Last week we did a project from our Zome Geometry book on Descartes’ Theorem for polyhedra:

Descartes’ Theorem from our *Zome Geometry* book

Shortly after publishing that project I received this comment from Youtube user mxlexrd about our torus video (the last video in the project above):

“I think you calculated this one wrong, for the corners in the centre of the ring the angles should be 90, 90, and 270. This means: Sum = 450 360 – sum = -90 Total = -90 x 8 = -720 Therefore the total for the whole shape is: 720 – 720 = 0 According to wikipedia, the total should be 360 x the Euler characteristic, which (according to wikipedia) for a torus is 0.”

I was really happy to get this comment and investigate the problem a little more. The investigation turned out to be pretty difficult, actually, and I’m still not 100% sure that I’ve understood the issue myself, though I understand it enough to know that the comment is absolutely correct. I’m glad mxlexrd took the time to provide the detailed feedback.

Because of my uncertainty about how to correct the mistake it took a few days to figure out how to design a good project to show the correction. Yesterday it occurred to me to build a slightly different “torus” out of our Zometool set to test the new calculation on a different shape. By Descartes’ Theorem, the calculation should be the same for two shapes that are topologically equivalent (and in this case should be zero). I hoped that if we were doing the calculation incorrectly we either wouldn’t get zero, or we would get different answers for the two different shapes. Fortunately we did get zero for the answer both times, so hopefully we are now on the right track.

Here’s the project:

Only my older son participated this morning because my younger son has a little stomach bug. We started by reviewing the calculation in Descartes’ Theorem for a dodecahedron. A better review of the ideas behind the theorem is in our prior project – the review here was just a way to remind my son about how the calculation worked:

Next we attempted to correct the previously incorrect calculation for our “square” torus. I hope that I have understood why the three angles in the middle are 90, 90, and 270, though I am still not 100% sure on this point.

Our last step was looking at our new “pentagon” torus shape. We walk through the calculation again and arrive at the same result as before! Maybe, possibly we’ve understood the calculation in Descartes’ Theorem correctly this time around đź™‚

So, a fun project correcting a mistake in a prior project. I’m happy to have been alerted to this mistake. Topology wasn’t my field in school, so I don’t have quite as many alarm bells going off when we head down an incorrect path in that area of math. Hopefully going back and correcting our mistake was a good lesson for the kids, too (and possibly a double good lesson, as figuring out how to build that pentagon torus was a pretty good challenge!). I definitely feel as though I’ve understood Descartes’ Theorem a little better going back through it a second time.

## Comments

Yes, you’ve understood the angle measures around the hole correctly. This is where it would have been easier to see with a solid model. You could also note that the “degenerate” corners that were along edges could have been included in the calculation and wouldn’t have changed the results as they all had angle sums of 360 (so 360 – angle sum = 0). This is one of the things you should expect to see from a topological invariant: refining (or changing) the network that tesselates/covers the space shouldn’t change the resulting calculation.

I think your two students would be able to see the result 360 * (V-E+F). In fact, I thought you were going to get there in the example of the pentagonal pyramid with the observation that the actual angles of the triangular faces didn’t matter.

So:

(1) recognize that every angle gets included once

(2) thus, your procedure gives 360 * V – sum of interior angle sums for all faces

(3) because the interior angles of planar polygons add to 180 * (#sides – 2), the sum of all those just depends on the total number of edges and faces (remember, an edge is a side shared by 2 faces).

One benefit of having this discussion is that it might help make the procedure of Descarte’s Theorem easier to understand. As presented, it might feel like a convoluted set of arbitrary steps that magically produce a consistent result. Another idea is to do some constructions with polydrons (or magnetiles). Playing with them has given me a greater appreciation of why the angle deficit is a natural thing to consider.

Oh yes – magnetiles!! – hadn’t even thought of that. Sad that the next couple of weeks is going to have so much travel – the list of projects that I want to do is backing up and up.