier, in defining rotations, he admits that “There’s a fancy set-theoretic definition: if you know it I don’t need to say what it is, and if you don’t, you know enough already without it.” If you find yourself outside the clubhouse, but wonder what goes on where “Galois Theory” is mentioned, I know of no other introduction as succinct and even at times delightful.
Topics such as permutations, sinusoidal waves, Rubik’s Cube, and more benefit from frequent, clarifying diagrams and tables. This includes knot theory and topology wherein symmetries arising from topological invariance are explored. This leads to sections on nature’s laws and patterns of symmetry with a particularly rich and fascinating discussion of gait among quadrupeds. Before shifting to the shifting sands of desert dunes, the cyclic group symmetry of gait patterns must be set aside “for reasons too extensive to get into here.”
For reasons too extensive to get into here, Tom Schulte is a lead software engineer by day and instructor in mathematics by night.
1. What is symmetry?
2. Origins of symmetry
3. Types of symmetry
4. Structure of groups
5. Groups and games
6. Nature's patterns
7. Nature's laws
8. Atoms of symmetry