## Wednesday, August 6, 2008

### Sonobe Phizz Duality

Sonobe and phizz modular origami units are assembled into polyhedral models in similar ways, but the models that they produce are "dual" to each other (vertices in one correspond to faces in the other). Assembling models out of sonobe and phizz units with this duality in mind provides a nice way of exploring duality and the relationships between edges, faces, and vertices.

The sonobe and phizz modules are both examples of edge modules that come together in groups of 3. In both cases, when the units come together in their groups of 3 they meet in a small triangular pyramid. These pyramids in turn come together in clusters of 3 to 6. The essential difference between the phizz and sonobe modules is in how these clusters form. In the phizz, there is a gap between the groups, so the resulting cluster seems to form the edges of a polygon. In the sonobe, the gap between the groups is small, so the cluster seems to form around a point. Consequently, in the phizz, the center of the cluster takes on the role of a polygonal face, while in the sonobe it is takes on the role of a vertex. Meanwhile, the pyramids formed by the groups of three units become raised vertices in the phizz unit, while in the sonobe they become cumulated faces of the resulting polyhedron.

We naturally interpret what phizz modules generate as polyhedral skeletons, while we see the shapes generated by sonobe as cumulated (or augmented) polyhedra. The fact that this different interpretation is based on the size of the gap that forms in the center of the module clusters suggests that seeing an origami model as a particular polyhedron or its dual is, to some extent, a matter of perception and interpretation.

In phizz models, the fact that the modules come together in groups of 3 dictates that the finished models have vertices of degree (or valence) 3, while in the sonobe case, the 3 units come together to make triangular faces. Dual models have the same number of edges, which corresponds to the number of modules required to build the model.

Some of the models that you can build are summarized below. The picture at the top of the post shows the 30-unit phizz dodecahedron and its dual, the 30-unit sonobe icosahedron.