Day 43: A Pyramidal Conjecture

A few weeks ago, a Geometry student of mine and I were chatting after class during our biweekly meeting. He’d been thinking some about the cut-and-paste challenge from Investigation #5, and this led us to talking about what the problem would be like in 3D.

As we started to put pyramids together using Geofix, we noticed that it seemed like some equilateral square-based pyramids and some regular tetrahedra might fit together perfectly to make a polyhedron.

Like so. And we brought by Metrocard dimpled octahedron over to compare. But the plastic pieces didn’t fit together so well, and so the student decided to makes some out of paper to better test our conjecture. He made nets for the pieces in Geogebra, printed them out on cardstock, and assembled them into the form shown in today’s pic.

It sure looks like it works. Later in the year we’ll have analytic techniques to further justify what’s going on here. Because maybe there are gaps?

This was a nice, unexpected foray into polyhedra and dissections. Lovely.


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