The Hallway - Section 1 Rotating a square

Step 1. Ask a question. Creating new mathematics begins with asking questions. Sometimes we ask good questions, and get good mathematics in return. Sometimes a question that's easy to ask is impossible to answer. Sometimes a question that sounds difficult turns out to be something you already know, just dressed up to look different. Sometimes the question leads, not to an answer, but to another question. It's like putting together a jigsaw puzzle where all the pieces were in the wrong box - you thought you were building the White House and you get the Golden Gate Bridge instead. So here's the first mathematical secret I want to share with you: You start with questions, and the answers are not in the back of the book. You're supposed to be surprised! In fact, if I had to choose, I'd say the questions are more important than the answers. The answers are finished, the end of the line. The questions keep calling you on to new possibilities, wondering what might be around the next corner. Remember, mathematics is not finished, it's work in progress.

We will start with this question, "How can we rearrange a square by turning?" If this seems like a silly or pointless question, I promise you that it isn't.

Step 2. Work with an example. Get a square piece of paper and some colors, and make yourself a model like this:

Step 3. Play with your model. A lot of great mathematics was very playful to begin with. We mathematicians do a lot of pretending, "I wonder what would happen if..." Remember, you don't usually know in advance what you're going to find, so you start by trying to "get the feel" of your problem. After playing with this square for a while, I decided that 'rearranging a square by turning" is not a good question to start with. How do you know how much to turn it? I can turn it a lot, or I can turn it just a very little. That's too much for me to decide. So I've already changed the question. That's allowed. When you are inventing mathematics, you can change your questions. Sometimes, we just need a question, even a real stinker, to start with, a preliminary question that we know we'll replace as soon as we get to the good question.

Step 4. Revise your question. I am going to simplify my question by saying that for now, I'm going to restrict myself to rearrangements where the sides of the square still "line up" after the motion. Make believe the square has to fit back into the same shadow when you're done. I can turn the square, but to fit back into the shadow, I have to end up with the corners in the corners.

Also, the square has to stay flat on my desk, I'm not going to flip it over face down. So, in this experiment, flipping is a no-no. We'll do that a little later.

Step 5. Keep things simple. Make sure you can explain your question, and your results, in simple, ordinary English. We'll need some fancy mathematical words later on, but for now, plain English will be just fine.

I think that's enough of the preliminaries, let's create some mathematics.

Now in order to talk about what happens to our square, we need to identify the corners somehow. Now, we could label them 1, 2, 3, 4, or A, B, C, D, or even @, $, %, &. The labels themselves don't mean anything, they just allow us to agree on which corner is which. So, I label the corners of the square by color -- Red, Green, Blue, Yellow.

Now, suppose we turn the square a quarter turn to the right, a move I'm going to call RIGHT FACE. This is a right face:

Suppose the square on your desk was in a different position when you started, so it doesn't look exactly like this example. It doesn't matter. We're not interested right now in the square, we're interested in the motion of a quarter turn to the right. So this is also a right face:

So your square may start from a different position, and end up in a different position, but every right face is exactly the same. What do I mean by that?

Let's stop for a minute and try to describe what we just did. One way to describe this motion is to say that the red corner moves to where the green corner used to be, the green corner moves to where the blue corner was, the blue corner moves to where yellow was, and yellow goes to where red was. No matter where you started with your square, if you turned it a quarter turn to the right, this description fits. Don't take my word for it. Play with your square until you're convinced that I got it right.

Depending on our focus, depending on what questions we're asking at the time, there are other descriptions of our right face. A fancy mathematician might call this a "rigid transformation of the plane". I like "right face" better. So, in plain English, right face means that Red goes to where Green was, Green goes where Blue was, Blue goes where Yellow was, and Yellow goes where Red was.

One mathematical notation, or "shorthand" to sum up this description looks like this:

which you read from the top down. The top row shows where each corner starts, and directly underneath it shows where that corner ended up.

Maybe this is a little clearer. Each little arrow shows the connection from where the corner started to where that corner went.

Here's the next mathematical secret I want to share with you: If the symbols start to look like meaningless squiggles, you're in real trouble. The notation is just shorthand, remember, and you have to fill in all the rest of the meaning that got left out. Don't be embarassed if you have to back up and refresh your understanding of what something means. Especially at first, there's a lot to remember, and nobody gets it perfect the first time through.

When you know you understand right face, it's time to add another move. This time we turn the square a HALF turn to the right, a move that I'm going to call ABOUT FACE. It looks like this:

Before you go on, describe this motion to yourself, in plain language, and then write it down in symbols like we just did for right face.

I looked at the motion and said "When I do about face,the Blue corner and Red corner change places, and the Green and Yellow trade places. In symbols, I wrote down a diagram that looks like this:

I hope you made it this far, because this is where it starts to get interesting. We're ready to take next step!

Step 6. Take a guess at the relationship between the two motions. How do right face and about face "fit together"? The fancy mathematician would call this "forming a hypothesis", but all that means is taking a guess. You look at the results you have so far, and make up a theory. Here's my theory: one right face followed by another right face is the same as one about face.

Step 7. Check your guess. Your theory might be right, and valuable, or it might be wrong, and valuable.

Okay, fair question. Remember, this is math in progress. You don't know if you guessed right or not until you check it out. If you guessed wrong, then finding out what went wrong with your guess is a chance to learn more about your question. What did you overlook? What happened that you didn't expect? Did your guess account for everything? Here's another secret: The fancy mathematicians guess wrong all the time. It's just that they cover their tracks, and erase all their mistakes, so no one will know. Good mathematicians guess wrong all the time, but they use their wrong guesses to feed new information into their questions. What went wrong, and why? This might take us back to step 1, asking a new question, and going in a different direction.

But not this time. I think I guessed right, and I'm going to convince you by showing all the details.

Look at where the square started, and where the square ended up. The net result of right face followed by right face is: the Blue corner and Red corner change places, and the Green and Yellow trade places. Does this sound familiar? It's the description of about face! Back up and refresh your memory if you need to. Don't take my word for it. Check this out for yourself.

In our notation, what we just did might look like this:

But I still don't like the arithmetic. Once you make that jog in the middle, how do you know what to put down for the answer? So let's make it clearer. Remember, the key information in the notation we're using is the pairs of corners, one in the top row where the corner started, and one in the bottom row where that corner ended up. As long as we keep each pair together, we can rearrange them to make the arithmetic a little easier.

What I mean is that instead of writing right face like this:

You should do this arithmetic with the other corners to convince yourself that right face followed by right face is exactly the same as about face. Of course, if you just turn the square on your desk, all this fancy arithmetic is just saying that a quarter turn plus a quarter turn is a half turn, which makes sense to me.

I know you're all way ahead of me by now, but I need to go slowly, step by step, or I get confused. There are just a couple more motions I need to tell you about.

If you rotate your square a three-quarter turn to the right, you accomplish the same motion as if you turned a quarter turn to the left, so I'm going to call this motion LEFT FACE. It looks like this:

There's one last motion we need to talk about. Suppose you turn your square one full turn. This is the motion I'm going to call STAY PUT. It looks like this:

Now it's time for your first assignment. You may remember the times table from when you learned multiplication. There was a column of numbers down the left hand side, a row of numbers across the top. You found the row and column for the numbers you wanted to multiply and their product was in the box where that row and column met. For example, the sample below shows the numbers 1 through 5, and I've marked how we pick out 2 times 4. You find 2 down the left hand side, and then go across to the column under the 4, and you find the result 8.

Now we're going to build a table like that for the rotations of our square. I'm going to abbreviate so I don't have to write so much in the little boxes. So, in the table we're going to build:

• SP means Stay Put
• RF means Right Face
• AF means About Face
• LF means Left Face
• + means "followed by" (as in, a motion down the left side followed by a motion across the top)

I got the table started for you. The Stay Put row and column were easy. Stay Put doesn't change anything. Any motion followed by Stay Put is the same motion, just as Stay Put followed by any motion is that motion.

I also filled in About Face followed by Right Face, which we showed already was the same as Left Face.

So here's the start of the table. Now you fill in the rest, and then click on the link at the bottom to compare your table with mine.

Continue on to Section 2 and check your answer

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