# Studying symmetry with Rubik’s cubes and Zometool

Over the last week we’ve been looking informally at symmetry. The topic came up because we were studying counting with symmetry in Art of Problem Solving’s “Introduction to Counting and Probability” book. Questions such as “how many different ways can 5 people sit around a table, if ways that differ only by rotation are considered the same?” let do some nice discussions, but the slightly different question: “how many different ways can 4 keys be put on a key chain, if ways that differ only by rotation and flips are considered the same?” was a little more challenging to talk about. Wanting to talk about that second question in a little more depth led to the fun diversion into symmetry that we’ve been doing this week.

We started with symmetries of simple polygons and moved on to the symmetries of a cube yesterday. That jump was much more difficult than I expected. We talked a little bit about cubes in the morning, they boys thought more about it during the day, and we had this nice conversation last night when I got home from work:

Today I wanted to move on to symmetries of an octohedron and got a nice surprise when they boys remembered looking at octohedron’s with our Zometool set earlier this summer.

As background, we go on a little vacation with college friends every year. This year the weather forecast for the first couple of days was lots and lots of rain (I think it was the remnants of a tropical storm), so I brought our Zometool set along hoping that it would be a fun way to pass the time with all of the kids. We also brought along this book that Patrick Honner had recommended on Twitter:

We did a few projects from the book as well as some other projects that probably would be best described as having the kids just play around. One of the neat projects that came from the kids playing around was this one from a 12 year old girl who assured me that she “hates math.”  It was really fun to talk to her about how her creation showed that you can fit 6 pyramids inside of a cube:

One of the projects from the book that we did involved building some Platonic solids.  I have to confess that I didn’t remember this project until the kids reminded me today if you want to build an octohedron, you start by making a “starburst.”  Bet you didn’t know that!

I’m glad that that the activity from the Zome Geometry book had stuck with them.  It was a much better starting point to talking about symmetries of an octohedron than what I had planned.   The “starburst” idea really helped them understand why those symmetries are the same as a cube:

I’m happy to see one more example of how the Zometool set helps kids understand shapes and geometry.  Happier still to see that they are also a great tool to help kids understand symmetries.  The list of useful recommendations from Patrick Honner is getting to be quite large!