Algebraic enumeration of polypolyhedra (Q6651115)

This fine piece of undergraduate research studies \textit{polypolyhedra}, certain collections of line segments in \(\mathbb R^3\) enjoying very strong symmetry properties.\N\NThis is motivated by the works of the origami artist Robert J.\ Lang, who attempted to define and classify polypolyhedra to study origami in a more rigorous fashion. The students at the University of Canterbury (New Zealand) revisit his studies, suggesting a precise mathematical definition of polypolyhedra and providing a provably correct classification of them.\N\NDue to the symmetry properties mentioned above, the authors quickly reduce the geometric question to one on finite reflection groups, that is, finite Coxeter groups. The enumeration problem is then turned into an analysis of conjugacy classes and applications of Burnside's lemmas.