It should probably be noted somewhere that this is more "papercraft" than "origami", and the thing you download is just the net of the solid, not instructions to make what's in the picture.
Also saddened by the lack of Catalan solids
Also saddened by the lack of Catalan solids
Even for the intersecting tetrahedra where instructions are available, I see that it's basically constructing the edges from 30 pieces of paper and then assembling them together. I wish they have more things that were built from a single square piece of paper.
Building from a single sheet is pretty limiting and beyond a cube build like the classic inflatable lantern you'd likely need glue to hold the folds down I think.