Here is a crease pattern for an origami Great Dodecahedron. This fascinating shape is something like a sunken icosahedron, and can also be seen as twelve intersecting pentagons with a raised star on each face. I tried several iterations of the layout because the details of forming tabs and pockets to close the model took some trial and error to get right. The basic idea is fairly straightforward. I use fivefold polar symmetry, and the whole pattern embedded in a single pentagon that takes up pretty much the entire square sheet. I was able to divide it into a grid of parallelograms using simple ratios. Each parallelogram then gets subdivided into the triangles that form the faces of the shape.
Since I just fold back the corners of the square to form the base pentagon, I tried a version folded from a pentagonal sheet, but this turned out not have enough extra paper around the edges to from the tabs and pockets. The pentagon’s height is slightly less than its width, which results in a then strip of unused paper at the bottom edge of a square sheet. I decided to try folding the strip around all five sides (except where it gets truncated at the corners), and that turned out be just the trick.
I’ve successfully folded a couple of these now that stay together well. Pictures as soon as I make one out of nice paper.
