# Descartes’ Theorem from our “Zome Geometry” book

We decided to kick of 2015 with a Zometool project. Flipping through Zome Geometry last night I found a neat section on Descartes’ Theorem for polyhedra. This project made for a fun start to the year.

We built the 7 shapes that we’d need for the project before we started filming. I didn’t tell the boys what we’d be doing with the shapes, though, it was just prep work.

In the first part of our talk this morning I explained the procedure that we needed to follow to explore Descartes’ Theorem and then we worked through the calculation for both a cube and a tetrahedron. There was a little bit of confusion getting going with the process, but we were able to complete the calculation for both shapes. In addition to the geometry, there is lots of good arithmetic practice in this project!

For the next part of the project we completed the calculation in Descartes’ Theorem for a dodecahedron, an icosahedron, and an octahedron. The one little bit of extra work we had in this section was completing the calculation for the dodecahedron – it wasn’t obvious what the angles in a pentagon where, but we figured it out:

Next we moved to a slightly more complicated shape – a pyramid having a pentagon for a base. The difficulty here is that we don’t know the exact values of the angles. However, we do expect that following the procedure for Descartes’ Theorem should end up with a value of 720 degrees. We went through this computation and discovered the surprising fact that it doesn’t matter what the angles are. Nice one, Zome Geometry authors 🙂 :

The last part of our project was studying at a solid that could not deform into a sphere. This talk was interesting because it wasn’t super easy for the boys to see the shape. I probably should have used paper or something to make the shape solid, though maybe the discussion about what intersections in the Zome set were actually corners was helpful. Not sure how this one will work on video, but hopefully it is useful. Even with the slightly disjointed discussion, we did get to a different answer than we’d gotten to before. That was fun and it showed the boys that this shape was somehow different than the prior shapes, and a fairly simple procedure of counting angles could help us understand the difference:

So, another fun project from Zome Geometry. The Zome construction in this one isn’t too difficult, which is nice. This project gives lots of opportunities to expand on some basic 2d and 3d geometric knowledge and also plenty of opportunities to build on number sense. Fun way to start 2015.