Yesterday we did a fun project based on Evelyn Lamb’s article about the Infinite Earring:

This morning the kids wanted to talk a little more about topology, so I chose to follow Lamb’s article a little more and talk first about the Cantor set and the the Long Line.

To discuss these topics, though, we first needed to have a brief discussion about two types of infinities – countable and uncountable. We’ve touched on infinity before, for example To Infinity and . . . to the next Infinity, but a 10 minute review seemed appropriate.

We started the project talking through some examples of countably infinite sets:

The next piece of the introduction talks about a larger infinity – “uncountable” infinity – and introduces Cantor’s diagonal argument. This argument was a little confusing to the kids, so we needed to go through it twice. I think it made much more sense to them the second time around:

With that introduction out of the way, we talked through the Cantor set. I loved hearing the various ideas from the kids here – does the Cantor Set have no points or infinitely many points? Does it have any length? What’s going on with this set!

Finally, one of my favorite fun topological spaces – the long line. The goal of introducing this example was just to show them a neat way to think of a regular line and then to show them a way to make something that seems “longer” somehow. They seemed to enjoy thinking about this strange space.

I’ll be interested to see if they are interested to talk more topology tomorrow – there are a couple more fun things that kids might find interesting to think about. Even if it just these two little projects, though, this diversion was definitely fun. It is so neat to hear kids think through these unusual ideas from math!