# Five tetrahedrons in a dodecahedron

Saw two different references to tetrahedrons in a dodecahedron yesterday.

First from the American Mathematical Society:

Second in section 3-27 of Henry Segerman’s new 3D printing book linked by Evelyn Lamb:

Seeing the first link made me wonder if we could make the shape out of our Zometool set. Seeing the second link made me think that we definitely had to try it!

I started off by showing the boys the article from the American Mathematical Society just so they could see the shape. It was fun to hear their thoughts on the shape evolve as they noticed more and more about it. It was pretty cool to hear that they wanted to try to build it and also great that they were immediately wondering whether or not it was even possible to build:

[. . . . and one again, sorry for the blank thumbnails on the youtube videos. This problem has been happening more and more lately, but it usually goes away in a day or so. I can’t figure out what’s going on but the videos should play just fine. Ugh.]

The next part of the project was seeing if it was even possible to put a tetrahedron inside of a dodecahedron with our Zometool set. It isn’t obvious how to do it and I think this section of our project shows how the Zometool set helps kids work on problem solving:

With the camera off I had the boys try to add a second tetrahedron to the shape. Here’s their description of the process and some thoughts about the new shape:

With the camera off the boys added the 3rd and 4th tetrahedrons, but I thought it would be fun to show the process of adding the 5th one. There’s not a lot of room, so putting the 5th one in is a pretty challenging exercise for kids. By total miracle we had exactly the right number of long green struts to make this shape!

Maybe showing the full build for this last tetrahedron is a little tedious, but I thought it was cool to see it all come together – including all of their thoughts along the way. At the end they noticed some pentagons formed by the intersection of the 5 tetrahedrons. That made me think to go back to the original AMS blog post since they had a nice picture of those pentagons.

So, going back to the AMC blog post we viewed the new pentagons they saw. They boys seemed pretty excited and even had some fun comments about the project.

Actually, that ended up not being quite the end. The boys wanted to try to make the new pentagons after we finished up, and it turned out you could make them with the Zometool set – here’s the picture:

A super fun project. So glad that I saw the two references to the 5 tetrahedrons yesterday!