Yesterday I saw an incredible video on a discovery about black holes made by a team led by the physicist Kip Thorne. The amazing thing about this discovery is that it came about through Thorne’s consulting work in the production of the new movie *Interstellar.* You just never know when inspiration is going to strike! The video I saw yesterday is here, though sorry I can’t get it to embed right:

http://player.cnevids.com/embed/5446f00e61646d41b4130000/5176e89e68f9daff42000013

What really struck me watching the clip above was the statement from Kip Thorne around 2:30 when he describes his reaction to seeing a picture of a properly rendered black hole: “I’d known it intellectually, but knowing intellectually is completely different than seeing it, than feeling it.”

This thought was in my mind the rest of the day yesterday and I couldn’t stop trying to think up other math-related ideas that would be easier to understand if you were able to hold them in your hand. At night I was flipping through my old Algebra book from college and found a great example by luck. This book is one of my all time favorite math books – Artin was a terrific teacher:

In Chapter 7 of the book Artin explains some basic group theory and uses the rotation group of the icosahedron as an example. One of the theorems shows that the group of rotations of the icosahedron is the same as second group – the alternating group . The details about aren’t important for this blog post, but what is important is that a critical geometric idea in the proof is that you can inscribe 5 cubes inside of a dodecahedron. Here’s the picture Artin gives on page 200 of the book:

I never really completely understood this proof, and I never understood at all how these cubes inside of a dodecahedron related to the rotations of an icosahedron. Thanks to Kip Thorne’s fun video and our Zometool set, though, I was determined to find out. The kids were going to come along for the ride, too 🙂

To start off the project I asked the boys to boys to build a large icosahedron out of our Zometool set while I was at work today. The sides of this icosahedron have length equal to two of the longest blue Zome struts. Here’s our starting shape:

The dodecahedron is a dual shape of an icosahedron and it turns out there’s an amazing way to add a few new Zome struts to the icosahedron to make a shape that combines the icosahedron with a dodecahedron. It is really cool to see this shape come together (another Zometool miracle!). Here is the shape (and this + all of the rest of the videos are published in 1080p HD, so you can watch in hopefully non-blurry full screen to get a better view of the shapes):

With the icosahedron and dodecahedron together, now all we need to do to get to the shape Artin was describing in his book is to add in the cube. I’m so happy that the Zometool set was able to help with this last step! With this final shape built, we are holding the picture from the book (plus the original icosahedron) in our hand. Again, hopefully the cube is visible in the video:

Now to see what the rotations of the icosahedron do to the cube. Of course this is a fun fact all by itself, and that’s what the boys are seeing. I’m seeing the critical step in showing that the Icosahedral group is isomorphic to , though, which was the group theory piece that was so hard for me to visualize in college. With the shape right in front of you it is easy to see how it all works – score one for Kip Thorne!

First we looked at what the rotations of the 20 triangles of the icosahedron do to the cube. To make what we were rotating a little easier to see on camera we put some lego figures next to each vertex of the triangle we were rotating. Hope that helps you to see the bottom triangle, and the rotations, a little easier in the video:

Next up, the 5 symmetries that come from the rotations of the icosahedron around a vertex. As an aside, these 5 rotations sort of help you see the similarities between the icosahedron and the dodecahedron. Again we lined up lego figures to help you see the rotations a little better in the video:

Finally rotating around and edge. This is the easiest symmetry of the icosahedron to see since we are just rotating by 180 degrees around the middle of an edge. It is interesting to see that although the icosahedron itself is unchanged by this rotation, the cube isn’t:

After we finished I asked the kids to help put away all of the Zometool pieces. My younger son told me that he didn’t want to take apart “this awesome shape” just yet. Yes!!

I’m glad the boys had as much fun with this project as I did. I definitely got a better understanding of a piece of group theory that I never really properly understood before. The boys were able to see (and build!) some amazing 3D shapes and also play around with rotations and symmetries a little. So so so much fun!!

My intuition is that this is where our kids get the most powerful lessons: when they see (and help) us in our own mathematical investigations.

My little ones have been recently spent a lot of time playing with polydrons. My middle son is especially taken with the icosahedron. It has also been a favourite of mine since D&D days long ago. Your post has convinced me to make sure I get a set of zometools.

Also, I second the recommendation for Artin’s Algebra.

It occurred to me that, if you wanted to make another version of these videos that emphasized the internal cube more, you could logically separate the construction into separate steps:

(a) show that the dodecahedron is the dual to the icosahedron

(b) show the dodecahedron with an internal cube (remove the icosahedron pieces)

(c) demonstrate the action of the dodecahedron symmetries on the internal cube

I understand that with the physical object in hand, it is cool to have all three shapes integrated into one model.