In Part III of our series on symmetry, we introduce symmetry groups, which fall within a branch of mathematics known as group theory. According to this Wikipedia document, "In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element." A very simple example of a group, as noted on the Wikipedia page, is that of ordinary numbers (integers, technically), with addition as the operation: "For any two integers a and b, the sum a + b is also an integer. In other words, the process of adding integers two at a time can never yield a result that is not an integer. This property is known as closure under addition."
This overview document from Minnesota-Morris professor Barry McQuarrie introduces mathematical groups and uses the example of how equilateral triangles can form a symmetry group of operations (turns) that leave the appearance of the triange unaltered.
Over the last year or two, I've been reading books on symmetry that I hoped would be accessible to a non-physicist, non-mathematician such as myself. When one reads a book on symmetry, one inevitably seems to confront the concept of Galois groups. It took a while, but I think I finally understand the general concept. At a very basic level, Galois groups can inform someone performing mathematical calculations if a given equation is solvable.
In particular, Galois groups are focused on algebraic equations with x raised to different powers. An equation with x-squared (x^2) as the highest power is known as a quadratic equation and can be solved using the quadratic formula. Mark Ronan's book Symmetry and the Monster contains a chapter ("Galois: Death of a Genius") that traces the history of how the quadratic formula was followed hundreds and thousands of years later with formulas for solving cubic (x^3) and quartic (x^4) equations. The ideas of Galois enter at the level of quintic (x^5) equations, as summarized in Ian Stewart's book Why Beauty is Truth: A History of Symmetry:
Now we begin to see the beauty of Galois's idea. Not only does it prove that the general quintic has no radical solutions, it also explains why the general quadratic, cubic, and quartic do have radical solutions and tells us roughly what they look like. With extra work, it tells us exactly what they look like. Finally, it distinguishes those quintics that can be solved from those that can't, and tells us how to solve the ones that can (p. 116).
As noted in the above-linked Wikipedia document, solving equations "by radicals" refers to "solutions expressible using solely addition, multiplication, and roots."
In other words, some quintics, such as the very simple x^5 = 1, can be solved (x = 1). Among other quintics, some can be solved by radicals and some cannot. Galois groups can distinguish the solvable from the unsolvable -- on the basis of symmetry. John Baez, who writes an online series, "This Week's Finds in Mathematical Physics," wrote the following in an entry about Galois and symmetry:
The basic idea is something like this. In general, a quintic equation has 5 solutions - and there's no "best one", so your formula has got to be a formula for all five. And there's a puzzle: how do you give one formula for five things?
Well, think about the quadratic formula! It has that "plus or minus" in it, which comes from taking a square root. So, it's really a formula for both solutions of the quadratic equation. If there were a formula for the quintic that worked like this, we'd have to get all 5 solutions from different choices of nth roots in this formula.
Galois showed this can't happen. And the way he did it used symmetry! Roughly speaking, he showed that the general quintic equation is completely symmetrical under permuting all 5 solutions, and that this symmetry group - the group of permutations of 5 things - can't be built up from the symmetry groups that arise when you take nth roots.
The moral is this: you can't solve a problem if the answer has some symmetry, and your method of solution doesn't let you write down a correct answer that has this symmetry!
An old example of this principle is the medieval puzzle called "Buridan's Ass". Placed equidistant between two equally good piles of hay, this donkey starves to death because it can't make up its mind which alternative is best. The problem has a symmetry, but the donkey can't go to both bales of hay, so the only symmetrical thing he can do is stand there.
Buridan's ass would also get stuck if you asked it for the solution to the quadratic equation. Galois proof of the unsolvability of the quintic by radicals is just a more sophisticated variation on this theme. (Of course, you can solve the quintic if you strengthen your methods.)
Not that solving quintics is easy. According to a different Wikipedia page, this one on quintics, sometimes “the solution requires about 600 symbols to write.”
I urge interested readers to check out the websites and books I've referenced. One other book, which I found too technical for my own knowledge level but others of you might find worthwhile, is entitled Fearless Symmetry: Exposing the Hidden Patterns of Numbers, by Avner Ash and Robert Gross.
Next time, we'll look at another type of symmetry group, Lie groups.