[ sorry to be a little rushed (and unedited) on this write up – I had to be out the door by 10:00 am and wanted to get this post finished before I left]
Yesterday I wrote about a problem I overheard Bjorn Poonen discussing earlier in the week:
That blog post has the full problem – which is super fun to think about. This post is an attempt to talk about the problem with kids. There are some neat geometric ideas in the problem and also lots of surprises.
So, here’s the first part of a two part set up for the problem – looking at a fairly straightforward situation in 2 dimensions/. The question here is how can we compute the radius of the small circle in the center of our picture?
This question led to a great discussion about symmetry and geometry.
The next question we tackled was the 3-dimensional version of the problem. The 2-d set up problem (and discussion) really helped my younger son understand the ideas required to solve the 3-d problem. The kids were able to see the generalization to n-dimensions pretty quickly!
At the end of this video we lay out the three questions in Poonen’s problem:
Next we looked at the solution to the first problem – does the inner n-dimensional sphere ever get bigger than the n-dimensional spheres inscribed in the n-dimensional cubes?
Working through this question turns out to be a really nice inequality / algebra exercise for kids.
The 2nd of Poonen’s questions is this – does the inner sphere ever get so big that it actually extends beyond the n-dimensional cube?
This question is a hint that some really strange things happen in higher dimensions. How could the inner sphere leave the confines of the cube?
It turns out that the solution involves looking at an inequality that is quite similar to the one we looked at in the first part of the problem:
The final part of the problem is a little tricky for kids, so I just presented the volume formula for even dimensional spheres and used that as a starting point.
From there we did a little algebra and arithmetic to come up with a fairly simple inequality that would tell us if our n-dimensional sphere had volume greater than the 2x2x2 . . . x2 n-dimensional cube:
Finally, we wrapped up the project by putting our inequaility into Mathematica and seeing what happened. The discussion about hunting around to find the first dimension when the sphere was bigger than the cube was really fun!
One thing to be careful about watching this video (that I was not as careful as I should have been making it!) is that our “k” value is equal to half the dimension. Since we are looking at even dimensions, we are calling the dimension “2k.”
So, a fun project that was full of surprises. Who would have thought that there was an interesting fact about the 1,206th dimension 🙂