A possible candidate for the “geometric origami” book I’m working on, although I’m not sure exactly what geometry properties it demonstrates!!

- crease 4×4
- valley twice to make a 4×1 strip
- add one diagonal in each small square
- form a tube, tucking 1 square inside
- twist!

Curiously, if you view it with a corner pointing upwards, the mind suddenly interprets it as an octahedron with sections missing….

Well, you mention in your last sentence there the main geometry concept it illustrates: The corners of the paper form an octahedron! But the real math there is proving that this is actually true, and not just an “it looks like an octahedron, but is it really?” kind of thing.

The trick to prove this is noticing that in a regular octahedron, if the length of one of the sides is 1, then the length of a diagonal (the line connecting two opposite corners of the octahedron) must be sqrt(2). (That’s the square-root of 2.) I’m not sure how easy that is to believe. But if you cut a regular octahedron in half to get a square-based pyramid, you can pretty easily see that this edge-length = 1 and diagonal length = sqrt(2) is the real deal.

Then look at your folded model, in particular your crease pattern. It’s has the side length =1 and diagonal length = sqrt(2) as well! So it’s an honest-to-goodness octahedron!

That’s some pretty good math, if you ask me!

I bow to your titanic expertise Tom ;)

I think the name of the model diagramed is “Hidding square” and it has been created by the genial Simon in June 2000.