Maybe don't start with the Greek idea of polyhedra.
Recognize that we're talking about space in general, i.e. the right brain, i.e. the visual imagination, as well as one's spatial abilities with tools, such as airplanes. Any four non-coplanar points define a tetrahedron or more generally, any four events. In this sense, they're inescapable.
Thinking coherently about volume, space, the relationships therein, involves thinking about the planet, stars, great circles. It also includes thinking about micro structures unseen by the naked eye. Not just crystal lattices but molecules of all kinds, viruses etc.
In building up spatial fluency, we have this backbone construct called the concentric hierarchy that's like a Swiss army knife of geometric relationships. You can anchor discussions about duals, relative volume, power law, space-filling, left and right handed, honeycombs, angle vs. frequency, and so forth. In other words, if a science teacher can simply take for granted that all these students have already internalized the concentric hierarchy, jitterbug included, then said teacher knows she or he doesn't have to start from scratch when explaining some architecture or chemistry.
This blog post might help as well:
Remember, it's not just any approach to polyhedra that you're promoting, but one that's supremely memorable, dense with information, and centered around the volume tables here:
In today's environment, I think a selling feature would be that these students would join a tiny elite receiving information in this form. The link to Bucky could be explicit. They'd know about the map too, and about how they're privileged in knowing one of our great minds wanted us to focus on the electrical grids, how they're joining together and spanning the globe. A really cogent curriculum would be generous with such global data, but of course such data isn't that easy to come by, even on the Web. We're not in the habit of doing serious place-based education to begin with. You'd think Google would be into it. I should apply as their electric grid specialist and chief curriculum writer eh? They do hire curriculum writers I've noticed.
:: global studies class, Portland ::