Consider a square hole and a square peg that sits inside the hole. You can remove the peg, rotate it by a quarter turn, and put it back in the hole, and it will fit. You can remove the peg, rotate it by half a turn, or 3-quarters of a turn, or a whole turn, and it will fit. You can do any of these actions followed by any other action, and the peg will fit, and you can undo any of these: if you rotate the peg by a quarter turn, followed by a 3-quarters turn, it will be in the original position, so it's like you never did anything.
So we have four positions that the peg can end up in, and thus four essentially different things we can do to the peg: rotate a quarter turn, rotate a half turn, rotate 3-quarters of a turn, or do nothing.
If we started with a triangular peg in a triangular hole, we'd have three positions, and thus three things we could do.
If we started with a circular peg in a circular hole, we'd have an infinite number of positions.
Now imagine a cubical box. It's not sitting in a hole, it's just lying on a table, but we're still supposed to pick it up, move it in some way, and then put it back so that it's occupying the same space that it was in before.
We can do a few things to the box. We can flip it over in various ways, and rotate it in various ways. In fact we have three axes in which we can rotate the box, and we can do a quarter turn, half turn, or 3-quarters turn around each of those axes. If we do several rotations, it matters which order we do the rotations in. If you rotate it a quarter turn around one axis and then a quarter turn around another, you get a different result than if you do the two turns in the opposite order.
Given an object and a hole that it fits in, we can rotate or flip the object in various ways so that it still fits in the hole. Call these things we can do to the object "symmetries" of the object. Normally when we say "symmetry" we mean an object looking the same on both sides, which corresponds to flipping the object over; since it looks the same on both sides it still fits in the "hole". Mathematicians use "symmetry" to refer to the act of flipping itself.
We've noticed some things about symmetries:
Doing nothing is a symmetry. A symmetry followed by another symmetry is a symmetry. Every symmetry is reversible, we can undo any symmetry.
If we write our symmetries symbolically, we can write the do-nothing symmetry as \(e\), and for symmetries \(f\) and \(g\), we write "doing \(f\) followed by doing \(g\) as \(gf\)(the reason for the ordering will come later), and that for a symmetry \(f\), there is another symmetry denoted \(f^{-1}\) such that \(f\) followed by its inverse is like doing nothing: \(f^{-1}f = e\).
We've noticed some things about symmetries:
Doing nothing is a symmetry. A symmetry followed by another symmetry is a symmetry. Every symmetry is reversible, we can undo any symmetry.
If we write our symmetries symbolically, we can write the do-nothing symmetry as \(e\), and for symmetries \(f\) and \(g\), we write "doing \(f\) followed by doing \(g\) as \(gf\)(the reason for the ordering will come later), and that for a symmetry \(f\), there is another symmetry denoted \(f^{-1}\) such that \(f\) followed by its inverse is like doing nothing: \(f^{-1}f = e\).
There is one more rule that symmetries follow. Remember that \(gf\) is a symmetry, and \(h\) is a symmetry, so doing \(gf\) followed by doing \(h\) is also a symmetry, \(h(gf)\). \(hg\) is also a symmetry, and doing \(f\) followed by doing \(hg\) is also a symmetry, \((hg)f\). The rule, which should sound trivial, is that \(h(gf) = (hg)f\), doing \(f\) then doing \(g\) then doing \(h\). So we can just write \(hgf\) without parentheses.
Any full set of symmetries of an object obeys these rules:
Any full set of symmetries of an object obeys these rules:
1): Closure: if \(f\) and \(g\) are in the set then \(gf\) is in the set.
2): Identity: there is a do-nothing called the identity \(e\), so that \(ef = fe = f\)
3): Inverses: there is an inverse of \(f\), denoted \(f^{-1}\), such that \(ff^{-1} = f^{-1}f = e\)
3): Inverses: there is an inverse of \(f\), denoted \(f^{-1}\), such that \(ff^{-1} = f^{-1}f = e\)
4): Associativity: \((hg)f = h(gf)\)
Any set with a composition operation that obeys the above four rules is called a group. The set of rotations that we can do to a square peg and still have it fit in the square hole is called the "group of symmetries of the square peg", and similarly for the triangular peg, the circular peg, and the cubical box.
As we go on, the "pegs" and the kinds of "holes" we want them to fit will become more complicated, less easily visualized, but these examples provide the basic intuition.
One last bit to note:
For the square peg, you have four symmetries, and if you do two of them in a row, it doesn't matter what order you do them in, the result is the same. We say thus say that the symmetries commute and that the group of symmetries of the square peg is "commutative" or "Abelian". For the cube, in contrast, it does matter sometimes, so we say that the group of symmetries of the cube is "noncommutative" or "non-Abelian". These notions, of commutativity and noncommutativity, play a big part in what's to come.
As we go on, the "pegs" and the kinds of "holes" we want them to fit will become more complicated, less easily visualized, but these examples provide the basic intuition.
One last bit to note:
For the square peg, you have four symmetries, and if you do two of them in a row, it doesn't matter what order you do them in, the result is the same. We say thus say that the symmetries commute and that the group of symmetries of the square peg is "commutative" or "Abelian". For the cube, in contrast, it does matter sometimes, so we say that the group of symmetries of the cube is "noncommutative" or "non-Abelian". These notions, of commutativity and noncommutativity, play a big part in what's to come.
No comments:
Post a Comment