Infinity, part 5

The first thing talked about today was why "infinity minus 1" doesn't make sense. Look at a list of numbers:

1,2,3,4,5,6,7,...

Adding 1 to a number means moving one number to the right in the list. Subtracting 1 from a number means moving one number to the left in the list. If we try to move one number to the left of infinity, where do we end up? We would have to land on a finite number, but which finite number? Gah! "Infinity minus 1" is undefined, because there is no answer that makes sense.

Then we talked about limits. The "limit" of a process can be thought of as what you end up with if you could actually get to the end of time. Here are several examples.

• Take a cookie. Eat half the cookie. Now eat half the cookie that remains. And eat half again. Continue this for all time. The limit is no cookie left!
• Inscribe a triangle in a circle. Then a square. Then a pentagon. Then a hexagon. Continue. The limit is a circle.
• Take a line segment. Draw dots to divide the line segment into thirds. Then erase the middle third (but keep the dots). With the two remaining line segments, draw dots to divide into thirds, then erase the middle third (but keep the dots). Continue. The limit is the Cantor Set.
• This one's easy to see, but hard to explain: the Koch Snowflake. And here's a good edible approximation.
• Draw a square. Divide into ninths, and remove the middle ninth. Divide each remaining square into ninths and remove the middle ninth. Continue. The limit is the Sierpinski Carpet. We didn't make Sierpinski cookies today, but I'm eyeing that for a future project.
• If you take the idea of the Cantor set (1 dimension) and the Sierpinski Carpet (2 dimensions) and go up to 3 dimensions, you get the Menger Sponge. Someone said this made them dizzy - I said it makes me a bit dizzy, too!

My favorite part of the class was when the kids figured out, all on their own, that the Cantor set contains no lines but infinitely many dots. Wow :)

We also had a discussion about how many sides a circle has. 0? 1? infinitely many? 2 (an inside and an outside)?