As I said in the last post, in group theory, you strip things down to a simple collection of values and one operation, with four required properties. The result is a simple structure, which completely captures the concept of symmetry. But mathematically, what is symmetry? And how can something as simple and abstract as a group have anything to do with it?
Let’s look at a simple, familiar example: integers and addition. What does symmetry mean in terms of the set of integers and the addition operation?
Suppose I were to invent a strange way of writing integers. You know nothing about how I’m writing them. But you know how addition works. So, can you figure out which number is which? Let’s be concrete
about this: here’s the set of numbers from -5 to 5, out of order, in my strange notation. {#, @, *, !, , ^, &, %, $ }. If you give me two of them, I’ll tell you their sum. What can you find out?
Well, you can figure out what zero is. With a bit of experimentation, you can see that adding
“&” and “>” gives you “>”, and adding “&” and “~” gives you “~”; the only integer for which that could
be true is zero. So you can tell that “&” is how I’m writing zero.
What else can you figure out? With enough experimentation, you can find an ordering. You can find two
values, “@” and “#”, where “@” plus “#” = “>”; “>” plus “#” equals “%”, and so on – each time you add “#” to something, you get another value, and if you start with “@”, and repeatedly add “>”, you’ll get a
complete enumeration of them: “@”, “>”, “%”, “$”, “!”, “&”, “#”, “*”, “~”, “^”, “>”.
So – which symbol represents one?
You can’t tell. “@” might be -5, in which case “#” is 1. Or “@” might be 5, in which case “#” is -1. You can tell that “*” must be either +2 or -2, but you can’t tell which. You can’t tell which numbers are the positives, and which are the negatives. If all you have available to you is the group operation of addition, then there is no way for you to distinguish between the positive and the negative numbers.
Addition of the integers is symmetric: you can change the signs of the numbers
but by everything you can do with the group operator, the change is invisible. I can write an equation in my representation: “%” plus “*” equals “!”, where I know that “%” is +3, “*” is -2, and “!” is +1. I can then switch the positive and the negative numbers, so that “%” is -3, “*” is +2, and “!” is -1, and the equation – in fact, any equation which relies on nothing more than the group operator of addition – can tell the difference. If you don’t know what the symbols represent, you wouldn’t even be able to tell that I’d changed my mind about what the symbols meant!
That’s a simple of example of what symmetry means. Symmetry is an immunity to transformation. If something is symmetric, then that means that there is something you can do to it, some transformation you can apply to it, and after the transformation, you won’t be able to tell that any transformation was applied. When something is a group, there is a transformation associated with the group operator which is
undetectable within the structure of the group. That “within the group” part is important: with the integers, if you have multiplication, you can distinguish between 1 and -1; 1 is the identity for multiplication, -1 is not. But if all you have is addition, that the transformation is invisible.
Think of the intuitive notion of symmetry: mirroring. What mirror symmetry means is that you can draw a line through an image, and swap what’s on the left-hand side of it with what’s on the right-hand side of it – and the end result will be indistinguishable from the original image. Addition based groups of numbers captures the fundamental notion of mirror symmetry: it defines a central division (0), and
the fact that swapping the objects on opposite sides of that division has no discernable effect.
For an example of how that can get more interesting that just the integers without changing the fundamental concept, you can look at the following example: given a hexagon, you can reflect it along many different dividing lines without recognizing any difference.
When some object is symmetrical with respect to a particular transformation, that means that you cannot distinguish between that object before the transformation, and that object after. There are, of course, many different kinds of symmetry beyond the basic mirroring ones that we’re all familiar with. A few examples of basic geometric symmetries:
Scale: scale symmetry means that you can change the size of something without altering it.
Think of geometry, where you’re interested in the fundamental properties of a shape – the number
of sides, the angles between them, the relative sizes of the sides. If you don’t have any way of
measuring size on an absolute basis, then an equilateral triangle with sides 3 inches long and an equilateral triangle with sides 1 inch long can’t be distinguished. You can change the scale of
things without creating a detectable difference.
Translation: translational symmetry means you can move an object without detecting any change. If you have a square grid, like graph paper, drawn on an infinite canvas, you can move it the
distance between adjacent lines, and there will be no way to tell that you changed anything.
Rotation: rotational symmetry means you can rotate something without creating a detectable change. For example, as illustrated in the diagram below, if you rotate a hexagon by 60 degrees, without any external markings, you can’t tell that it’s rotated.
There are, of course, many more, and we’ll talk about some of them in later posts.
For a fun exercise, look at the Escher image at the top of this post. It contains numerous kinds of symmetries; several different kinds of reflective symmetries, translational symmetries, rotational symmetries, color-shift symmetries, and more. See how many you can find. I’ve been able to figure out at least 16, but I’m sure I’ve missed something.
It depends what you mean by a “kind” of symmetry. The group is $D_3ltimesmathbb{Z}^2$ — a dihedral group (to flip around one of the triangular shapes) along with a rank-2 lattice to slide the triangular shapes among each other, and a semidirect product to combine them together. So what’s a “kind”? An element? A subgroup? A normal subgroup? Some sort of equivalence class of subgroups?
[ Addition of the integers is symmetric: you can change the signs of the numbers but by everything you can do with the group operator, the change is invisible. I can write an equation in my representation: “%” plus “*” equals “!”, where I know that “%” is +3, “*” is -2, and “!” is +1. I can then switch the positive and the negative numbers, so that “%” is -3, “*” is +2, and “!” is -1, and the equation – in fact, any equation which relies on nothing more than the group operator of addition – can tell the difference. If you don’t know what the symbols represent, you wouldn’t even be able to tell that I’d changed my mind about what the symbols meant!
That’s a simple of example of what symmetry means. ]
How does group theory then capture the language of symmetry? If 2+3=3+2 indicates a symmetry, then we have symmetries outside of group theory. Do you mean to say that commutative or Abelian group theory captures the notion of symmetry?
A couple of thoughts:
in a group, if x + y = x
then -x + x + y = -x + x
y = 0
So, once you know that you have a group and x + y = x, then you know that y = 0. Nothing further needed (inverses are unique in a group)
Symmetry and group theory: “kind” of symmetry: I think he means a non-trivial automorphism of a group.
How group theory captures symmetry: a group can act on a set and that group action can be thought of as “a symmetry”.
example: consider that “devil angel” tiling of a disk. rotational symmetries of the disk can be obtained by letting the group S^1 (think of the set of complex numbers of modulus 1 with the operation of complex multiplication); this can give various rotations of that disk, some of which correspond to the symmetries of the images on the disk.
Symmetries are elegant, and groups are elegant; but fields, whose two operations form a group and a monoid, have always seemed less so to me, for all their utility… But, the most symmetrical geometry (in their group-theoretic classification) is projective geometry, with its points at infinity; and adjoining a related infinity to a field yields a more symmetrical pair of structures (symmetrical in the mirroring sense), each of which is a commutative generalization of an abelian group. (I just thought I’d throw that in, in case it interests anyone:-)
[A couple of thoughts:
in a group, if x + y = x
then -x + x + y = -x + x
y = 0
So, once you know that you have a group and x + y = x, then you know that y = 0. Nothing further needed (inverses are unique in a group)]
Let ‘+’ stand for ordinary multiplication. Consequently {1, -1}, with 1 and -1 representing the regular numbers ‘1’ and ‘-1’, qualifies as a group under ‘+’, with x+1=x. But, since we let 1 equal the regular number 1, it doesn’t equal 0.
I think you mean to say that if x+y=x, then y indicates the identity element of the group. It does so, specifically because the identity element gets defined as the element such that x”*”y=x=y”*”x, where “*” indicates any sort of operation that satisfies the group axioms.
[I think he means a non-trivial automorphism of a group.]
I don’t see why you think so, since he didn’t bring up the automorphism concept, at least not explicitly. I don’t recall it myself. I don’t think he meant to assume his readers would know that concept.
[How group theory captures symmetry: a group can act on a set and that group action can be thought of as “a symmetry”.]
Perhaps, but Mark didn’t talk about symmetry in such a sense. He used a much broader concept. The operator ‘max’ on {0, 1/2, 1} qualifies as logically symmetric or commutative, has an identity of ‘0’, and works associatively. I can think of 0 min 1 as having 1 min 0 as its symmetrical counterpart. This corresponds to changing “the signs of numbers”, as for a set {-1, 0, 1} I could write -1 min 1 and apply the transformation to get 1 min -1, and no change in the structure involved happens. But, there exists no inverse for min, so neither of those sets qualify as a group under min.
“I can then switch the positive and the negative numbers, so that “%” is -3, “*” is +2, and “!” is -1, and the equation – in fact, any equation which relies on nothing more than the group operator of addition – can tell the difference”
I think that should say, “can’t tell the difference”.
Nod. He was just using “0” as the name of the identity element. That’s common, iirc.
If we were talking about multiplication specifically, then yeah, we’d say “1”. However, in more general group terms 0 is used. This is also because they use 1 as the name of the identity of the second operation in a field.
What does ‘iirc’ mean?
Yes, by “0” I meant “group identity element”; of course many might have thought I meant the identity with respect to the “addition” operation in a ring.
Sorry for the confusion.
Doug, “iirc” is ‘”f I recall correctly”.
Doug, about your example: I like it. But I don’t think that your set with binary operation (a semigroup) qualifies for symmetry. Here is why:
If one makes a multiplication table for abstract elements a, b, c; the first row of which looks like this:
a*a=a, a*b=b, a*c =c
Then it is clear that a
[Then it is clear that a then b*c = c means that b]
I think you meant to write more here, but I don’t know what.
I don’t get your point, since if we write out a table with for {a, b, c}, with a
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Hi Mark,
Nice post on Escher images and other symmetries.
There are also Penrose Tiles
http://mathworld.wolfram.com/PenroseTiles.html
related to Polyforms.
http://mathworld.wolfram.com/Polyform.html