A system for constructing relatively small polyhedra from Sonobé modules

Anyone familiar with the Sonobé module has probably assembled both the augmented octahedron and the augmented icosahedron Instructions for both appear in both Kasahara and Takahama and in Mukerji, and can also be found on most of the innumerable websites discoverable by googling the word “Sonobé”. Kasahara and Takahama give some sketchy instructions on how to build some other polyhedra with these modules. Some of their models are relatively small but their concentration, as does that of the websites, turns rapidly to monstrously large polyhedra containing hundreds or even thousands of units. Thus the casual explorer of the wonders of the Sonobé module, who perhaps has no desire to fold thousands of units before construction can even begin, is left without many new models small enough to be built in half an hour or so. If you have an augmented octahedron at hand, look at it carefully and notice that it consists of eight equilateral triangles capped off by low 3-sided pyramids. The key insight here is that other polyhedra all of whose faces are equilateral triangles can also be constructed in their augmented versions out of Sonobé modules. Such polyhedra are called deltahedra after the Greek capital letter delta: ∆. The icosahedron is a deltahedron, which is why an augmented version can be constructed of Sonobé modules. It turns out that there are six other convex deltahedra besides the octahedron and the icosahedron [see 1, 6]. These references are useful because they include instructions for making paper models of the deltahedra, which in turn are useful for designing augmented Sonobé versions. As you become more familiar with the system described here you will be able to dispense with the paper models. There are also infinitely many nonconvex deltahedra. In theory augmented versions of any of them could be constructed from Sonobé modules. In practice, however, not all are constructible because steep angles between the faces of a given deltahedron can force the pyramidal caps to intersect which (unfortunately) isn’t possible with standard origami paper. Now, as you stare at the augmented octahedron, notice that there are two kinds of vertices: The kind at the tips of the 3-sided pyramids and the kind where four pyramids come together. There are six of the second kind, which correspond to the six original vertices of the octahedron. For any given deltahedron it is the pattern in which these original vertices are connected along with the number of pyramids at each one which determine the construction of the augmented version. Given any deltahedron it is possible to draw a diagram which will serve as a blueprint for the