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I copied the tables here again, to refresh my memory.
I think the story goes like this:
If you picture yourself standing on a giant clock face, looking at the 12, and you do a Right Face (a quarter turn to the right), you end up pointed toward the 3. Another Right Face and you'd be pointed at the 6. Another right face and you'd be looking at the 9. One more Right Face and you'd be back where you started from, looking at the 12.
To say this another way:
Try this experiment:
On the table for the square, everywhere you see a Stay Put, replace it with a 12. Replace every Right Face with a 3, every About Face with a 6, and every Left Face with a 9. You will have transformed the table for the square into the table for the clock just by changing the names.
This is so important that mathematicians have a special name for this relationship. The motions of the square and arithmetic on the 12-3-6-9 clock are ISOMORPHIC. Don't let the fancy word frighten you. It comes from two Greek words, isos meaning same, and morphe meaning form.
These two operations that we have been looking at, rotating a square and adding hours on a clock, appear very different on the surface, but they are two different examples of the same underlying mathematical structure. They have the same form.
This is a good time to mention again the idea of mathematical abstraction. By ignoring the superficial differences, a mathematician can see that the clock and the square have the same structure. They have the same form.
--Big deal! What are you getting so worked up about?
Here's your first payoff for mathematical abstraction. If you understand the rotations of the square, then you already know the 12-3-6-9 clock. Once you see that it has the same form as the square, you already know it. You don't have to figure it out all over again. You just change the names.
You only have to do the arithmetic once, and then just keep track of how to change the names. For example, we already know that the opposite of Right Face is Left Face (Right Face followed by Left Face leaves you where you started) so we also already know that the opposite of adding 3 hours is adding 9 hours (adding 3 hours followed by adding 9 hours leaves you where you started).
I am the laziest person I know, and every time I find a way to avoid wear and tear on my brain, I get worked up.
--What about the rectangle? You haven't mentioned the rectangle lately.
Good question. In fact, that's what I want to ask you about. Put your rectangle table next to your square table, and determine whether they have the same form, whether they are isomorphic. Now remember, we've already seen that things that look different, the clock and the square, may have the same form. Just because a square looks different from a rectangle, that doesn't mean that they can't have the same form. Whatever answer you come to, make sure that you have good, mathematician-in-training reasons for your decision, okay?
I'm going to lay down for a while. These Greek words have worn me out. When you have made your decision about the square and rectangle, come over to the next section and wake me up.
Continue on to the next section [***not finished yet***]
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