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Folding & Unfolding in Computational Geometry: Introduction Joseph ORourke Smith College (Many slides made by Erik Demaine)

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Folding and Unfolding in Computational Geometry z1D: Linkages z2D: Paper z3D: Polyhedra yPreserve edge lengths yEdges cannot cross yPreserve distances yCannot cross itself yCut the surface while keeping it connected

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Characteristics zTangible zApplicable zElementary zDeep zFrontier Accessible

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Outline Topics: z1D: Linkages z2D: Paper z3D: Polyhedra

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Lectures Schedule Sunday 7:30-8:300Introduction and Overview Monday 9:00-9:501Part Ia: Linkages and Universality Monday 10:00-10:502Part Ib: Pantographs and Pop-ups Monday 1:30-2:30Discussion Monday 2:40-3:303Part Ic: Locked Chains Monday 3:40-4:304Part IIa: Flat Origami Tuesday 9:00-9:505Part IIb: One-Cut Theorem Tuesday 10:00-10:506Part IIIa: Folding Polygons to Polyhedra Tuesday 1:30-2:30Discussion Tuesday 2:40-3:307Part IIIb: Unfolding Polyhedra to Nets Tuesday 3:40-4:30Guest Lecture: Jane Sangwine-Yeager Wednesday 9:00-9:508Part Id: Protein Folding: Fixed-angle Chains Wednesday 10:00-10:509Part Ie: Unit-Length Chains: Locked? Thursday 9:00-9:5010Part IIc: Skeletons, Roofs, Medial Axis Thursday 10:00-10:5011Part IId: Medial Axis Models Friday 9:00-9:5012Part IIIc: Cauchys Rigidity Theorem Friday 10:00-10:5013Part IIId: Bellows, Volume, Reconstruction

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Outline: Tonight Topics: z1D: Linkages z2D: Paper z3D: Polyhedra Within each: zDefinitions zOne application zOne open problem

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Outline 1 1D: Linkages zDefinitions yConfigurations yLocked chain in 3D yFixed-angle chains zApplication: Protein folding zOpen Problem: unit-length locked chains?

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Linkages / Frameworks zLink / bar / edge = line segment zJoint / vertex = connection between endpoints of bars Closed chain / cycle / polygon Open chain / arc TreeGeneral

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Configurations zConfiguration = positions of the vertices that preserves the bar lengths Non-self-intersecting configurations Self-intersecting zNon-self-intersecting = No bars cross

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Locked Question zCan a linkage be moved between any two non-self-intersecting configurations? ? zCan any non-self-intersecting configuration be unfolded, i.e., moved to canonical configuration? yEquivalent by reversing and concatenating motions

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Canonical Configurations zChains: Straight configuration zPolygons: Convex configurations zTrees: Flat configurations

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Locked 3D Chains [Cantarella & Johnston 1998; Biedl, Demaine, Demaine, Lazard, Lubiw, ORourke, Overmars, Robbins, Streinu, Toussaint, Whitesides 1999] Cannot straighten some chains, even with universal joints.

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Locked 2D Trees [Biedl, Demaine, Demaine, Lazard, Lubiw, ORourke, Robbins, Streinu, Toussaint, Whitesides 1998] zTheorem: Not all trees can be flattened yNo petal can be opened unless all others are closed significantly yNo petal can be closed more than a little unless it has already opened

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Can Chains Lock? zCan every chain, with universal joints, be straightened? Chains Straightened? 2DYes 3D No: some locked 4D & beyondYes Polygonal Chains Cannot Lock in 4D. Roxana Cocan and J. O'Rourke Comput. Geom. Theory Appl., 20 (2001)

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Open 1 : Can Equilateral Chains Lock? Does there exist an open polygonal chain embedded in 3D, with all links of equal length, that is locked?

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Protein Folding

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Fixed-angle chain

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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.

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Unflattenable fixed-angle chain

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Open Problems 1 : Locked Equilateral Chains? (1)Is there a configuration of a chain with universal joints, all of whose links have the same length, that is locked? (2)Is there a configuration of a 90 o fixed- angle chain, all of whose links have the same length, that is locked? Perhaps: No? Perhaps: Yes for 1+ ?

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Outline 2 2D: Paper zDefinitions yFoldings yCrease patterns zApplication: Map Folding zOpen Problem: Complexity of Map Folding

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Foldings zPiece of paper = 2D surface ySquare, or polygon, or polyhedral surface zFolded state = isometric embedding yIsometric = preserve intrinsic distances (measured along paper surface) yEmbedding = no self- intersections except that multiple surfaces can touch with infinitesimal separation Flat origami crane Nonflat folding

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Structure of Foldings zCreases in folded state = discontinuities in the derivative zCrease pattern = planar graph drawn with straight edges (creases) on the paper, corresponding to unfolded creases zMountain-valley assignment = specify crease directions as or Nonflat folding Flat origami crane

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Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line

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Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line

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Easy?

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Hard?

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Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line

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Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line

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Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line 1 7 6

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Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line 7 6

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Map Folding zMotivating problem: yGiven a map (grid of unit squares), each crease marked mountain or valley yCan it be folded into a packet (whose silhouette is a unit square) via a sequence of simple folds? ySimple fold = fold along a line 6 9 zMore generally: Given an arbitrary crease pattern, is it flat-foldable by simple folds?

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Open 2 : 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?

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Outline 3 3D: Polyhedra zEdge-Unfolding yDefinitions xCut tree: spanning tree xNet yApplications: Manufacturing yOpen Problem: Does every polyhedron have a net?

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Unfolding Polyhedra zCut along the surface of a polyhedron zUnfold into a simple planar polygon without overlap

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Edge Unfoldings zTwo types of unfoldings: yEdge unfoldings: Cut only along edges yGeneral unfoldings: Cut through faces too

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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. o spanning: to flatten every vertex o forest: cycle would isolate a surface piece o tree: connected by boundary of polygon

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Commercial Software Lundström Design,

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Open 3 : 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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Archimedian Solids

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Nets for Archimedian Solids

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Cube with one corner truncated

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Sclickenrieder 1 : steepest-edge-unfold Nets of Polyhedra TU Berlin, 1997

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Sclickenrieder 3 : rightmost-ascending-edge-unfold

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Open 3 : 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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