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Folding & Unfolding in Computational Geometry: Introduction. Joseph O’Rourke Smith College (Many slides made by Erik Demaine). Folding and Unfolding in Computational Geometry. 1D: Linkages. Preserve edge lengths Edges cannot cross. 2D: Paper. Preserve distances Cannot cross itself.

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## Folding & Unfolding in Computational Geometry: Introduction

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**Folding & Unfolding in Computational Geometry:Introduction**Joseph O’Rourke Smith College (Many slides made by Erik Demaine)**Folding and Unfolding in Computational Geometry**• 1D: Linkages • Preserve edge lengths • Edges cannot cross • 2D: Paper • Preserve distances • Cannot cross itself • 3D: Polyhedra • Cut the surface while keeping it connected**Characteristics**• Tangible • Applicable • Elementary • Deep • Frontier Accessible**Outline**Topics: • 1D: Linkages • 2D: Paper • 3D: Polyhedra**Outline: Tonight**Topics: • 1D: Linkages • 2D: Paper • 3D: Polyhedra Within each: • Definitions • One “application” • One open problem**Outline1 ― 1D: Linkages**• Definitions • Configurations • Locked chain in 3D • Fixed-angle chains • Application: Protein folding • Open Problem: unit-length locked chains?**Linkages / Frameworks**• Link / bar / edge = line segment • Joint / vertex = connection between endpoints of bars Closed chain / cycle / polygon Open chain / arc Tree General**Configurations**• Configuration = positions of the vertices that preserves the bar lengths • Non-self-intersecting = No bars cross Self-intersecting Non-self-intersecting configurations**Locked Question**• Can a linkage be moved between any twonon-self-intersecting configurations? • Can any non-self-intersecting configuration be unfolded, i.e., moved to “canonical” configuration? • Equivalent by reversing and concatenating motions ?**Canonical Configurations**• Chains: Straightconfiguration • Polygons: Convexconfigurations • Trees: Flat configurations**Locked 3D Chains [Cantarella & Johnston 1998; Biedl,**Demaine, Demaine, Lazard, Lubiw, O’Rourke, Overmars, Robbins, Streinu, Toussaint, Whitesides 1999] Cannot straighten some chains, even with universal joints.**Locked 2D Trees[Biedl, Demaine, Demaine, Lazard, Lubiw,**O’Rourke, Robbins, Streinu, Toussaint, Whitesides 1998] • Theorem: Not all trees can be flattened • No petal can be opened unless all others are closed significantly • No petal can be closed more than a little unless it has already opened**Can Chains Lock?**• Can every chain, with universal joints, be straightened? “Polygonal Chains Cannot Lock in 4D.” Roxana Cocan and J. O'RourkeComput. Geom. Theory Appl., 20 (2001) 105-129.**Open1: Can Equilateral Chains Lock?**Does there exist an open polygonal chain embedded in 3D, with all links of equal length, that is locked?**Flattenable**A configuration of a chain if flattenable if it can be reconfigured, without self-intersection, so that it lies flat in a plane. Otherwise the configuration is unflattenable, or locked.**Open Problems1 : Locked Equilateral Chains?**• Is there a configuration of a chain with universal joints, all of whose links have the same length, that is locked? • Is there a configuration of a 90o fixed-angle chain, all of whose links have the same length, that is locked? Perhaps: No? Perhaps: Yes for 1+e?**Outline2 ― 2D: Paper**• Definitions • Foldings • Crease patterns • Application: Map Folding • Open Problem: Complexity of Map Folding**Foldings**• Piece of paper = 2D surface • Square, or polygon, or polyhedral surface • Folded state = isometric “embedding” • Isometric = preserve intrinsic distances (measured alongpaper surface) • “Embedding” = no self-intersections exceptthat multiple surfacescan “touch” withinfinitesimal separation Nonflat folding Flat origami crane**** Structure of Foldings • Creases in folded state =discontinuities in the derivative • Crease pattern = planar graph drawn with straight edges (creases) on the paper, corresponding tounfolded creases • Mountain-valleyassignment = specifycrease directions asor Nonflat folding Flat origami crane**6**7 1 5 8 2 4 9 3 Map Folding • Motivating problem: • Given a map (grid of unit squares),each crease marked mountain or valley • Can it be folded into a packet(whose silhouette is a unit square)via a sequence of simple folds? • Simple fold = fold along a line**6**7 1 5 8 2 4 9 3 Map Folding • Motivating problem: • Given a map (grid of unit squares),each crease marked mountain or valley • Can it be folded into a packet(whose silhouette is a unit square)via a sequence of simple folds? • Simple fold = fold along a line**5**8 2 4 9 3 Map Folding • Motivating problem: • Given a map (grid of unit squares),each crease marked mountain or valley • Can it be folded into a packet(whose silhouette is a unit square)via a sequence of simple folds? • Simple fold = fold along a line 6 7 1**5**8 2 Map Folding • Motivating problem: • Given a map (grid of unit squares),each crease marked mountain or valley • Can it be folded into a packet(whose silhouette is a unit square)via a sequence of simple folds? • Simple fold = fold along a line 6 7 1**Map Folding**• Motivating problem: • Given a map (grid of unit squares),each crease marked mountain or valley • Can it be folded into a packet(whose silhouette is a unit square)via a sequence of simple folds? • Simple fold = fold along a line 1 7 6**Map Folding**• Motivating problem: • Given a map (grid of unit squares),each crease marked mountain or valley • Can it be folded into a packet(whose silhouette is a unit square)via a sequence of simple folds? • Simple fold = fold along a line 7 6**Map Folding**• Motivating problem: • Given a map (grid of unit squares),each crease marked mountain or valley • Can it be folded into a packet(whose silhouette is a unit square)via a sequence of simple folds? • Simple fold = fold along a line 9 6 • More generally: Given an arbitrary crease pattern, is it flat-foldable by simple folds?**Open2: Map Folding Complexity?**Given a rectangular map, with designated mountain/valley folds in a regular grid pattern, how difficult is it to decide if there is a folded state of the map realizing those crease patterns?**Outline3 ― 3D: Polyhedra**• Edge-Unfolding • Definitions • Cut tree: spanning tree • Net • Applications: Manufacturing • Open Problem: Does every polyhedron have a net?**Unfolding Polyhedra**• Cut along the surface of a polyhedron • Unfold into a simple planar polygon without overlap**Edge Unfoldings**• Two types of unfoldings: • Edge unfoldings: Cut only along edges • General unfoldings: Cut through faces too**Cut Edges form Spanning Tree**Lemma: The cut edges of an edge unfolding of a convex polyhedron to a simple polygon form a spanning tree of the 1-skeleton of the polyhedron. • spanning: to flatten every vertex • forest: cycle would isolate a surface piece • tree: connected by boundary of polygon**Commercial Software**Lundström Design, http://www.algonet.se/~ludesign/index.html**Open3: Edge-Unfolding Convex Polyhedra**Does every convex polyhedron have an edge-unfolding to a net (a simple, nonoverlapping polygon)? [Shephard, 1975]**Sclickenrieder1:steepest-edge-unfold**“Nets of Polyhedra” TU Berlin, 1997**Open3: Edge-Unfolding Convex Polyhedra**Does every convex polyhedron have an edge-unfolding to a net (a simple, nonoverlapping polygon)? [Shephard, 1975]

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