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Map Folding by Rahnuma Islam Nishat B.Sc.Engg., Bangladesh Map Folding by Rahnuma Islam Nishat B.Sc.Engg., Bangladesh University of Engineering and Technology, 2009 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in the Department of Computer Science c Rahnuma Islam Nishat, 2013 University of Victoria All rights reserved. This thesis may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. ii Map Folding by Rahnuma Islam Nishat B.Sc.Engg., Bangladesh University of Engineering and Technology, 2009 Supervisory Committee Dr. Sue Whitesides, Supervisor (Department of Computer Science) Dr. Frank Ruskey, Departmental Member (Department of Computer Science) iii Supervisory Committee Dr. Sue Whitesides, Supervisor (Department of Computer Science) Dr. Frank Ruskey, Departmental Member (Department of Computer Science) ABSTRACT A crease pattern is an embedded planar graph on a piece of paper. An m×n map is a rectangular piece of paper with a crease pattern that partitions the paper into an m×n regular grid of unit squares. If a map has a configuration such that all the faces of the map are stacked on a unit square and the paper does not self-intersect, then it is flat foldable, and the linear ordering of the faces is called a valid linear ordering. Otherwise, the map is unfoldable. In this thesis, we show that, given a linear ordering of the faces of an m × n map, we can decide in linear time whether it is a valid linear ordering or not. We also define a class of unfoldable 2 × n crease patterns for every n ≥ 5. iv Contents Supervisory Committee ii Abstract iii Table of Contents iv List of Tables vi List of Figures vii Acknowledgements xi Dedication xii 1 Introduction 1 1.1 Map Folding . 1 1.2 Problem Statement . 3 1.3 Our Contributions . 4 1.4 Thesis Outline . 5 2 Background and Related Work 6 2.1 History of Mathematical Origami . 6 2.2 Applications of Origami . 7 2.3 Mathematics of Flat Folding . 8 2.3.1 Single-Vertex Crease Patterns . 9 2.3.2 Simple Folding . 10 2.4 Conclusion . 18 3 Valid Linear Orderings 19 3.1 Preliminaries . 19 v 3.2 Unassigned Crease Patterns . 20 3.2.1 Checkerboard Pattern . 20 3.2.2 Butterflies . 21 3.2.3 Recognizing Valid Linear Orderings . 23 3.3 Mountain-Valley Patterns . 27 3.3.1 Directed Networks . 27 3.3.2 Recognizing Valid Linear Orderings . 29 3.4 Enumerating Valid Linear Orderings . 31 3.5 Conclusion . 33 4 Unfoldable Maps 35 4.1 Preliminaries . 35 4.1.1 2 × n Maps . 35 4.1.2 Symmetric Closure . 36 4.2 Folding Small Patterns . 37 4.3 2 × 5 Mountain-Valley Patterns . 42 4.3.1 Flat Foldable 2 × 5 Patterns . 42 4.3.2 Unfoldable 2 × 5 Patterns . 48 4.4 Generalized Unfoldable Patterns . 53 4.5 Conclusion . 59 5 Conclusions 60 Bibliography 63 vi List of Tables Table 5.1 Count of flat foldable 1D mountain-valley patterns. 62 vii List of Figures Figure 1.1 A mountain-valley pattern on a 3 × 3 map. 3 Figure 1.2 A flat folded state of P with four layers l1; l2; l3 and l4. 3 Figure 2.1 (a) A flat foldable mountain-valley pattern on a paper P , and (b) the disk r cut out from P where v is the center of r. 9 Figure 2.2 (a) A flat folded state of a 1D paper with four layers l1; l2; l3; l4. The creases are shown in black dots. We assume that the creases are labeled valley when seen from above (i.e., positive Y-axis), (b) a one-layer simple fold of the topmost layer l1, (c) a some- layers simple fold of the top two layers l1; l2, and (d) an all-layers simple fold. 11 Figure 2.3 (a) A 1D piece of paper P with a crease pattern C = c1; c2; : : : ; c6. The creases with the labels mountain and valley are shown as blue and red dots, respectively. (b) A flat folded state P (2) of P C 1 1 1 with the crease pattern (2) = c1; c2; : : : ; c5 that is obtained by a pair of simple folds (called a \crimp") along the pair of creases c4; c5 in C. ............................. 12 Figure 2.4 (a) A 1D piece of paper P with a crease pattern C = c1; c2; : : : ; c6. The creases with the labels mountain and valley are shown as blue and red dots, respectively. (b) A flat folded state P (1) of C 1 1 1 P with the crease pattern (1) = c1; c2; : : : ; c5 that is obtained from P by an end fold operation on the leftmost crease c1 in C. 13 Figure 2.5 An m × n map with two homogeneous crease lines. 15 Figure 3.1 (a) A 3 × 3 map with crease pattern C, where the vertices of C are shown as black disks and the dummy vertices of P are shown as red disks. (b) The faces of C. 20 viii Figure 3.2 A 3 × 4 crease pattern C. The faces pointing light side up in the final flat folded state are shown in white and the other faces are shown in gray. 21 Figure 3.3 A pair of butterflies B1;B2, where (a) B2 is stacked on B1, (b) B1 is stacked on B2, (c) B1 nests in B2, and (d) B2 nests in B1. 21 Figure 3.4 The unit square u on the XY -plane. The 3 west butterflies of C in Figure 3.2 are shown to the left of u. The 6 east butterflies of C are shown to the right of u. The 4 north butterflies of C are shown above u and the 4 south butterflies of C are shown below u. .................................. 22 Figure 3.5 (a) The north butterflies of the crease pattern in Figure 3.2. (b){ (h) The steps of Algorithm 3. 26 Figure 3.6 (a) A 2 × 2 mountain-valley pattern, (b) the checkerboard pat- tern, and (c) the directed network. 27 Figure 3.7 (a) A 3×3 mountain-valley pattern, and (b) its directed network that contains a cycle f0;0 ≺ f0;1 ≺ f0;2 ≺ f1;2 ≺ f2;2 ≺ f2;1 ≺ f2;0 ≺ f1;0 ≺ f0;0........................... 28 Figure 3.8 (a) A 2 × 3 mountain-valley pattern, and (b) its directed network. 32 Figure 3.9 (a) B1, (b) B2, (c) e and u1; u2, (d){(f) invalid orderings of faces in a flat folded state. 33 Figure 3.10(a) B1, (b) B2, (c) e and u1; u2, (d){(f) invalid orderings of faces in a flat folded state. 34 Figure 4.1 A 2 × 7 mountain-valley pattern. 36 Figure 4.2 (a) A 2×3 mountain-valley pattern C, (b) after applying MirrorX on C, (c) after applying MirrorX and Switch on C, (d) after applying switch on C, (e) after applying MirrorY on C, (f) after applying MirrorY and Switch on C, (g) after applying MirrorX and MirrorY on C, (h) after applying MirrorX, MirrorY and Switch on C. ............................ 37 Figure 4.3 (a) A 2 × 3 mountain-valley pattern, where the faces are labeled, (b) the same mountain-valley pattern, where the ribs and spinal creases are labeled. 38 ix Figure 4.4 (a) A 2 × 4 mountain-valley pattern C. C is flat foldable but the final flat folded state cannot be achieved by simple folding, (b) the directed network of C...................... 40 Figure 4.5 (a) The west butterflies, where the labeled arcs represent hinges, (b) the east butterflies, and (c) the south butterflies of the 2 × 4 mountain valley pattern in Figure 4.4(a). 41 Figure 4.6 (a) A partially labeled 2 × 5 mountain-valley pattern C1, the black lines represent creases without labels, (b) another partially labeled 2 × 5 mountain-valley pattern C2, (c) directed network of C1, (d) directed network of C2. 43 Figure 4.7 (a) A 2 × 5 mountain-valley pattern, where the pre-spine folds fu2; l2g and fu3; l3g have different labels, (b) the corresponding 1D pattern with only the creases u2 and u3. 46 Figure 4.8 (a) A 2×5 mountain-valley pattern C3, (b) another 2×5 mountain- valley pattern C4, (c) directed network of C3, (d) directed net- work of C4.............................. 46 Figure 4.9 (a) The west butterflies, where the labeled arcs represent hinges, (b) the east butterflies, and (c) the south butterflies of the 2 × 5 mountain valley pattern in Figure 4.8(a). 47 Figure 4.10(a) An unfoldable 2 × 5 mountain-valley pattern C1, (b) another unfoldable 2×5 mountain-valley pattern C2, (c) directed network of C1, and (d) directed network of C2. 48 Figure 4.11(a){(b) Intersections in the linear ordering of C1, (c){(e) inter- sections in the linear ordering of C2. 49 Figure 4.12(a) C1, and (b) MirrorX = C1, (c) MirrorY, (d) (MirrorY + Mir- rorX) = MirrorY, (e) Switch, (f) (Switch + MirrorX) = Switch, (g) MirrorY + Switch, (h) (MirrorY + Switch + MirrorX) = (MirrorY + Switch) applied on C1. 50 Figure 4.13(a) C2, and (b) MirrorX, (c) MirrorY, (d) MirrorY + MirrorX, (e) Switch, (f) Switch + MirrorX, (g) MirrorY + Switch, (h) MirrorY + Switch + MirrorX applied on C2. Note that the mountain-valley patterns in (b), (d), (f), (h) are the same as the patterns in (c), (a), (g), (e), respectively. 51 x Figure 4.14(a) An unfoldable 2×n mountain-valley pattern C. (b) Checker- board pattern of the first five columns c0; : : : ; c4, and (c) the corresponding directed network.
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