October 25 was the birth date of Évariste Galois(October 25, 1811 – May 31, 1832). Who he was? Well! he was Albert Einstein of Algebra. You can easily find people (And I am beginning to realize this) who consider Galois to be one among the greatest geniuses who ever lived (Probably among top 5 in the ranking). His out of the box thinking resulted in a completely new branch of Mathematics. I would not go into much detail here, but to tell you at an abstract level, he used group of permutations to show that any general polynomial equation of degree 5 (In fact Abel proved it only for the case 5) and above cannot be solved by any formula in radicals. What I mean is that you cannot have something of
[-b±√(b²-4ac)]/2a
as the value of roots in a general polynomial equation of degree 2(quadratic equation) for any polynomial of order 5 and above.
Not only this, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals.
Now if you would suppose that reasoning for such a proof will require and consist of only manipulations of coefficients and variable, you are completely mistaken. No sir, he used permutations of putative solutions. This in itself is out of the box thinking and extraordinarily a work of a genius. Who would have thought of such a thing. Many greats before him and Abel tried to find such formulas(in radicals) for degree 5 and above, and failed gloriously.
If you are wondering what a group(Group Theory) really is: it is just a set with a defined operation. The members of this set follow 4 rules with respect of this operation:
1. Closure: The combination of any 2 members by the group operation has to produce another member of the set.
2. Associativity: When three ordered members combine, the result does not depend on which two are combined first.
3. Identity: There exists one such a member, that when it is combined with any other member by the group operation, the operation itself results in second member. This is true even when the order of Association is reversed. This is the only member, with which commutativity is guaranteed to work.
4. Inverse: Every member when combined with its inverse by group operation, the operation itself results in identity member. So there is an inverse member for every member with respect to the group operation.
Group Theory is also something which at an elementary level allows you to understand symmetry at a greater abstraction, and not just mirror symmetry(reflection symmetry). A thing is symmetrical if there is something you do to it and after you have finished doing it, it looks exactly the same as before (This is not an adult talk :-P).
For example an equilateral triangle is symmetric with respect to n(π/3) rotation, where n is an integer.
Some of the kinds of symmetry one can expect:
1. Reflection Symmetry: As in Butterfly
2. Rotational Symmetry: The symmetry of an equilateral triangle, when we rotate it by n⨯60° (n[π/3]).
3. Helical Symmetry: Screw movement.
4. Glide Symmetry: reflection in a line or plane combined with a translation along the line in the plane
5. Magnification Symmetry: Scale Symmetric under magnification, as in fractals.
6. Translation Symmetry: Repeating Patterns on a bed sheet or a saree.
One can argue that if modern definitions of Symmetry and Groups were known at the time of Sir Issac Newton, probably we would have needed less of Einstein. The above statement only emphasizes the importance of identifying symmetry in natural phenomenons and operations.
I would not go any further, so please read some books to explore more on this subject. You can start with a book titled "The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry" by Mario Livio. This book will definitely set you on the right course, but it is not an exhaustive treatise on the subject. Still it is a great inspiration for Non-Mathematicians like me and inspires us to learn more of pure Mathematics, exhaustively.
![[info]](http://l-stat.livejournal.com/img/userinfo.gif){kind=small}
contemplative
naughty