I first posted about an origami model based on a Star Polyhedron in November 2012, but I knew at the time that many more models were possible based off of a similar idea of taking a Star Polyhedron and removing parts of each face such that some underlying topology of the polyhedra was maintained, but the faces no longer intersect. These representation of a star polyhedron can be thought of as ‘topological’ as they emphasize the internal connectedness of these self-intersecting figures.
Since then, I have folded some more polyhedra following the same idea, and started naming them after actual stars (seeing as they are “star” polyhedra). I will presenting on some of the math and design thoughts for this series at the 6th International Meeting on Origami in Science, Mathematics and Education (6OSME) in Tokyo and displaying two of them at Bridges 2014 in Seoul, Korea later this summer.
Unukalhai, below, is a ‘topological’ model based on the Small Triambic Icosahedron and is composed of 60 rectangular sheets of paper folded into identical origami units. Each face is represented by a 3-pronged spiral, and can be thought of as a subsection of a small triambic icosahedron, chosen such that the model can be joined together without self-intersection. Unukalhai was made out of five colors of paper to highlight the natural five-coloring of the icosahedron.
Tania Australis was the first star polyhedra origami piece that I folded, and has been mentioned on this blog before. It is composed of 30 identical ‘S’ shaped pieces that each represent one face of a Great Rhombic Triacontahedron and was inspired by George Hart’s Frabjous sculpture.
Tania Borealis is composed of 30 ‘S’ shaped pieces, put together in the shape of a Medial Rhombic Triacontahedron. It complements Tania Australis, but is much more fragile, so I do not plan to put it on display in Asia.