Orientation. Not the position and direction of objects in space, but whether objects are left-handed or right-handed. Orientation is what dyslexic people have a hard time telling apart. This involves a number of related ideas. Mirrors. Chirality. Handedness. Knit and purl stitches. British people driving on the wrong side of the road. Geometry. Higher dimensions. Let us talk about these things, which really are more interesting than you might have thought at first. Interesting especially because everyone is a little dyslexic if you give them a complicated enough situation.
We begin with an old chestnut. Why is it that mirrors reverse left and right, but not up and down?. This might sound like a silly question. In fact, it is, because the real question is this: there seem to be three pairs of spatial dimensions that the mirror can reverse. Up and down, left and right, front and back. How come the mirror can only reverse a pair, but not two pairs? That is, shouldn't there be more than two orientations? If there are three pairs of spatial directions, and each pair can either be straight or reversed, shouldn't there be 2 × 2 × 2 = 8 possible orientations for an object in three dimensions?
The (surprising?) answer is no. An object can only be in one of two possible orientations, conveniently named "left-handed" and "right-handed" (or "positive" and "negative" if you want to be more mathematical about it). So if the mirror reversed more than one pair of spatial dimensions than the front-back reversal it makes (left-right and up-down aren't reversed by mirrors!), it wouldn't have reversed anything at all. Even more surprising is that this is not a peculiarity of three-dimensional space: in any number of dimensions, objects can only exist in one of two possible orientations. If you can believe it, group theory has something to do with it. But let's first explore more familiar ground.
God does not create (many) dice
Do you have a die handy? One of those little cubes with black dots on them. Looking at one while you read this might help, although it isn't essential if you have a good geometric imagination (most of us are strangely bad at geometry, by the way).
Ok, look at opposite faces of the die. If your die is properly built, it should have the property that opposite faces of the die add up to seven. Three dots opposite to four, two to five, and one to six. Now ponder the following question. You have a cube on which you must put six numbers such that opposite faces of the cube add up to seven. How much freedom does this give you? How many different possible dice can you build?
The answer is, not many. Only two. One for each orientation. So whatever die you picked up must either be a right-handed (positive, shall we say) die, or a left-handed (negative) die. The way to think about it is this: essentially, the restriction that opposite sides have to add up to seven means that you have three pairs of numbers to put on the die in some fashion, and each pair can either be "reverse" or "backwards". So it's essentially the same problem as the mirror chestnut: three directions, and one flipping for each. Because there are only two spatial orientations, there are only two possible dice that can be built. Here they are, if you'll pardon the ASCII art:
/1 /| - 5 /1 /| - 5
/___/ | /___/ |
4- |2 |3/ 4 - |3 |2/
| |/ | |/
Notice that the only essential difference is that 2 and 3 have swapped places. Let us look at this die-building process a little more slowly, shall we?
How dice are built (well, not really)
With a minor mental modification, instead of thinking of cubes, we are going to be thinking about three mutually perpendicular directions, one direction for each pair of opposite faces. Rods, if you will, joined perpendicular at their centres. Now, it doesn't make a difference in which order we decide to pick the first two rods. You might think that it does, but I invite you to look at the following piece of art, where diagonals are meant to suggest lines coming out of the computer screen.
------.------ X ------.------ Y
------.------ X ------.------ Y
\ \ Y
-------.------ X -------.------ X
\ Y \
\ \ X
-------.------ Y -------.------ Y
\ X \
The point is that ALL of these configurations are the same one, because they can all be rotated in three dimensions until they are identical to the first configuration. Try it. It's not hard. There's a theorem lurking here, but I won't state until we get to talk about higher dimensions. The interesting bit comes when we add one extra perpendicular direction, the Z-rod, which we might as well just add to the first configuration, where the X- and Y-rods are flat on the screen. And there are only two ways that this can be done:
\ | Z \ |
\ | \ |
------.------ X ------.------ X
| \ | \
| \ Z | \
I'm happy with calling this a proof that there are only two orientations in three dimensions. QED, shall we say.
This rod-placing exercise tells us more. To every object in three-dimensional space, we can assign either of the above two rod configurations, that is, a positive or a negative orientation. What a reflection does, through a mirror, is to transform a positively-oriented object into a negatively-oriented object and vice-versa. Two reflections undo the reversal of orientation (although they might move the object in space in some funny way). That is to say, after two reflections, it is always possible to bring back an object to its original position only through translations and rotations. A reflection undoes itself. To undo a swap, simply swap again.
Chiral molecules and purl stitches
There are interesting applications about orientation, in chemistry, and in knitting. Disclaimer: I'm not a chemist; I'm more of a geometer and a bit of a knitter.
First, what I don't know much about: chirality. It turns out that geometry has a lot to do with how molecules bond with each other. This really shouldn't be surprising. What did surprise chemists at one point, is that the geometry of some molecules comes in two varieties: left and right. While this doesn't affect physical properties of the material such as its boiling point or its colour, it does affect how the material reacts chemically. Pairs of molecules that are identical except for their orientation are called stereoisomers. I hear that a particularly interesting pair of stereoisomers exist in milk, such as lactic acid pairs, and that there is a science fiction story out there (entitled Technical Error by Arthur C. Clarke, thanks rootbeer277!) about a man who got mirror-imaged and had problems because his stomach couldn't digest a number of improperly-oriented molecules.
More information on chirality can be found elsewhere in the nodegel.
Now, knitting, and truth be told, my entire motivation for noding this writeup. First, some basics: in knitting there are two basic kinds of stitches, the knit (positive) and the purl (negative). Actually, stitches by themselves are self-chiral, meaning equal to their own mirror image, but when they sit on the needle they acquire positive and negative orientations (the needle-stitch compound has a definite orientation).
Everyone has different knitting styles. I think combined knitting is what I do. Remember what I said about everyone being a little dyslexic given a complicated enough situation and how people are surprisingly bad at geometry? This applies to me when I'm knitting. I never noticed the difference until I tried to follow knitting patterns that called for knitting through the back loop in order to twist a stitch. I thought that was the only way to knit?
Orientation comes in at three or four levels when knitting. The stitch itself may either be a knit or a purl stitch. It can sit facing right or facing left on the needle. It can be knitted or purled with a clockwise or counterclockwise wrap of the yarn on the other needle. And finally, if you want to take into account left-handed knitters, that's a fourth possibility of reversing the orientation of the work.
The cool thing is that most of these things don't matter, because in the end there are only two ways a stitch can be oriented! The moral of the story is that you can knit however you want to and not worry about orientation as I once did (and seems that hapax did too, as the node combined knitting suggests). So long as you aren't inadvertently twisting your stiches (which is then no longer a simple purl or knit), and as long as you keep straight what the front and back of the stitch is, regardless of whether it's facing left or right on the needle, all possible sources of orientation-reversal collapse into one of two orientations: positive and negative. So relax. There isn't a wrong way to knit.
This whole orientation business gets more interesting if we look at what happens in three dimensions and then cast our imagination to loftier dimensions. After all, I feel cramped in three dimensions, don't you? Lots of marvelous low-dimensional weirdness happens in three or four dimensions: two or three more platonic solids instead of the lonesome three in dimensions higher than four, the fundamental group of the three-dimensional sphere wasn't determined to be the only trivial one until recently, but all other dimensions had been solved. A sphere acquires maximum volume in six dimensions and then its volume decreases to zero as the dimensionality increases. The complete symmetry group of rigid motions of the six-dimensional canonical basis is the only one with an outer automorphism; for every other dimension all automorphisms are inner.
Right. Sorry if some of those examples don't make sense to you. What I'm trying to say is that three spatial dimensions is a very remarkable and strange number of dimensions to live in, because it's a small number, and that in higher dimensions the weirdness starts to fade. It is therefore not a stupid question if there are more than two possible orientations in higher dimensions; perhaps two orientations in three dimensions is a low-dimensional exception. Or perhaps there are only two orientations in three dimensions because we are cramped, and we can actually find 42 different orientations if we had the freedom to move around in 1729 dimensions.
All I'm trying to do here is to surprise you when I say that in any number of dimensions there are only two orientations. A positive and a negative one. The boring answer happens to be remarkable.
The proof of this fact is easy and essentially contained in the proof I gave above for three dimensions. Imagine that we are sitting in six-, seven-, or sixty-nine-dimensional space, where we can find sixty-nine mutually perpendicular directions. Then as in the three-dimensional case we can pick sixty-eight mutually perpendicular directions in any way we want to and pointing however we want to. But then we can rotate those sixty-eight directions inside the surrounding sixty-nine dimensional space and bring them all to look alike. Only the choice of the last direction will determine an orientation for sixty-nine dimensional space.
This discussion about orientation in higher dimensions shouldn't convince nitpicky mathematicians by itself, although I do trust that it convinces most ordinary folks who read it. I hope it convinced you.
Other fun and geeky facts
It turns out that orientation isn't a concept all that tied down to perpendicularity and the rigid, zero-curvature Euclidean space that has been implicit throughout this writeup. In fact, every manifold, however curvy and high-dimensional, can be locally oriented in one of two ways (but not always globally: the Möbius strip has only has one side!) That means, for example, that we can assign a left-handed or right-handed orientation to the hyperbolic plane.
Also, a nice side effect contained in the proof that there only exist two orientations is that to change orientation of an object in any number of dimensions, we can perform a rotation in a space with one more dimension. That is, p's and q's can be transformed into each other if we are allowed to lift them off the page and rotate them three-dimensionally; your left and right hands could be superimposed on each other and have the same three-dimensional orientation if only you could rotate your arms four-dimensionally. There are no two-dimensional chiral molecules in three-dimensional space. And so on.
I said at the very beginning of this writeup that group theory has something to do with orientation. The involvement of group theory with all this business is that the symmetry group of n-dimensional Euclidean space (or of n mutually perpendicular rods, each rod distinguishable and directed) is the alternating group on n symbols if we only allow n-dimensional rotations, and the complete symmetric group on n symbols if we allow the odd permutations (odd permutations correspond to orientation-reversing symmetry transformations).
There. That's all I have to say about orientation. Wasn't that fun?