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Recently I've been reading about symmetry and group theory. About a year ago I tried to teach myself group theory from a set of notes I found on the web (well actually a SUPA course, for those that know of it) but couldn't figure out what the point of it was. Now I have a much better idea of what group is and how it works. Group theory has many interesting applications and is the language of something fundamental about the universe: it is the language of symmetry.
So my (most) recent exploration into symmetry started when a friend gave me a copy of Fermat's Last Theorem. He hadn't read it but didn't want to carry it back home to the Netherlands (apparently he has a dutch version at home), so he gave me it and I read it the same afternoon. That was the shortest time that I've completed book in. A fascinating book about an incredible tale of mathematics and a recurring theme of romanticism. By romanticism I don't mean that crappy soppy garbage called 'Romance' that you find in the book store next to the sci-fi section (is that a rule of book stores?) but rather romanticism proper like the romantic era which is full of tales about tragedy and Faustian dichotomies. Fermat's Last Theorem was the longest unsolved mathematical puzzle (358 years) but was solved in 1995 (completed and correct in '95?) by Andrew Wiles. A great achievement in itself and hence very noteworthy but the story surrounding it is also worthy of note. Many prominent characters of the story come to a grim end: Niels Henrik Abel dies at an early age from disease, Evariste Galois at the age of 20 (became one of the most important mathematicians) was killed in a duel by a compatriot during the times of the French Revolution and the suicide of Japanese mathematician Yutaka Taniyama. The book about this tale by Simon Singh is definitely worth reading and not solely for the mathematically inclined. My fascination with the mathematics as well as the story inspired me to find more books about Galois, Group theory, Symmetry and Fermat's Last Theorem. So while on vacation I took a trip in a book store, I briefly glanced over the Sci-Fi & Fantasy section as well as the Physics & Astronomy section before settling into the Mathematics section for quite some time. I picked out three books that seemed interesting: "The equation that couldn't be solved" by Mario Livio, "Fearless Symmetry" by Avner Ash and Robert Gross, and finally "The Poincare Conjecture" by Donal O'Shea. I've read the first book in its entirety, half of the second book and nothing of the third. The second book is heavy mathematically but culminates in explaining the proof of Fermat's Last Theorem, the third book is more obscure and something I'm less familiar with. However, it is the first book that I'd like to talk more about... I realise that this article is reading like a blog post but I'll try to cut down on the Blogianity from here onward. To appreciate the usefulness and power of group theory and its role in describing symmetry then try reading "The equation that couldn't be solved" by Mario Livio. I now have a much clearer picture of both. Livio writes "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." Symmetry seems to be a fundamental property of our universe although, as Livio notes, that could be a selection effect (a bias/preference within our results). The book shows the history of Group theory coming from trying to solve the quintic polynomials. The French mathematician Evariste Galois is the one that invented it: in doing so he was able prove that quintic polynomials could not be solved by a general formula that relied upon the 'normal' algebraic operations and the roots of the polynomial in equation. The creativity used in developing this method is a 'proof' of something that Schopenhauer said: "Talent can hit a target that no one else can but genius can hit a target that no one else can see." Talent, is something I equate to skill. If you train harder enough then you can improve your skill level (hit those hard targets); unfortunately genius is not something we can train. Genius may be ill-defined but Schopenhauer's comment still stands: Galois's breakthrough makes sense in hindsight but few have the ability to devise it from scratch. Symmetry is present in many classical works of music (eg Bach and Mozart), in DNA, animals (humans have vertical bilateral symmetry) and in fact most objects in the universe (at least approximately). Furthermore, it appears to be of utmost importance in evolutionary biology: predator detection as well as finding a mate. I'm going to rely upon some of the ideas from the later chapters about evolutionary biology to suggest something bold, something that suggests to me a way to differentiate humans from the rest of the animal kingdom: Individuals are supposed to select the most appropriate mate from the gene's point of view. Most appropriate being those genes that increase the chances of survival and reproduction, id est high quality of genes and capability of parantal care. Without having a DNA-testing kit then animals resort to symmetry; attractive mates are more symmetric in appearance. So my point? Humans can, and do, go against the notions of symmetry: we don't chose partners of highest biological quality but pick the one we feel 'happiest' with at that moment in time. I doubt this is true for other animals but I can't substantiate this claim yet. The answer is inductive. That said, the animals most like humans would be my guess about where to look to try and disprove my suggestion (it is only a suggestion/conjecture at the moment). All of which corroborates with the fact that humans are massively imperfect and prone to making a mess of everything. [/misanthropy] As far as music goes, I'd wager that symmetry is present in all forms of popular music. It helps to make it popular by being easy to listen to and easy to understand. A cyclic structure of A-B-A-B, or verse-chorus-verse-chorus, is simple and a demonstration of translational symmetry. As Livio expounds in his book: symmetrical patterns are easy to recognise. Harmonics are, as Plato noted, a statement of symmetry too. Then it comes as no surprise that easy to digest music like Britney Spears is popular. Bach is harder to 'digest', it is harder to understand. Why? because the symmetry is less obvious; symmetry exists in Bach's music but it isn't as simple as A-B-A-B. I recently spotted a video on Youtube of an electronic version of Bach's Toccatta and Fugue that highlights translational well (synchronization is out). In physics: symmetry implies a law of conservation, that is something stays constant and 'unchanging'. From dictionary.com, Greek summetri?, from summetros, of like measure : sun-, syn- + metron, measure; we see that symmetry is 'of like measure': eg comparable in size or shape. All of the previous considerations suggest to me that agreement in debate and the type of personalities that we bond with best are those 'of like measure'. That's obvious you say but we should have a tool of how to logically analyze agreement and attraction (not just physical) between two people. Such tools already exist. The political compass and the Jungian tests are constructions, ways of displaying information, that aid people's ability to see agreement. This is something that I will explore in another post. For now I leave you thinking about the importance of symmetry. :-)
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