S is for Symmetry

29 October 2009

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?

Two Colored SquaresWell, 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.

Thought Experiment: Talking to the Other Aliens

22 October 2009

This is a direct continuation of the previous post, so read that one first if you haven’t yet.  In some sense this post is simpler than the previous, in that it uses simpler concepts and doesn’t involve understanding of the real number system.  But it may be harder for many readers, because I’m asking you to imagine an alien race which does not understand certain things that you probably can’t remember a time when you didn’t understand.  And it’s hard to imagine what it would be like not to know what we know.

It’s interesting, isn’t it, how people are much better at temporarily adding an unfamiliar concept to their working context than they are at temporarily subtracting a familiar one?

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Thought Experiment: Talking Math with the Aliens

20 October 2009

Though the connection may not at first be apparent, this is part of my promised (threatened?) attempt to put the fundamentals of Galois theory in terms suitable for readers of this blog.  It will be a slow build, because there are a lot of a pieces to put into play.

Today, a thought experiment.  Imagine you have made contact with another form of intelligent life.  Communication is still at a primitive stage, but you’ve devised a way of sending each other signals, and you and the alien are in the process of building up your shared vocabulary in this new language.  (I’m imagining some sort of IM window, your imagination may vary.)

Well you’ve heard that the universal language is mathematics, and you want to establish a shared vocabulary for basic math.  With some effort, you establish an agreement on the concepts of “addition” and “multiplication” (think about how you might do this, how you might distinguish these two operations from one another).  You figure out what name they have for what you call “zero” and “one” easily enough.  (For example, you could ask what number plus itself equals itself to nail down zero, then ask what number multiplied by itself equals itself, other than zero, to nail down one — think about it.)  Once you have zero and one, addition and multiplication, you can get 2, 3, 4, etc., then the negative integers, and then fractions.

It would take some time, but suppose you eventually get sufficient communication to have shared language for the real number line (maybe you explain Dedekind cuts, whatever, I don’t care). (Actually this isn’t essential, and it’s just as interesting to suppose you don’t establish shared vocabulary for the real numbers; we’ll explore that elsewhen.)

So now you’re feeling ambitious, and you want to know how the alien talks about imaginary numbers.  What does the alien call your i?  You assume (reasonably) that such a developed race would also have some corresponding concept, so you ask for a number which multiplies by itself to give negative one, and the alien says “blarg: blarg times blarg plus one is zero”.  Victory!

But then doubt sets in.  Are you really sure his blarg is your i?  After all, (-i)^2=-1 too.  Maybe blarg is negative i?  How would you know?  Think about it as long as you like, but the answer is, you wouldn’t.  There are no questions you could ask that would say for sure whether blarg was i or -i.

(You might try to say something about “the one on the upper half of the complex numbers”, but that’s no good.  You have no reason to believe that they visualize complex numbers anything like how you do, and anyway that distinction is happening only in your mind, not in the math.  It’s no more constructive than defining “three” as “the number that looks like half an eight”.  That’s not math, not even arithmetic.  It’s trivia about our way of writing numbers.)

We could rephrase this whole thing without aliens (but why would you ever prefer not to include aliens?).  Suppose that I had misunderstood my teacher the day she defined the complex plane; suppose I had thought that i was one unit below the origin, the opposite of the convention you’re probably used to.  What would happen when I try to talk math with the people like you who learned it the usual way?  Nothing interesting!  You and I believe all the same statements about numbers!  We both think (3+2i)+(4-i)=6+i and we both think (3+2i)(4-i)= 14+5i.  If we visualize these facts geometrically, then the picture in my head doesn’t match the picture in yours (it’s upside down).  As long as we stick to the numbers and equations, as long as nobody explicitly mentions the pictures we are thinking about, we’ll be in perfect agreement about complex numbers.

You may have learned in high school that, if you have a polynomial with real coefficients and a+bi is a root, then so is a-bi.  Now we see the reason that underlies this truth: no algebraic statement in terms of real numbers can distinguish a\pm bi from one another.  The point in your mind I call a+bi, I call a-bi, and vice versa.

In fancier talk: the complex numbers have a symmetry, usually called complex conjugation, which preserves all the real numbers and which preserves any facts and relationships which can be expressed in terms of basic algebra.  The numbers a+bi and a-bi are interchangeable because they have to be, because they are bound by the symmetry.  Symmetries are magical things.

As we shall see, symmetries are powerful tools for understanding many kinds of situations, and the language of mathematics is the right language for getting at symmetries.

But there is more to the story.  We’ll talk to the aliens a little more next time.

Now there’s completeness and then there’s completeness…

15 October 2009

This post achieves a fortuitous segue from the last post into my serious of articles on the beauty of Galois theory.

In the previous post I introduced Dedekind cuts as a means of constructing the real number line, and I said that this perspective is responsible for the completeness of the real numbers \mathbb{R}.

Now, that was completeness in the topological sense.  There is another, very different notion of algebraic completeness.

A number system is called algebraically complete if every polynomial equation in one variable with coefficients from that number system can be solved in that number system.

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Dedekind cuts

15 October 2009

I am currently teaching a course in geometry for teachers (Euclidean, nonEuclidean, projective, the whole ball of wax), and we were recently discussing the need for the Dedekind axiom for plane geometry, which guarantees in effect that the points on a geometric line behave the  same , way as real numbers on a real number line.  What was interesting to me was that, even after all we’d said about all the ways that geometry might behave in unexpected ways if we don’t make certain assumptions, somehow the idea that geometric lines were real number lines was more deeply ingrained.  The idea that there might be a world were there were no line segments with length \pi was harder to imagine than the idea of a world where there are multiple lines through a point parallel to another point.

It got me thinking, why is that?

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Return to Purple Squares

10 October 2009

In the last post, we show that the following simple diagram provides all the information needed to prove that \sqrt{2} is irrational.


But as it true so often in mathematics, there is much more to see beyond the surface-level observations, and this time I cannot resist going back to this picture to say more.

Our main points last time were:

  1. the big square has the same area as the two light squares together if and only if the dark square has the same area as the two white squares together
  2. a square of side m has the same area as two squares of side n if and only if m/n = \sqrt 2.

We know we can’t get equality in either case though, which motivates the following approximate version.

  1. the big square has almost the same area as the two light squares together if and only if the dark square has almost the same area as the two white squares together
  2. a square of side m has about the same area as two squares of side n if and only if m/n \approx \sqrt 2.

In other words, if the ratio of the  sides of the large and light squares is “about” \sqrt 2, then the ratio of the sides of dark and white squares is also “about” \sqrt 2.  Which one is a better approximation?  The one involving the larger squares.  The absolute discrepancy in area between the squares is the same, but the relative discrepancy will be smaller if the areas are larger.  (The same reason I was so much more dramatically older than my sister when I was 6 and she was 2 than I will be when I’m 82 and she’s 78.)

Let’s add some letters to simplify the statements.  If  m is the side of the dark square and n is the side of the white square, then m+2n is the side of the big square, and m+n is the side of the light square.  (Make sure you can see this in the picture.)  Then our claim is that if m/n is a reasonable approximation to \sqrt 2, then (m+2n)/(m+n) will be a better one.

It’s too much to hope for m^2=2m^2, but if we take m=1,n=1, then they’re only off by 1.  So we can take 1/1 as a starting point.  Then we expect 3/2=1.5 to be a better approximation.  But why stop here?  Taking m=3,n=2, 7/5=1.4 is a better estimate.  We can keep this up forever, giving the following sequence of increasingly good rational approximations to $\sqrt 2$:
1/1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, \cdots
These approximations are getting very close very fast (the last one is right to six decimal places, enough for any practical application I can think of), and we’re not working very hard to get them!

Actually, more still is true.  If we start with any fraction m/n, even one which is nowhere near \sqrt 2, repeatedly applying the rule m/n \mapsto (m+2n)/(m+n) will give us a sequence of numbers that, in the long run, will converge to $\sqrt 2$.  Since the absolute area discrepancy doesn’t change, but the squares get larger and larger, the approximation is eventually as close as we might like.  The sequence from the previous paragraph is still the best one, though, because there the area discrepancy is 1, which is the best we can hope for since we proved last time that 0 is impossible.

Actually, it can be proven, without anything fancier than stuff we’ve already said, that all the solutions in positive integers of the equation m^2-2n^2=\pm 1 come from the sequence two paragraphs back. Can you see how?

It can also be proven that the approximations will be alternately overestimates and underestimates.  Can you see why?

There is a rich theory of  rationally approximating irrational numbers, including a method based on continued fractions for finding optimal approximating fractions to any real number.  What is amazing is that in this case we can get exactly the same answers predicted by the general theory without knowing anything sophisticated.  We don’t need continued fractions or even a precise definition of “good rational approximation”.  All we need is the picture.

(In case  you either don’t like pictures or really like algebra, then the corresponding algebra fact is (m+2n)^2 - 2(m+n)^2 = -(m^2-2n^2), but that’s so much less colorful…)


I am aware that the triangle diagram in the previous post somehow got removed from my WordPress uploads.  I can’t fix this until I get back in my office on Monday, but I will do it at that time.

Back from Maine with Squares and Triangles

6 October 2009

Ah, how I’ve missed my little blog.  Sorry about the hiatus, but now I’m back from the number theory conference with a head full of new ideas.  Unfortunately, most of the topics of the conference have far too many prerequisites to fit in this blog.  Let’s just say I saw many beautiful things and was reminded (in case I had forgotten) why I am a number  theorist.

There was one historical talk, in which David Cox lectured on Galois theory according to Galois.  If you aren’t a math major, don’t worry, you’ve probably never heard of Evariste Galois.  So inspired was I by this talk, and by the beauty of  the ideas at play in what Galois brought to light, that I want to share the heart of Galois theory with all of you. This will take quite a few posts to realize, working our way there one vignette, one thought experiment at a time. Fasten your seatbelts, ladies and gentlemen . . . the next few weeks will be interesting.

I did learn one extremely clever thing which is suitable for this audience “right out of the box”.  The inimitable Steve Miller showed me the following purely graphical proof that \sqrt{2} is irrational.

What would it mean for \sqrt{2} to be rational? It would mean that \sqrt{2}=m/n for some integers m and n, which we can choose to be in lowest terms.  In other words, there is a square of integer side length (m) whose area is the same as two squares of another integer side length (n), and furthermore we couldn’t find smaller integer squares with this relationship.  Place the two smaller squares in opposite corners of the larger square as sshown in the picture.


By our setup, the two light purple squares together have the same area as the large square.  This means that the uncovered area (the two white squares) must account for the same area as the doubly-covered area (the darker purple square).  If the original squares have whole-number sides, then so do these.  And the new squares are obviously smaller than the new ones, since they’re physically inside the new ones.  But we had supposedly chosen the smallest possible integer squares with this property.   Contradiction.

Neither is this trick is limited to \sqrt{2}.  The following picture can be seen as a demonstration of the irrationality of the square root of 3, if you look at it right.  I leave that to you.


If you want a further challenge, try to find proofs in the same spirit that \sqrt{6} and \sqrt{10} are irrational.