Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain be straightened?

author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Acknowledgments My time as a graduate student has been the best period of my life so far, and for that wonderful experience I owe many thanks. I had two excellent advisors, Anna Lubiw and Ian Munro. I started working with Anna after I took her two classes on algorithms and computational geometry during my Master’s, which got me excited about both these areas, and even caused me to switch entire fields of computer science, from distributed systems to theory and algorithms. Anna introduced me to Ian when some of our problems in computational geometry turned out to have large data structural components, and my work with Ian blossomed from there. The sets of problems I worked on with Anna and Ian diverged, and both remain my primary interests. Anna and Ian have had a profound influence throughout my academic career. At the most

Origami, the human art of paper sculpture, is a fresh challenge for the field of robotic manipulation, and provides a concrete example for many difficult and general manipulation problems. This thesis will present some initial results, including the world’s first origami-folding robot, some new theorems about foldability, definition of a simple class of origami for which I have designed a complete automatic planner, analysis of the kinematics of more complicated folds, and some observations about the configuration spaces of compound spherical closed chains. Acknowledgments Thanks to my family, for everything. And thanks to everyone who has made the development of this thesis and life in Pittsburgh so great. Matt Mason. Wow! Who ever had a better advisor, or friend? Thanks to my colleagues, for all that I’ve learned from them. Jeff Trinkle, “in loco advisoris”. My academic older siblings and cousins, aunts and uncles, for their help and guidance in so many things: Alan Christiansen, Ken Goldberg, Randy

Abstract. We prove that all single-vertex origami shapes are reachable from the open flat state via simple, non-crossing motions. We also consider conical paper, where the total sum of the cone angles centered at the origami vertex is not 2π. For an angle sum less than 2π, the configuration space of origami shapes compatible with the given metric has two components, and within each component, a shape can always be reconfigured via simple (non-crossing) motions. Such a reconfiguration may not always be possible for an angle sum larger than 2π. The proofs rely on natural extensions to the sphere of planar Euclidean rigidity results regarding the existence and combinatorial characterization of expansive motions. In particular, we extend the concept of a pseudo-triangulation from the Euclidean to the spherical case. As a consequence, we formulate a set of necessary conditions that must be satisfied by three-dimensional generalizations of pointed pseudo-triangulations. 1

Origami is the centuries-old art of folding paper, and recently, it is investigated as computer science: Given an origami with creases, the problem to determine if it can be flat after folding all creases is NP-hard. Another hundreds-old art of folding paper is a pop-up book. A model for the pop-up book design problem is given, and its computational complexity is investigated. We show that both of the opening book problem and the closing book problem are NP-hard. Keywords: Computational Complexity, Origami, Paper folding, Pop-up book. 1

This paper presents recent research that was conducted in the field of computational origami. First, it examines previous work done by Kan Chu Sen in 1721 on the idea of a fold-and-cut problem. Second, the paper will look at Kawasaki’s Theorem, which states that the sum of the alternating interior angles of a given figure is 180°. Then the sum of the fold pattern of a folded origami figure explained in Maekawa’s Theorem will be explored. By combining these three ideas a new problem arises: given multiple folds that bisect the vertex and making a complete straight cut of the vertex, can the sum of the alternating fold-cut angle be determined by just the number of folds of the paper alone? This paper will present and prove a formula for solving this

Abstract — Many practical multi-body systems involve loops. Studying the kinematics of such systems has been challenging, partly because of the requirement of maintaining loop closure constraints, which have conventionally been formulated as highly nonlinear equations in joint parameters. Recently, novel parameters defined by trees of triangles have been introduced for a broad class of linkage systems involving loops (e.g., spatial loops with spherical joints and planar loops with revolute joints); these parameters greatly simplify kinematics related computations and endow system configuration spaces with highly tractable piecewise convex geometries. In this paper, we describe a more general approach for multi-body systems, with loops, that allow construction trees of simplices. We illustrate the applicability and efficiency of our simplex-tree based approach to kinematics by a study of foldable objects. We present two sets of new parameters for single-vertex rigid fold kinematics; like the parameters in the triangle-tree prototype, each has a geometrically meaningful and computationally tractable constraint formulation, and each endows the configuration space with a nice geometry. I.

The Carpenter’s Rule Theorem states that any chain linkage in the plane can be folded continuously between any two configurations while preserving the bar lengths and without the bars crossing. However, this theorem applies only to strictly simple configurations, where bars intersect only at their common endpoints. We generalize the theorem to self-touching configurations, where bars can touch but not properly cross. At the heart of our proof is a new definition of self-touching configurations of planar linkages, based on an annotated configuration space and limits of nontouching configurations. We show that this definition is equivalent to the previously proposed definition of self-touching configurations, which is based on a combinatorial description of overlapping features. Using our new definition, we prove the generalized Carpenter’s Rule Theorem using a topological argument. We believe that our topological methodology provides a powerful tool for manipulating all kinds of self-touching objects, such as 3D hinged assemblies of polygons and rigid origami. In particular, we show how to apply our methodology to extend to self-touching configurations universal reconfigurability results for open chains with slender polygonal adornments, and single-vertex rigid origami with convex cones.

The Carpenter’s Rule Theorem states that any chain linkage in the plane can be folded continuously between any two configurations while preserving the bar lengths and without the bars crossing. However, this theorem applies only to strictly simple configurations, where bars intersect only at their common endpoints. We generalize the theorem to self-touching configurations, where bars can touch but not properly cross. At the heart of our proof is a new definition of self-touching configurations of planar linkages, based on an annotated configuration space and limits of nontouching configurations. We show that this definition is equivalent to the previously proposed definition of self-touching configurations, which is based on a combinatorial description of overlapping features. Using our new definition, we prove the generalized Carpenter’s Rule Theorem using a topological argument. We believe that our topological methodology provides a powerful tool for manipulating many kinds of self-touching objects, such as 3D hinged assemblies of polygons and rigid origami. In particular, we show how to apply our methodology to extend to self-touching configurations universal reconfigurability results for open chains with slender polygonal adornments, and single-vertex rigid origami with convex cones.