Also the dissection into 12 pyramids is beautiful. There’s also the cube contained in the dodecahedron, with the 6 “roof” shapes on the side. A model made of Zometool is one great way to see and understand this.

Speaking of dodecahedra, rather than the Platonic regular one we see so much, I’ve recently fallen in love with the rhombic dodecahedron. Added bonus is that its surface area and volume are a lot easier to understand (and compute exactly) with elementary techniques!

]]>Inspired by George Polya (How to solve it) I would also suggest he looks at cubes and octahedrons in the “lots of little pyramids” way, as a clue to the 3D triangle arrangement. ]]>

I think that at J’s level he doesn’t need trigonometry, he would be perfectly well of with the idea that he can draw a pentagon (construct or copy) and measure stuff, and use ideas of symmetry for scaling. I have had some fun with this problem already.

And his figuring out the pentagonal pyramid thing is the REAL math. ]]>