Snub Cube with Windows, from Unit Origami: Multidimensional Transformations by Tomoko Fuse.

The snub cube is one of those fancy Archimedean solids, for when you’re bored with the Platonic solids.  Each vertex lies at the intersection of one square and four triangles.  All together, there are 24 vertices, 60 edges, 32 triangular faces, and 6 square faces.  And can you imagine, it’s only made of 12 sheets of paper?

The Snub Cube I think is particularly characteristic of Tomoko Fuse’s style.  It’s kind of a simple no-frills model–make a snub cube, no flowers or decorative designs, just 12 sheets of paper, bam!  12 sheets of paper isn’t enough to cover 38 faces?  Let’s leave 8 of the faces uncovered, call them windows.  And let’s have each sheet of paper span 6 different faces.  So simple!

But the thing is, if you chain together two triangles and a square, the total angle is 60 + 60 + 90 = 210.  Man, who wants to make 210 degree corners?  But that’s what’s needed here.  And as it turns out, the best method is to start with paper in a 1:sqrt(2) ratio.  I start with square pieces of paper, but I can cut them down to the right size, fine.

So that’s Fuse’s style.  Lots of kludges in the service of simplicity.