Here’s an idea I’ve been working on for a while. I’ve seen this kind of thing done as modular and thought it was doable as a single sheet, and I figured I’d go for it. Single-sheet stellated polyhedra are pretty advanced, but the color change brings it to a whole ‘nuther level of complexity. It turns out to be a very rewarding shape to fold, and the design is replete with all kinds of interesting symmetries.
The first challenge was to work out how to achieve the arrangement of alternating colors. Once I’d worked that out the resulting (flat) shape would serve as the base for the 3-d phase. I needed two corners to come to the center like a blintz, but offset. Working out the amount of offset for the grid to be the right size was the key problem. It turns out the key angle is 67.5 degrees, which is 3/4 of 90 degrees and easily derived. It also turns out the angle has a slope of 3/2, which is also easily derived from a square grid. From this I was able to work out the arrangement of the squares in the inner rotated grid and the outer triangular grid areas, which correspond to the blintzed flaps. The 3/2 slope made was convenient because the grid is has an integer relation of the unit the whole. Each square of the grid has a length of 2/13 the edge of the paper, as you can see in the crease pattern. Neat, huh?
Another interesting property of the model is that once you’ve made the base and put in the color change squares, the easiest way to deal with the leftover paper on the two remaining corners is to flip the model over and do the same thing. The result is the pattern is the same on both sides, although made with opposite colors, and either side can be used for the outside.
The finished model fold together well and looks really good. I went ahead and made a few.
As an added bonus Lizzy and Michelle were folding bowls and picture frames, so here’s one.
