Monday, October 27, 2008


Sometimes seeing mathematical truth is as easy a seeing your hand in front of your face, but sometimes the language of mathematics can even make that hard. Mathematicians use their own language which is often off-putting and frequently disguises everyday truth. This has occurred in the case of handedness.

Mathematicians recognize a phenomenon, give it a name, and then we can study it. This is how we operate. If we give it a Greek name we can attempt to leave the impression that we know Greek and are more educated than is actually the case. The name we give handedness is chirality. If we called it handedness, which is what it is, you might not think it was very important, but we are hoping to hook into your native pomposity by using the Greek word.

Are you hooked?

If not, then you don’t know what this sentence is saying. Or this one.
But I digress.

Chirality, that is to say handedness, is a mathematical phenomenon not a physical one, though it does manifest itself in physics because of the geometrical nature of the universe. But we needn’t get fancy, as this isn’t a truth that college professors alone own. I grew up hearing stories from the oilfield. A new, inexperienced member of the crew would be sent on an errand to bring back a left-handed monkey wrench. Hours of amusement would be gained as the neophyte went around asking for a left-handed money wrench.

As you’ve no doubt figured out (and if you haven’t let me tell you about a little bird called the snipe...) there is no such thing as a left-handed monkey wrench. Monkey wrenches work exactly the same way in either hand. They are not affected by handedness. The pompous way we denote this is to say they are achiral.

The reason the practical joke works is there are objects which are affected by handedness. A baseball glove would be a good example of this. You will not be able to get a catcher’s mitt which had been designed to go on the left hand to fit properly on the right. It simply will not happen. The geometry of the universe prevents it. (For latex surgical gloves, this is not an issue because latex is a lot more flexible than leather, so it doesn’t have to be fitted to a particular hand.)

You don’t have to actually worry about gloves to see this. Consider the following experiment. Put your hands out in front of you in the manner of traditional prayer. You will notice one is a reflection of the other. Now put one atop of other so as the middle fingers line up and the nails of each hand face you. Notice that the pinkie of one hand is aligned with the thumb of the other and vice versa. You cannot line arrange your hands so that thumbs align with thumbs and pinkies with pinkies. Not even if you try real hard, not even if you hack one off them off with a machete like they did in the “Blood Diamond” episode of Law and Order.

Chirality, handedness, is related to the mathematical concept of orientation. This is a technical mathematical term that does correspond to its usual English meaning. If I tell you that I am newly arrived in town and have yet to become oriented, you will discern that I am trying to figure out where the important places in the area are located so that I can get around more easily.

The root word in orientation is orient which originally meant east. When this becomes the verb “to orient,” I have to wonder whether it comes from the fact that at one time most European maps had the east at the top. (Jerusalem, the origin of the Christian religion was in the east.) They realized that by settling on particular direction, i.e. east, all of the other directions were determined. If you know east, you can find west pretty easily. If you know east, you can figure out north. If you know north, you can find south.

Orientation is a phenomenon that occurs in all dimensions. I would like to take a little time in order to discuss dimensions one, two, and three.

When I speak of dimension one, I am talking about a line, or objects that are line-like such as circles and curves. To illustrate dimension one, consider the plight of a man who has awoken in a tube underground, having been placed there by an evil genius. (Bwah-hah-hah.) On one end of the tube there is a beautiful woman with all of the assets that are traditionally considered the due of a hero-type, but, of course, on the other end is painful, embarrassing death, something too hideous for a gentle person like me to describe in detail, but rest assured it’s bad.

You might say, wait a minute, that’s a three-dimensional example because you’ve got a person in your example and people are three-dimensional. Well, you are correct, of course, but from the mathematical perspective it’s one dimensional because the map, which the hero has secreted in one of the orifices of his body, is a line. At one end of the line there is the hero’s reward (think Jessica Alba) and the other a picture (shudder) of the hideous form of death that awaits him there.

On that map, there is an arrow at the middle of the line segment that points from one end toward the other. The tunnel that our hero is in has also been marked with arrows inside at points spaced at regular intervals all aimed in the same direction. The arrows establish an orientation on the tunnel, and it is left to our hero to determine whether they match the orientation on his map. One might suggest that he say “Hey, baby, is that you Jessica?” at various points just to be sure.

Another example of orientation would be the use of Uptown and Downtown in New York City, being up the river and downtown down the river. It sort of makes sense, eh?
We have a lot more experience navigating two dimensions than one, and, in this case, there is more structure so it makes for a richer phenomenon.

We live on the surface of Planet Earth. In spite of the fact that we are three dimensional creatures, because of gravity, much of our navigation is done in two dimensions. Every map is based on a two-dimensional plane; every globe is based on a two-dimensional sphere. (In case you are confused about a globe being two-dimensional, I would suggest a trip to Wichita for you.)

In order to aid our comings and goings on Planet Earth, our forebears coordinatized it, and, in doing so, used a particular orientation. The coordinate system consists of latitude and longitude lines. In order to assign numbers to latitude and longitude, orientation was required.

From a practical aspect, the whole enterprise was helped by the fact that the earth is spinning on an axis. This gives us not only a direction that the sun comes up in each and every day, but poles where the axis of rotation meets the surface of the planet.

Though it sort of breaks the flow, I feel I need to make the point that those of us on planet earth have competing systems. I’m not talking about the metric system versus the British system—which the British no longer use by the way—but something more basic. In giving directions, one can either deal with cardinal directions, i.e. north, south, east, and west, or with relative directions, i.e. forward, backward, right, and left.

I’ve spent most of my life on the plains. Our here on the flat, we’ve a large number of section-line roads to coordinatize the world. As a consequence, we like our cardinal directions, thank you very much. Others, of a weaker, quiche-eating breed, are forced to deal with the matter in a power way, giving directions in terms of right and left, forward and backward.

To return to my original line of thought, everyone knows how to determine north from east, but I will describe it anyway. Face directly east and north is on your left hand. A subtle point here is that making this determination requires each of us to have our own personal orientation, i.e. you have to know right from left.

Another subtle point is that you don’t need hands to do this, a clock will do. (I know that a clock has hands but push on.) Though our beloved forebears did not realize it at the time I am sure, when they decided which direction clocks would turn they also gave us a device for determining orientation. As I understand it, this was taken from the direction shadows move on a sundial in the northern hemisphere. I don’t know whether the meaning of “clockwise” was set in this fashion or not, but I do know for a fact that’s how shadows from the sun move in the northern hemisphere. If you don’t believe me, go outside and watch the shadow of the flagpole move. If you are working in a government office, a flagpole is easy to find. In fact, I think this is what government employees do most of the time; there might even be a job which is described that way.

If determining an orientation seems too easy, we shouldn’t take it for granted. There are certain surfaces where, by taking the wrong path (or the right path depending upon you intentions), you could make a clock go counter-clockwise. One such surface is the Mobius strip. I am required by the Ancient and Hollowed Guild of Mathematicians to mention it as a price of them not assassinating me for circulating this article.

Having talked about clocks allows us to move to orienting three dimensions. In three dimensions, we have not only left and right and forward and backward, we also have up and down. Adding up and down doesn’t seem like much of a challenge to us because we’ve an old enemy, gravity, to make the determination. (Those of you who are over forty know why I classify gravity as an enemy.)

But if you ever escape the confines of Planet Earth—perhaps only in your imagination—gravity is not necessarily there to help us. However, you can use a clock to help. Take your left hand and allow the palm to close in the direction that the hands on the clock are moving; this would be the natural way. Your thumb is pointing up.
I now have brought hands back into the picture. Think now about your right hand. If you point your fingers out in front of you and close them across your palm, the thumb on that hand is pointing upward as well. This is called the right hand rule. You can use it to take the lids off of mayonnaise jars and to unscrew screws.
It is a very handy thing to know.

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