On the Design of Physical Folded Structures by Jason S. Ku B.S., Massachusetts Institute of Technology (2009) S.M., Massachusetts Institute of Technology (2011) Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2016 c Massachusetts Institute of Technology 2016. All rights reserved. ○ Author................................................................ Department of Mechanical Engineering May 18, 2016 Certified by. Sanjay E. Sarma Professor Thesis Supervisor Accepted by . Rohan Abeyaratne Chairman, Department Committee on Graduate Theses On the Design of Physical Folded Structures by Jason S. Ku Submitted to the Department of Mechanical Engineering on May 18, 2016, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering Abstract Folding as a subject of mathematical, computational, and engineering study is rel- atively young. Most results in this field are hard to apply in engineering practice because the use of physical materials to construct folded structures has not been fully considered nor adequately addressed. I propose a three-fold approach to the design of folded structures with physical consideration, separating for independent investigation (1) the computational complexity of basic folding paradigms, (2) the automated accommodation of facet material volume, and (3) the design of folded ge- ometry under boundary constraints. These three topics are each necessary to create folded structures from physical materials and are closely related. Thesis Supervisor: Sanjay E. Sarma Title: Professor 2 Acknowledgments This thesis is dedicated to my wife and parents who have supported me so much throughout my time as a doctoral candidate. I also can’t even begin to thank my thesis committee and all the other collaborators that have shared in my research journey. You inspire me everyday to learn more about the world and to teach others what we may find. 3 Contents 1 Introduction 13 1.1 Scope . 14 1.2 Background . 15 1.2.1 Intersection . 18 1.2.2 Material . 19 1.2.3 Geometry . 21 2 Complexity 25 2.1 Box-Pleating is Hard . 25 2.1.1 Definitions . 26 2.1.2 Bern and Hayes and k-Layer-Flat-Foldability . 31 2.1.3 SCN-Satisfiability . 31 2.1.4 Unassigned Crease Patterns . 33 2.1.5 Assigned Crease Patterns . 36 2.1.6 Generating Instances . 39 2.1.7 Remarks . 42 2.2 Simple Folding is Hard . 42 2.2.1 Definitions . 43 2.2.2 Description of a Simple Folding . 45 2.2.3 Orthogonal Paper/Orthogonal Creases . 48 2.2.4 Inapproximability . 52 2.2.5 Assigned Square Paper/45∘ Creases . 53 2.2.6 M/V Unassigned Square Paper/45∘ Creases . 55 4 2.2.7 Infinite, Orthogonal Paper/Orthogonal Creases . 61 2.2.8 Infinite, Rectangle Paper/Orthogonal Creases . 65 3 Material Thickness 67 3.1 Existing Techniques . 68 3.1.1 Hinge Shift . 68 3.1.2 Volume Trimming . 70 3.1.3 Offset Panel . 70 3.1.4 Offset Crease . 70 3.2 Definitions . 71 3.3 Algorithm . 73 3.3.1 Crease Width . 74 3.3.2 Polygon Construction . 75 3.3.3 Refinement . 77 3.3.4 Scale Factor . 78 3.3.5 Final Construction . 80 3.3.6 Adding Thickness . 82 3.4 Models . 83 3.4.1 Implementation . 83 3.4.2 Simulations . 83 3.4.3 Physical . 86 3.5 Remarks . 89 4 Geometry 90 4.1 Definitions . 92 4.1.1 Necessary Condition . 94 4.1.2 Bend Lines . 95 4.1.3 Split Points . 98 4.1.4 Partitions . 100 4.2 Algorithm . 101 4.2.1 Existence . 101 5 4.2.2 Constructing Partitions . 102 4.2.3 Triangles . 103 4.2.4 Combining Partitions . 103 4.2.5 Edge Insetting . 105 4.3 Implementations . 108 4.4 Remarks . 109 4.5 Folded Quadrilateral Boundaries . 111 4.5.1 Flat Foldability . 111 4.5.2 Boundary Condition and Flat Foldability . 113 4.5.3 Two Boundary Conditions . 115 5 Conclusion 122 6 List of Figures 1-1 Images of transformers in popular culture. The left is Optimus Prime from the Transformers series, and the right depicts a T-1000 from the Terminator series. 14 2-1 Topologically different local interactions within an isometric flat fold- ing. Forbidden configurations are shown for Face-Crease and Crease- Crease Non-Crossing. 28 2-2 Local interaction between overlapping regions around two distinct creases. 29 2-3 SCN Gadgets. [Left] A Complex Clause Gadget constructed from the Not-All-Equal clause on variables v, w, and y of a NAE3-SAT instance on six variables. [Right] The five elemental SCN Gadgets. 32 2-4 Elemental SCN Gadgets simulated with unassigned crease patterns. 34 2-5 Elemental SCN Gadgets simulated with assigned crease patterns. 37 2-6 A crease pattern generated by our software for an unassigned Complex Clause Gadget. The gadget relates the yellow, red, and green variables with a satisfying M/V assignment, (yellow = True, red = False, green = false). 40 2-7 A crease pattern generated by our software for an assigned Complex Clause Gadget. The gadget relates the yellow, red, and green variables with a satisfying M/V assignment, (yellow = True, red = False, green = false). Dots indicate the layer that folds below. 41 7 2-8 Example folding steps demonstrating the differences between simple folding models. L is a directed dotted line in the direction of a, U is textured, and the fold line f −1(L) @U is a thick line with the number \ of layers # specified. 47 2-9 An orthogonal simple polygon with orthogonally aligned mountain- valley creases (drawn in red and blue respectively) constructed from an instance of 3-Partition that can be folded using simple folds if and only if the instance of 3-Partition has a solution. 50 2-10 Process to check the Partition solution: 1) pleat variables to change height of bar by 2t, 2) fold along the rightmost wrapper crease around the column, 3) fit the bar through the cage folding the bar to theleft along the next wrapper crease, 4) repeat until n=3 triples adding to 2t have been checked. 51 2-11 Turn gadgets for assigned case. Red/blue lines represent the M/V assignment. 54 2-12 Figure 18 from [4]. Corrections marked in red creating reflections of c1 and c2 on the covering flap, and trimming the covering flaps so that c1 and c2 do not intersect v0 or v1 within the covering flap. 56 2-13 (Top-left) Crease pattern for the Wrapper in the unassigned model. Red lines show unassigned creases. (Top-right) Creases are colored according to their folding order. (Bottom) Folding sequence showing the creases that are being folded. 57 2-14 Unassigned turn gadgets. Creases must be folded according to color order on left. Input and output creases are labeled with arrow heads, forward signals in black and return signals in red. 59 2-15 Example collection of turn gadgets connected in series demonstrating forward and return signal propagation. 60 8 2-16 An orthogonal simple polygon with mountain-valley assigned paper- aligned orthogonal creases (drawn in red and blue respectively) con- structed from an instance of 3-Partition that can be folded in the infinite one-layer model if and only if the instance of 3-Partition has a solution. 62 3-1 Some existing thick folding techniques: (A) Hinge Shift, (B) Volume Trimming, (C) Offset Panel, and (D) Offset Crease. 69 3-2 From left to right: (1) generic crease pattern Ξ0, (2) locally flat foldable crease pattern Ξ with layer ordering graph Λ, (3) with reduced layer ordering graph Γ, and (4) flat folding fΞ(Ξ). 72 3-3 Polygon construction. A generic internal crease pattern vertex showing relationship between offsets and angles. 75 3-4 A non-simple vertex polygon and refinement by clipping crossings. 76 3-5 Trimming intersecting region. 77 3-6 Unbounded intersection for inside touching creases in input flat folded state. 78 3-7 Scale factor calculation showing relevant quantities. 79 3-8 Construction process. 81 3-9 A screenshot of our offset crease implementation in action. The model shown is a traditional bird base with uniform thickness offset. 84 3-10 Numerical folding simulation of two thickened crease patterns using Freeform Origami. 85 3-11 Parameterized thick single vertex construction in Mathematica. 86 3-12 An acrylic physical model constructed using the offset crease technique presented. 87 3-13 An acrylic physical model constructed using the offset crease technique presented. 88 9 4-1 (Left) A boundary mapping that might be used to design a color-change checker board model. (Right) An unfinished crease pattern with parts of the crease pattern unknown. 91 4-2 Points f(u); f(v); f(q); f(p) with spheres S0;S1;S2. The shaded area S S S is the region in which f(p) may exist if p; u; v is non- 1 \ 2 ⊂ 0 f g expansive under f. ............................ 93 4-3 Input and output to the hole problem showing notation. Given polygon P R2 and mapping f : @P Rd, find isometric g : P Rd such ⊂ ! ! that g(@P ) = f(@P )............................ 94 4-4 The bend points of (P; f; v) showing relavent angles 휃; 휑, 훽 , points f g u; v; w; p; f(u); f(v); f(w); q , and sets R; S . The upper figures f g f g show only the boundary mapping, while the lower images show filled, locally satisfying mappings of the interior. 96 4-5 Visibility of p. If x X is not visible from v, one of a; b; y; z X 2 f g 2 will be.