Banach-Tarski, Hilbert curves, and infinite sets

I have my kids write short reports every day on chapters they select in Cifford Pickover’s amazing “Math Book.”  (sorry I don’t know Latex well enough to format the title correctly).

These reports give them a chance to see fun math outside of the standard stuff covered in their school books.   Last week my younger son stumbled across the section on the Banach-Tarski theorem and it really intrigued him.  I finally got around to talking a little more about that theorem with the boys today, though it obviously isn’t the easiest subject to cover with younger kids!

The first thing we talked through was the two different statements of the theorem.  A short, and excellent as usual, summary of the two theorems can be found on the Cut the Knot website:

I covered the the two different statements of the theorem and moved on to a much easier to understand example of infinite sets – why there are the “same number” of positive integers and positive even integers.

With the example with integers and even integers showing us how to compare infinite sets, I moved on to showing them that a line segment of length 1 has the “same number” of points as a line segment of length 2.    The ideas in this proof at least let you see how one object could somehow be the “same size” as something that seems to be twice as large.


The next thing we talked about was how we could see that a line segment of finite length could have the “same number” of points as an infinitely long line.    We approach this idea using stereographic projection:

Next we moved on to 3D and I showed them that the sphere has the same number of points as the plane.  The idea here was also to look at stereographic projection, though luckily for this example we have a special prop designed by Henry Segerman that we found on Laura Taalman’s 3D printing blog:

Goes without saying that holding the model in your hand is quite an improvement over a sketch on the board!

So, by this time we’d seen that a line segment has the “same number” of points as the whole line, and a sphere has the “same number” of points as the plane.  Now we show something really amazing – a line can fill up a square, and hence the plane.  That means that a line segment has the “same number” of points as the whole plane.  Wow.

The approach here took much longer than what is on camera.  We found this great website that gave a tutorial on Hilbert Curves:

We also found some space filling curves on Laura Taalman’s blog:


So, the 5 minutes on camera was actually preceded by a couple of hours of printing and drawing Hilbert curves on our own.  It made for a really fun morning:

Lots of people to thank for this one – Clifford Pickover,  Kerry Mitchell, Alexander Bogomolny, Laura Taalman, and Henry Segerman.  So glad to have resources like theirs online to help kids learn about this kind of fun math.

** Addendum **
After finishing up this post we were playing around with our 3D printed Hilbert curve and took it off the base.  After we did that, we found that we could stretch it out into almost a line.  Cool!!  I think it helps kids get a better feel for the fact that it is all one long line segment twisted up into a curve: