## S is for Symmetry

If your basic education was like mine,  you learned about symmetry in elementary school, and it was pretty much limited to telling which shapes were symmetric and which ones weren’t.  Of course symmetry isn’t just a matter of yes-or-no, and some objects are more symmetric than others.  A square is more symmetric than a rectangle, say, and a circle is more symmetric than either.  (This can be made precise, of course; in this case, even the crudest way is adequate: a square has 8 symmetries but a non-square rectangle only has 4, and a circle has infinitely many.)

And as the previous two posts show, symmetries are not just geometric in nature.  Structures of all sorts, in wildly varying contexts, have interesting symmetries; moreover, these symmetries can explain some strange phenomena and put them into their proper perspective.

A good working definition of symmetry is a transformation of an object which preserves its important features (the shape of a geometry object, the algebra of a number system, etc.)

You may not like my count of 4 symmetries for a rectangle.  For the record, these are the four.

1. flip it over horizontally
2. flip it over vertically
3. do both (in other words, half-turn)
4. do nothing

Doing nothing certainly is , and it turns out to just be better to include it in our lists. For one thing, the numbers 4 and 8 are more suggestive of the notion that a square is “twice as symmetric” as a non-square rectangle than 3 and 7 would be.  There’s a better motivation, which we’ll see next time.  For now, let’s just agree that every object has at least the trivial symmetry, and that to be “symmetric” means to have at least 2 symmetries (at least one that is interesting).

So if you have only one symmetry, it’s the trivial one, and you’re asymmetric. Just like if only one person shows up to your birthday party, it’s you, and you’re lonely.

So we can classify objects based on how many symmetries they have.  Loosely speaking, the more symmetric something is, the more symmetries it has.  Actually there’s a lot to learn even from this low level of sophistication.

Lots of things in life have exactly two symmetries.  The human face, the grid of a traditional crossword puzzle, the shape of a piece of bread, etc.  And this particular sort of symmetry, which seems to be intrinsically aesthetically pleasing to most people, means that these objects all have something deep in common.

But we can say more.  For example: how might you compare and contrast the following shapes?

Well, both have four symmetries, assuming we’re counting colors. Here are the lists (left first).

1. quarter-turn
2. half-turn
3. three-quarter-turn
4. nothing (or a full turn)

And for the right square:

1. flip it over horizontally
2. flip it over vertically
3. do both (in other words, half-turn)
4. do nothing

But these collections of symmetries have a deeper difference.   If you just look at the shapes a minute, move them around in your head, you’ll probably notice they “feel” different.  It’s hard to nail down why exactly, but we can.

In the purple/cyan case, there was really only one kind of symmetry—rotation.  All the symmetries just come by repeating the basic quarter turn.  But there’s no one symmetry of the blue-green square that gives rise to all the others.  Also, every symmetry of the blue-green square has the property that if you do it twice, you’re back where you started, and the purple-cyan squre has other kinds of symmetry.

If this seems like this is too confusing or somehow too “high-level” mathematically, then as my six-year-old daughter always says, be brave in your heart.  I firmly believe that, like the mathematical abstraction that is counting, the mathematical abstraction that is symmetry is something that is instinctive for humans.  The deficit is not in human faculties, rather it is in human language.  Standard English provides a pretty poor language to talk about these issues.  (The analogy with our sense of smell, I think, is apt; experiment suggests that our latent sense of smell is as good as any dogs, but we think about what we talk about, and our words to differentiate smells are pretty limited and clumsy, so our sense of smell is highly impaired in practice, but not by lack of latent ability.)  But it turns out that mathematics, like it or not, is an ideal setting to verbalize gut notions of symmetry.  Like that smell you can’t quite identify and don’t know how to describe to your friend, like why you enjoyed that movie much, like the difference between how normal coffee and decaf taste, like so many things in life, symmetry seems ineffable.  But as I always say, eff the ineffable. Mathematics provides a comprehensive language for formulating and expressing ideas about symmetry, the language of group theory.  Stay tuned for the next post, where I’ll talk about this extensive language mathematicians have developed to discuss and understand symmetries.

So I’ll concede that I was forced into awkward language in that paragraph distinguishing the two shapes, but I think it’s fair to say that we understand the difference between the two shapes better than we did before.  The mind’s eye knows they’re differently symmetric, and now the mind’s mouth can make that precise.

Symmetries are one of those things that, once you know to look for them, you see them everywhere.  There are lots of important examples of symmetry issues in physics, but let’s look at a very simple one: the so called arrow of time.  That is, does time have an intrinsic forward and backward?  There is the direction that we think of as forward, but is there an intrinsic difference?  Of course at the level of human experience, they are different.  I remember the year 2000 but not the year 2020.  But at the level of physical laws, it’s much less clear.  If I showed you a movie of interactions of electrons, would you be able to tell whether you were watching it playing forward or backward?  Do you see how this is like the $i$ vs. $-i$ question from a recent post?  The question is this: is there a symmetry of space-time which interchanges past and future, but preserves physical laws?  If there isn’t, then that means the arrow of time, our perception of the direction it flows, is intrinsic to the universe.  If there is, then it’s perfectly plausible to imagine, say, some other creatures which perceive themselves as moving through time the other way.  What’s funny here is that on a small enough level, say at the level of subatomic particles, most theories have past-future symmetry (an electron can gain a photon, or it can lose a photon, which is like gaining a photon backwards).  But on the larger scale of time and space, we do not see past-future symmetry.  Thermodynamics, for example, is not symmetric.  Entropy increases over time.  If I made a video of a breaking egg, you’d know which way was future.  Eggs break, eggshells don’t assemble.  The correct reconciliation of these ideas is, I think, an important open issue in physics.

There are also examples that are far less serious.  Ever play rock-paper-scissors?  Against a computer that just picks its move at random with equal probabilities? (You can do just that at eyezmaze, a site with outstanding games, if you don’t count this one.)  If you have, then you probably lost interest pretty fast, because you realized that your decisions don’t matter.  And why don’t they matter?  Because rock-paper-scissors has symmetries, three symmetries which preserve the rules about which throws beat which and which also preserve the computer’s “strategy”, just enough to interchange all the possible throws and guarantee that rock, paper, and scissors are always exactly equally good throws.  This is different from RPS against a person, which is interesting, because your opponent’s psychology doesn’t have symmetry.

Okay let’s stop there, because if you’re me it doesn’t get any better than explaining in precise mathematical terms why one game is more fun than another.

P.S. (at least a bit heavier than what precedes)

Symmetries are at the heart of the so-called Erlangen Program for geometry developed by Felix Klein. The sound-byte version of which is “If you want to understand a geometry, understand the symmetries that preserve it.”  In the case of ordinary plane geometry (the kind you learned in Mrs. Gunderson’s algebra class in high school), this means understanding the transformations of the plane which preserve lengths and angles.  There are some obvious types of transformations that work, such as the follow.

• rotations around a point
• reflections across a line
• translations

It turns out that all the symmetries of the usual geometric plane are given by rotations or reflections, possibly followed by a translation.  All the fundamentals of Euclidean geometry can be recovered by really understanding these families of symmetries.  You may have heard of something called hyperbolic geometry, where parallel lines behave differently.  How might someone get a concrete handle on how the hyperbolic plane works?  We can characterize its symmetries, and see that this plane has different kinds of symmetries than the ordinary Euclidean plane.  And when I say compare them, I don’t mean that in a fuzzy, hand-wavy way. All these symmetries can be expressed in concrete numerical ways (using matrices).  The power of the method leads to an increase importance of understanding various matrix groups (whatever that means) in geometry, and a closer relationship between algebra and geometry.  But this is a subject for another day.