# Tetrahemihexahedrons and other things that violate Euler’s Criterion

In math 2 we’re working with three-dimensional figures. I like to start with this problem, inspired by this sign spotted outside of Labyrinth in Capitol Hill:

I LOVE Labyrinth, and think this is an excellent way to get the students to start thinking about the connection from two-dimensional shapes to three-dimensional objects.

Later I gave them a task involving toothpicks and marshmallows. I handed them each a measured-out quantity of each, as well as a table with two out of the three qualities in Euler’s Criterion* filled in – the number of edges (toothpicks), faces, and vertices (marshmallows). The class struggled with but eventually succeeded in creating cubes, icosahedra, and dodecahedra with this limited information about what they were supposed to look like. If we hadn’t run out of time I was going to challenge them to create a tetrahemihexahedron.

I was only going to ask them to try it because it would almost definitely be impossible with toothpicks – not all of the edges can be the same length for it to work. I’d love to have them build a net for one on the group part of the final though – definitely more exciting that a rectangular prism.

I’ll probably be back tomorrow; fractals have been very exciting lately, plus I made a sweet quadrilateral flow chart.

*For all convex solids, V – E + F = 2. For the Tetrahemihexahedron it yields 1 instead.