FOLDING EQUILATERAL PLANE GRAPHS

We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

Weight functions, double reciprocity laws, and volume formulas for lattice polyhedra

We extend the concept of manifold with boundary to weight and boundary weight functions. With the new concept, we obtained the double reciprocity laws for simplicial complexes, cubical complexes, and lattice polyhedra with weight functions. For a polyhedral manifold with boundary, if the weight function has the constant value 1, then the boundary weight function has the constant value 1 on the boundary and 0 elsewhere. In particular, for a lattice polyhedral manifold with boundary, our double reciprocity law with a special parameter reduces to the functional equation of Macdonald; for a lattice polytope especially, the double reciprocity law with a special parameter reduces to the reciprocity law of Ehrhart. Several volume formulas for lattice polyhedra are obtained from the properties of the double reciprocity law. Moreover, the idea of weight and boundary weight leads to a new homology that is not homotopy invariant, but only homeomorphic invariant.