I started in on a geometry course with my older son this week. On Wednesday we were discussing some basic shapes and he asked a neat question: If you have a circle in a plane, are there more lines in the plane passing through the circle or more lines in the plane that don’t pass through the circle? Fun!

I told him that the answer to the question was a little more complicated than it seems, but we’d talk through it over the weekend. Well, the weekend is here and we talked through it this morning!

We started by introducing the question and talking about some of the non-intuitive properties of infinity. I thought the easiest place to start would be comparing the set of positive integers with the set of positive even integers since this comparison is a nice way to show that infinite sets are a little strange! I think that kids can understand some of the basic ideas about infinite sets, even if some of the concepts my be a little over their heads:

Next we moved on to a slightly more difficult question – comparing the set of positive integers with the set of positive integers that are powers of 2. In this case it looks like the second set is much smaller than the first one, and finding a way to see that these two sets have the same size did prove to be a challenge. However, with a little nudge, they were able to find a way to map the two sets to each other and even sort of answer the question “what is the opposite of powers?”

Probably the next natural step would be to show that the rational numbers are also countable, but I decided to skip that proof because I was worried that it would be more of a distraction and wouldn’t help so much with the question about lines and circles. Instead the next thing we talked about was comparing the real numbers to the integers via Cantor’s diagonal argument. This argument shows that there are more real numbers than integers. Although I didn’t necessarily want to focus on the different infinities, I thought it was important to help them understand the idea that just because two sets are infinite, they may not be the same size. In retrospect, I wish I wouldn’t have called this the “next infinity,” I guess we’ll have to correct that little slip the next time we talk about infinity.

With all of this background behind us, we moved on to answering the original question about lines and circles. We began by looking at a problem that is a little easier – what happens if we look only at vertical lines? Restricting our attention to this slightly easier problem allows us to see a surprising result – the number of points between 0 and 1 is the same as the number of points between 1 and infinity!

Now with the discussion of the vertical lines out of the way we can solve the general problem if we can figure out how to deal with lines that aren’t vertical. As luck would have it my older son thought looking at horizontal lines would be a good way to start. That idea got the boys thinking about rotational symmetry and led them to the solution to the original problem! Unfortunately I got confused on one of the pictures, but hopefully that 30s of confusion didn’t cause too much confusion – the perils of illustrating some of these ideas early on Saturday morning!

This was a really enjoyable project and the boys seemed to have a lot of fun and stayed engaged all the way through. I’m extra happy that this project came from a question that my son asked earlier this week. It is nice to talk about some of these ideas from pure math every now and then. It helps show younger kids that math isn’t just about playing around with numbers.

There’s something I don’t quite get – in looking at the specific vertical lines within the circle, you used only rational numbers – 1/2, 1/3, 1/1,000,000, presumably 3/5 etc, which mapped to reciprocals that were also rational numbers – 2, 3, 5/3.

Is there something in the jump from talking about numbers to putting lines on a plane that means you can’t have a line thru a point at an irrational number? I’m phrasing this badly… there’s a point on the number line for 1/sqrt(2), right? But is there no point on the plane at 1/sqrt(2) through which a line can be drawn?

Because if there is a 1/sqrt(2) through the circle that maps to sqrt(2) not through the circle, wouldn’t that mean both the lines thru and the lines not thru would be uncountable infinities, but they’d somehow map to each other… that doesn’t sound possible (then again, most of this is so far beyond what seems possible, what’s one more thing).

I definitely could have (and probably should have) used an irrational number in the example there. The calculation would work exactly the same way -> for an irrational number x between 0 and 1, the reciprocal would be an irrational number greater than 1.

You are right that the number of vertical lines passing through the circle is uncountable, and the number of vertical lines not passing through the circle is also uncountable.

Thanks – great post. Wish I’d had this kind of thing when I was in school.