Infinite All-Layers Simple Foldability

A classic problem in computational origami is flat foldability : given a crease pattern (planar straightline graph with n edges) on a polygonal piece of paper P , can P be folded flat isometrically without selfintersection while creasing at all creases (edges) in the crease pattern? The problem can also be defined for assigned crease patterns, in which every crease is labeled mountain or valley depending on the direction it is allowed to fold. The decision problem (for both assigned or unassigned) is NP-hard [5], even when the paper is an axis-aligned rectangle and the creases are at multiples of 45◦ [2]. But even when a crease pattern does fold flat, the motion to achieve that folding can be complicated [6], making the process impractical in some physical settings. Motivated by practical folding processes in manufacturing such as sheet-metal bending, Arkin et al. [3] introduced the idea of simple foldability—flat foldability by a sequence of simple folds. Informally, a simple fold is defined by a line segment and rotates a portion of the paper around this segment by ±180◦, while avoiding self-intersection. The problem generalizes to d-dimensions. In particular, for 1D paper, P is a line segment and creases are defined by points in P . In [3], they defined several models for simple folds and, for many models, showed that deciding simple foldability is polynomial for 1D paper, polynomial for rectangular paper with axisaligned creases, weakly NP-complete for rectangular paper with creases at multiples of 45◦, and weakly NP-complete for orthogonal paper with axis-aligned creases. In particular, they provided an algorithm to determine simple foldability of a 1D paper in O(n log n) deterministic time and O(n) randomized time in the all-layers model, requiring that a simple fold through one crease, also folds through all layers overlapping that crease. Akitaya et al. [1] extended the list of simple folding models, and for many models showed simple foldability to be strongly NP-hard for 2D paper. In particular, they introduced the infinite all-layers model of simple folds for 2D paper which is studied here, requiring that each simple fold be defined by an infinite line, and that all layers of paper intersecting this line must be folded. This model is probably the most practical simple folding model; for example, Balkcom’s robotic folding system [4] is restricted to this model. In this paper, we improve on [3] giving a deter-