Have a Happy Halloween!

No, wait. It's past Halloween....Happy Thanksgiving?

No?

Okay, then. Merry Christmas.

I am substantially late in delivering my Halloween project this year, but I do not apologize for it. I've been quite busy in the last few months and only had time to work on this during the weekends. What's more, this project was much more ambitious than what I usually fold for Halloween. The shape itself has no formal name, but if I had to give it one, I'd stick with the usual conventions in Stewart Toroid nomenclature and call it an "icosaugmented near-miss rhombicosidodecahedral toroid"—that is, a 20P₃(20Y₃ + 30P₃ ≈ E₅/12P₅(D₅)). With stars.

Intricate though this structure may seem, it is really no harder to make than a "standard" 210-unit tetrahedral symmetry toroid. The only real difference is that I capped each tetrahedron with an open, 9-unit triangular prism "cage" instead of the usual 3-unit triangle.

The final breakdown is as follows:

• Black units: 330 Open Frame Units, from Tomoko Fuse's Unit Origami: Multidimensional Transformations, pages 65-66. Each unit was folded from a quarter-sized sheet of paper (75x75 mm².) Online instructions for these units can be found on page 4 of this PDF file.

• Orange units: 180 Star Modules, from Miyuki Kawamura's Polyhedron Origami for Beginners, pages 74-75. Although they may not look the part, these really are edge modules, which means that I needed nine of them to create each orange "star" (actually a cumulated triangular prism). Each module was folded from quarter-sized paper.

• Green units: 30 Star Modules folded out of full-sized sheets of paper (150x150 mm²). This double-size Star Module dodecahedron (which very closely resembles a small stellated dodecahedron) just happens to fit in the dodecahedral void at the center of the toroid. This was an unexpected result; before starting the project, I thought that any Star Module polyhedron would fit inside an analogous Open Frame Unit polyhedron folded out of sheets having the same starting size. It was only after I finished folding a miniature Star Module dodecahedron out of 30 quarter-sized sheets that I realized it would be too small to fit snugly in the center.

Big star or small, that central dodecahedron was more or less an afterthought.