Related papers: Rigid Foldability is NP-Hard
We show that minimally 3-rigid block-and-hole graphs, with one block or one hole, are characterised as those which are constructible from $K_3$ by vertex splitting, and also, as those having associated looped face graphs which are…
We consider several quadrilateral origami tilings, including the Miura-ori crease pattern, allowing for crease-reversal defects above the ground state which maintain local flat-foldability. Using exactly solvable models, we show that these…
Circle packings are arrangement of circles satisfying specified tangency requirements. Many problems about packing of circles and spheres occur in nature particularly in material design and protein structure. Surprisingly, little is known…
Soft robots employing compliant materials and deformable structures offer great potential for wearable devices that are comfortable and safe for human interaction. However, achieving both structural integrity and compliance for comfort…
Simple closed curves in the plane can be mapped to nontrivial knots under the action of origami foldings that allow the paper to self-intersect. We show all tame knot types may be produced in this manner, motivating the development of a new…
Origami as a deployable structure offers the unique advantage of achieving compact stowage via flat-folding while forming a well-defined surface composed of rigid panels upon deployment. However, since origami consists of flat facets, it is…
Contacts between particles in dense, sheared suspensions are believed to underpin much of their rheology. Roughness and adhesion are known to constrain the relative motion of particles, and thus globally affect the shear response, but an…
We consider the zero-energy deformations of periodic origami sheets with generic crease patterns. Using a mapping from the linear folding motions of such sheets to force-bearing modes in conjunction with the Maxwell-Calladine index theorem…
Origami structures have been widely explored in robotics due to their many potential advantages. Origami robots can be very compact, as well as cheap and efficient to produce. In particular, they can be constructed in a flat format using…
Periodic origami patterns made with repeating unit cells of creases and panels bend and twist in complex ways. In principle, such soft modes of deformation admit a simplified asymptotic description in the limit of a large number of cells.…
Twisting sheets as a strategy to form functional yarns relies on millennia of human practice in making catguts and fabric wearables, but still lacks overarching principles to guide their intricate architectures. We show that twisted…
We study geometric rigidity of a class of fractals, which is slightly larger than the collection of self-conformal sets. Namely, using a new method, we shall prove that a set of this class is contained in a smooth submanifold or is totally…
Folding a sheet of paper along a curve can lead to structures seen in decorative art and utilitarian packing boxes. Here we present a theory for the simplest such structure: an annular circular strip that is folded along a central circular…
Discrete tomography deals with reconstructing finite spatial objects from lower dimensional projections and has applications for example in timetable design. In this paper we consider the problem of reconstructing a tile packing from its…
Like a cr\^epe resting on a plate, a thin elastic sheet can fold smoothly under its own weight, forming reversible shapes without creases or imposed hinges. Such soft folds arise from a balance between elastic bending and gravity, yet their…
We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard,…
We prove the #P-hardness of the counting problems associated with various satisfiability, graph and combinatorial problems, when restricted to planar instances. These problems include \begin{romannum} \item[{}] {\sc 3Sat, 1-3Sat, 1-Ex3Sat,…
Hybrid logic with binders is an expressive specification language. Its satisfiability problem is undecidable in general. If frames are restricted to N or general linear orders, then satisfiability is known to be decidable, but of…
Traditional origami starts from flat surfaces, leading to crease patterns consisting of Euclidean vertices. However, Euclidean vertices are limited in their folding motions, are degenerate, and suffer from misfolding. Here we show how…
We study the three-dimensional equilibrium shape of a shell formed by a deployed accordion-like origami, made from an elastic sheet decorated by a series of parallel creases crossed by a central longitudinal crease. Surprisingly, while the…