The Hallway - Section 3 Clock Arithmetic

Here's the table I built for the rectangle.

I double-checked this, but I may be wrong. If your table doesn't look like mine, send me an e-mail and convince me what I did wrong, and I'll send you a gold star.

Now we have 2 tables, one for rotating the square, and the one for the rectangle that we just looked at. Before we move on to something new, it's time for:

Step 9. Analyze your data. Now that we have a couple of samples, let's compare them, and see what we can see.

Just for convenience, I'll put them here side by side.

I need to go back to the observations we already started in the previous lesson, but this time we're looking at both tables.

Here are some questions that I think we should consider. For each table on its own:

• Is it self contained? Is each possible combination of "followed by's" recorded in the table?
• Is there a motion that acts like a zero element?
• Does every motion have an opposite? For every motion, is there another motion we can combine with it to make the zero element.

What are your answers to these three questions?

My answer is "YES" to each question. Does this make sense to you? There are differences between the 2 tables, important differences. BUT, if we narrow our focus, and abstract only the details we need to answer these questions, then we say yes to each question for both tables.

There are some other similarities, and some other differences, but I want to take a little detour before we get to those.

We need to talk about numbers for a while. Don't worry, the arithmetic is easy, and it's something you already know. Think for a minute about how you add time on a plain, old, 12 hour clock face. If you go to a movie at 10 o'clock, and you arrange to meet your friends out front in 3 hours, then you all know to meet there at 1 o'clock, not at 13 o'clock.

If you put dinner in the crock pot at 7 o'clock and it has to simmer for 10 hours, then dinner will be ready at 5 o'clock, not at 17 o'clock.

What's going on here? When we're adding time on a clock, we add up to twelve then we start over again. Mathematicians take this simple idea and come up with a foreign sounding fancy word, and say that we're doing addition modulo 12. Don't let the fancy name confuse you. It's just ordinary clock addition like I just described.

--What does clock arithmetic have to do with the square and rectangle?

Excellent question. Bear with me for a minute. Let's play "what if", the national pasttime of all good mathematicians. What if our clock had only 4 numbers, like this:

Now suppose we made a table with just these numbers, and we add them doing plain old clock arithmetic. Here's a blank table with the labels filled in to get you started:

Now you take a few minutes to fill in your table before scrolling down.

I put my table next to the table we already made for the square. Check your "clock" table against mine, and then take a close look at the 2 tables, and see if you notice anything.

You look at what's going on with these 2 tables, really study them for a while, and then we can compare notes. Just click on the link to continue when you're ready.

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