On his journey north, Basho stopped at Matsushima and was spellbound by its beauty. The town itself was small, but the bay is studded with some 260 tiny islands. The white stone of the islands has been eaten away by the sea, leaving a multitude of endlessly different shapes, pillars and arches, all crowned with pine trees. Each island is different and unique, and each, with its sculpted white cliffs tufted with pine, is beautiful. It is, however, the whole bay, the combined diversity, that ultimately makes Matsushima one of the three great scenic locations in Japan. So far on our journey we have been admiring the beauty of the islands; the subtlety and intricacy of the different algebras that arise from different patterns, different symmetries. It is time to step back and begin to admire the whole — and in so doing gain a deeper and richer perspective on all the sights of our journey so far.
In the previous entries Shifting Patterns and Permutations and Applications we have looked at how patterns and symmetry can be abstracted into algebras. Each different pattern provided its own unique algebra, with different algebraic rules. These are, in a sense, our islands; each different and unique and beautiful. What we need to explore now is the entire bay. To do this we need to take a step back, or more accurately up, and try to examine the whole. The aim is to abstract across all the different algebras that different patterns generate. We could, perhaps, liken this to the process of developing algebra from numbers by abstraction; here, however, we have different pattern-algebras standing in place of different numbers. In abstracting up from numbers to basic algebra we looked for those properties that held true regardless of the particular numbers under consideration — each different number has its own unique properties (indeed, there is a vast richness of material here that is studied in number theory) — but there are certain properties that are common to all. What we seek is a set of basic algebraic rules that are true for all pattern-algebras; each different pattern algebra may have its own idiosyncratic rules, but hopefully we can find a basic set of underlying rules. In practice we’re going to hope for even more than that. Not only do we want the algebraic rules we determine to be common to all pattern-algebras, we would very much like it if any algebra that satisfied the basic rules turned out to be the pattern-algebra for some pattern. That is, we want a two way correspondence: every pattern-algebra should satisfy these rules, and everything that satisfies these rules should be a pattern-algebra.
So where to start? First and foremost we know that every pattern always has the null symmetry — the do nothing symmetry: doing nothing to a pattern will always preserve the pattern. Thus it follows that our rules should somehow express this fact. The question then is how to express the existence of a null symmetry in terms of algebraic rules. That, in turn, leads to the question of what algebraic rules make a given symmetry element a null symmetry. Now, the algebraic rules we can have for pattern-algebras are all expressed in terms of combining together different symmetries, so a rule for the null symmetry will be one that expresses how it interacts with other symmetries in combination. How does the null symmetry interact with other symmetries? Since it is the “do nothing” symmetry, combining it with other symmetries gives the same result as if we hadn’t done it at all. That is, we know that a symmetry e is a null symmetry if, for any symmetry a we have
ae = ea = a
Thus our first requirement, or rule, for general pattern-algebras is that there must be some element e of the algebra such that ae = ea = a for any (i.e. every) element a of the algebra.
Next it would be nice to start to characterise the symmetry aspect of the pattern-algebras. Ultimately this can be reduced to just a couple of properties, though such a reductionist approach does diminish the transparency of the relationship to symmetry. The first, and easier to grasp, property boils down to the reversibility of symmetries. That is, if a certain action is a symmetry, then doing it’s opposite and taking everything back to how you started is also a symmetry. This is perhaps a little non-obvious since the “opposite” action doesn’t appear to necessarily be a pattern preserving symmetry in its own right; however, since the the result of our initial action results in a state that preserves the pattern, and the initial state to which we go back via the “opposite” action also necessarily preserves the pattern (by definition essentially) it follows that the opposite action necessarily takes pattern preserved states to pattern preserved states (that is, as long as we start in a pattern preserved state, we’ll always end up at another one) and is thus a symmetry. Looking at it from a broader perspective this is really a relatively deep statement about the nature of what we are calling symmetries: they are actions that move from one one state to another preserving some pattern that we have deemed it relevant to conserve; it is in the nature of such actions that they be reversible and that the inverse or opposite action is also a pattern preserving action. How do we write that in terms of algebraic rules? The opposite action is going to be one that combines with initial action to result in effectively a null action. Thus our second requirement is that for any symmetry a there exists some symmetry b such that
ab = ba = e
where e is the null symmetry whose existence we were guaranteed by the first requirement.
The final property we require is the hardest to explain in clear practical terms. It will be easier to just state it, and then try and discuss exactly what it means, and why it might be relevant. The property we require is that for any three elements of the pattern-algebra, call them a, b, and c, we have
a(bc) = (ab)c
This is the associative law which you should recall from A Fraction of Algebra. What we are essentially saying by applying it here is that how we group together composition of symmetries is unimportant to the end result. That this is true of symmetries is relatively clear: given a sequence of symmetry actions the order of the actions matters, but how we group them does not; we can think of some pair, or group of actions in the sequence, as a single atomic action and the end result will be the same. For a more explicit example of this we can think of our example of the symmetries of a square from Shifting Patterns. There we expressed things in terms of two basic actions: a rotation by 90 degrees, r; and a flip about the vertical axis, f. These combined to provide other symmetries, for example fr was the symmetry action of flipping the square about its trailing diagonal. Now, given a sequence of such actions, it didn’t matter whether we thought it as a diagonal flip about the trailing diagonal axis followed by a rotation by 90 degrees, or a flip about the vertical axis followed by a rotation by 180 degrees, or simply as a flip about the horizontal axis; all amount to the same result, the same rearrangement of corners of the square. Stretching your mind to abstract this to general symmetries will let you see that they too will have the same property. Seeing that this property is the last piece we need to characterise symmetries is a little harder, and perhaps beyond the scope of this entry. Suffice to say this third requirement is the last one we need.
There are, of course, a few unstated assumptions that we’ve been getting away with here. For the purposes of informal discussion that’s fine, but when we get into hammering out the specifics for mathematical purposes we can’t afford to let such ambiguity stand. So, let’s spell out these final details…
In assuming that there are symmetries of pattern we are assuming that there are a set of actions, that is, rearrangements, and that those actions are composable — that we can combine two actions together one after the other. In terms of our pattern algebra that amounts to assuming the existence of some algebraic objects, which we can denote by letters (or other characters if we prefer) and some sort of binary operation for those algebraic objects. A binary operation is essentially just a rule that allows us to combine together any two elements and arrive at a third. Addition is a binary operation on numbers; you add together two numbers to get a third. Likewise multiplication is a binary operation for numbers; you multiply two numbers together and get a new number. So what is the binary operation for pattern-algebras? There isn’t one in particular. Indeed, the binary operation is essentially defined by the algebraic rules unique to that pattern-algebra, so different pattern-algebras have different binary operations. There are a set of rules that help narrow down binary operations for pattern algebras in general, and those are, of course, precisely the rules we’ve been discussing previously. What matters, however, is that there exist some binary operation that can act on any pair of algebraic objects.
Putting all of this together we can arrive at a formal description of what it takes to be a pattern-algebra. Using abstraction to pare it down the the minimum set of requirements we have: a pattern-algebra is a set of objects, with a binary operation on those objects, such that the following hold:
- There is an object e such that, for any other object a in the set, ea = ae = a.
- For each object a in the set, there is some object b, also in the set, such that ab = ba = e (Where e is the special object mentioned in requirement 1).
- For any three objects a, b, and c in the set, we have (ab)c = a(bc).
and that’s it. This sort of very abstract and minimalist definition is exactly what you’ll find at the very beginning of most books on Group Theory, since this is the formal definition of what mathematicians call a group. Indeed a group is really just a pattern-algebra, though in the more rarefied areas of group theory the patterns they relate to can be so hideously complex as to be effectively unimaginable. Since using the same terms as mathematicians will help keep us on the straight and narrow when referring to any outside sources I’ll henceforth be using “group” to mean the sort of pattern-algebra we’ve been discussing throughout Shifting Patterns, Permutations and Applications and this entry. You can, of course, mentally translate “group” to mean “pattern-algebra” to help keep the mental connections to patterns and symmetry clearer.
So what was all this abstraction for? What exactly have we gained by reducing things to this very abstruse definition in terms of sets and binary operations and algebraic rules? One may as well ask what the point of abstracting over different collections of objects to arrive at the abstruse notion of numbers and arithmetic is. We’ve dropped away all the fine grained particularity, and reduced things to a simple matter of clearly defined rules. Any set of objects with a binary relation that meets these three simple rules can be thought of as a group relating to some pattern. We don’t have to know about the pattern however, we can simply work within the specific rules of the group. More importantly, however, we now have an overview of the entire bay of islands rather than finding ourselves inspecting each unique island. These three simple rules provide the minimal basis for any group. Any group must have at least these three rules, and we can add any extra rules we like to these three base rules (as long as the resulting set of rules is self-consistent) and arrive at a new group. We aren’t bound by patterns any longer, we can explore the unique complexity of any island in the bay by simply building the group of our choosing; by imposing the extra rules, the extra structure, we wish. We need only choose a set of some size, and rules regarding how elements interact with one another.
This, in turn, begins to draw us full circle. The perceptive reader may have noticed that each of the three rules defining a group is listed amongst the algebraic rules for addition of numbers back in A Fraction of Algebra; we had the existence of an additive identity (the number zero) which covers requirement 1, the existence of additive inverses (negative numbers) which covers requirement 2, and the associativity of addition which covers requirement 3. Furthermore, you will hopefully recall that, as discussed in The Slow Road, higher order operations such as subtraction, multiplication, and division, could be built from addition. Numbers form a group; they are a pattern-algebra!
What pattern does the group of numbers under addition relate to? If we are thinking of the integers then you can think of an infinite line of marbles (infinite in both directions), with the various symmetries being horizontal shifts of varying sizes. The fact that such a shift is indeed a symmetry, that it results in exactly the same pattern of marbles, is something we discussed and resolved in our parallel road considering the nature of the infinite. And so now, having scaled the heights, we find we can look back, out across the vast bay with an infinite number and diversity of unique and interesting islands, and see that what we had thought of as the vast plain of numbers with all its intricacies, is, in fact, just a single island amidst a sea of many many more. Certainly there is much to explore on that original island, but that is just the beginning; we now have the perspective to see how much wider the world is; to put our narrow beginnings in their proper place. And still we have only just begun! There is a vast bay of islands yet to explore, and once we have begun to comprehend some of their mysteries we can strike out for higher ground again from which we can look back and see even how narrow our current vista is.