If you are so inclined, I uploaded two different crease patterns for this design: 3.6.3.6 Waterbomb/Flagstone Tessellation, crease pattern (with grid) 3.6.3.6 Waterbomb/Flagstone Tessellation, crease pattern (no grid, as pictured above) I’m really at somewhat of a loss on what to name these tessellations. If folded fully they become flagstone tessellations, ala Joel Cooper; if left three-dimensional, they are “waterbomb” style tessellations (although waterbomb is the wrong term for us to use here, but we’ll dispense with that argument for the moment.) Regardless, this is a 3.6.3.6 tessellation- the old standard, triangles and hexagons together. When you fold “normal” tessellations, the twists are always the dual of the tessellation you are folding. (For example, the 3.6.3.6 tessellation has a dual made up of rhombic stars- and the rhombic star tessellation, when folded, has hexagon and triangle twists, which most people actually think of as a 3.6.3.6 tessellation even though it’s really the dual of that…) However, with these “waterbomb” tessellations, there seems to be a little bit of change due to the geometry involved. I …

This is a crease pattern of one of my designs[1][2] (which for the sake of convenience I have dubbed ‘Arms of Shiva’.) However, this takes that design and throws it through the flagstonization machine, hinge-pleating all the folds to make the pattern you see above. While it might look like it is very wasteful of space (it is) there’s still less wasted paper doing it via this method than our normal straight pleat-and-twist methodology. Go figure! I would love to see this folded, in case there’s anyone out there feeling intrepid enough to give it a try… (update: I folded it, eventually!)

Here’s a rudimentary sketch of a 3.4.6.4 “Flagstone” tessellation. Formed by creating the initial “waterbomb” type collapses, and then twisted to form the familiar flagstone style tiling. I did not draw the lines for the WB collapses but I’m guessing you can figure this out if you have any idea what I’m talking about, right? More info on all this soon. I’m writing something down but I keep finding myself hamstrung by lack of proper wording and also some gaps in understanding. In the little box sketches in the upper right, you can find two examples of a rhombus tiling (the dual of the 3.6.3.6 tessellation) that have been done as a normal straight-pleat tessellation, and then as a flagstone style tessellation. If we adhere to using just the grid and it’s main offset lines (in this case, 30 degree angles) the flagstone style pleating is more efficient in terms of total area that one can tessellate given a particular number of pleats. Of course, the hinges that connect the flagstone polygons together can be …