Folding & Unfolding Folding & Unfolding in Computational in Computational

Geometry:Geometry:IntroductionIntroduction

Joseph O’RourkeJoseph O’RourkeSmith CollegeSmith College

(Many slides made by Erik (Many slides made by Erik Demaine)Demaine)

Folding and Unfolding in Folding and Unfolding in Computational GeometryComputational Geometry

1D: Linkages

2D: Paper

3D: Polyhedra

Preserve edge lengths Edges cannot cross

Preserve distances Cannot cross itself

Cut the surface while keeping it connected

CharacteristicsCharacteristics

TangibleApplicableElementaryDeepFrontier Accessible

OutlineOutline

Topics:1D: Linkages2D: Paper3D: Polyhedra

Lectures ScheduleLectures Schedule

Sunday 7:30-8:30 0 Introduction and Overview

Monday 9:00-9:50 1 Part Ia: Linkages and Universality

Monday 10:00-10:50 2 Part Ib: Pantographs and Pop-ups

Monday 1:30-2:30 Discussion

Monday 2:40-3:30 3 Part Ic: Locked Chains

Monday 3:40-4:30 4 Part IIa: Flat Origami

Tuesday 9:00-9:50 5 Part IIb: One-Cut Theorem

Tuesday 10:00-10:50 6 Part IIIa: Folding Polygons to Polyhedra

Tuesday 1:30-2:30 Discussion

Tuesday 2:40-3:30 7 Part IIIb: Unfolding Polyhedra to Nets

Tuesday 3:40-4:30 Guest Lecture: Jane Sangwine-Yeager

Wednesday 9:00-9:50 8 Part Id: Protein Folding: Fixed-angle Chains

Wednesday 10:00-10:50

9 Part Ie: Unit-Length Chains: Locked?

Thursday 9:00-9:50 10 Part IIc: Skeletons, Roofs, Medial Axis

Thursday 10:00-10:50 11 Part IId: Medial Axis Models

Friday 9:00-9:50 12 Part IIIc: Cauchy’s Rigidity Theorem

Friday 10:00-10:50 13 Part IIId: Bellows, Volume, Reconstruction

Outline: TonightOutline: Tonight

Topics:1D: Linkages2D: Paper3D: Polyhedra

Within each:DefinitionsOne “application”One open problem

OutlineOutline1 1 ― 1D: Linkages― 1D: Linkages

Definitions Configurations Locked chain in 3D Fixed-angle chains

Application: Protein foldingOpen Problem: unit-length locked

chains?

Linkages / FrameworksLinkages / Frameworks

Link / bar / edge = line segmentJoint / vertex = connection between

endpoints of bars

Closed chain / cycle / polygon

Open chain / arc

Tree General

ConfigurationsConfigurations

Configuration = positions of the vertices that preserves the bar lengths

Non-self-intersecting configurations Self-intersecting

Non-self-intersecting = No bars cross

Locked QuestionLocked 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 ConfigurationsCanonical Configurations

Chains: Straight configuration

Polygons: Convex configurations

Trees: Flat configurations

Locked 3D Chains Locked 3D Chains [Cantarella & [Cantarella & Johnston 1998; Johnston 1998; Biedl, Demaine, Demaine, Lazard, Lubiw, Biedl, Demaine, Demaine, Lazard, Lubiw, O’Rourke, Overmars, Robbins, Streinu, Toussaint, O’Rourke, Overmars, Robbins, Streinu, Toussaint, Whitesides 1999]Whitesides 1999]

Cannot straighten some chains, even with universal joints.

Locked 2D TreesLocked 2D Trees[Biedl, Demaine, Demaine, Lazard, Lubiw, O’Rourke, [Biedl, Demaine, Demaine, Lazard, Lubiw, O’Rourke, Robbins, Streinu, Toussaint, Whitesides 1998]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 Chains Lock?

Can every chain, with universal joints, be straightened?

Chains Straightened?

2D Yes

3DNo:

some locked

4D & beyond

Yes“Polygonal Chains Cannot Lock in 4D.”Roxana Cocan and J. O'RourkeComput. Geom. Theory Appl., 20 (2001) 105-129.

OpenOpen11: Can Equilateral Chains : Can Equilateral Chains Lock?Lock?

Does there exist an open polygonal chain embedded in 3D, with all links of equal length, that is locked?

ProteinProteinFoldingFolding

Protein FoldingProtein Folding

Fixed-angle chainFixed-angle chain

FlattenableFlattenable

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.

Unflattenable fixed-angle Unflattenable fixed-angle chainchain

Open ProblemsOpen Problems11 : : Locked Equilateral Locked Equilateral Chains?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 90o fixed-angle chain, all of whose links have the same length, that is locked?

Perhaps: No?

Perhaps: Yes for 1+?

OutlineOutline2 2 ― 2D: Paper― 2D: Paper

Definitions Foldings Crease patterns

Application: Map FoldingOpen Problem: Complexity of Map

Folding

FoldingsFoldings

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 Flat origami crane

Nonflat folding

Structure of FoldingsStructure 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 as or

Nonflat folding

Flat origami crane

Map FoldingMap 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 6 72 5 83 4 9

Map FoldingMap 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 6 72 5 83 4 9

Easy?Easy?

Hard?Hard?

Map FoldingMap 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

2 5 83 4 9

1 6 7

Map FoldingMap 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

2 5 8

1 76

Map FoldingMap 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 76

Map FoldingMap 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

76

Map FoldingMap 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

69

More generally: Given an arbitrary crease pattern, is it flat-foldable by simple folds?

OpenOpen22: Map Folding : Map Folding Complexity?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?

OutlineOutline3 3 ― 3D: Polyhedra― 3D: Polyhedra

Edge-Unfolding Definitions

Cut tree: spanning treeNet

Applications: Manufacturing Open Problem: Does every polyhedron

have a net?

Unfolding PolyhedraUnfolding Polyhedra

Cut along the surface of a polyhedron

Unfold into a simple planar polygon without overlap

Edge UnfoldingsEdge Unfoldings

Two types of unfoldings: Edge unfoldings: Cut only along edges General unfoldings: Cut through faces too

Cut Edges form Spanning Cut Edges form Spanning TreeTree

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 vertexo forest: cycle would isolate a surface pieceo tree: connected by boundary of polygon

Commercial SoftwareCommercial Software

Lundström Design, http://www.algonet.se/~ludesign/index.html

OpenOpen33: Edge-Unfolding Convex : Edge-Unfolding Convex PolyhedraPolyhedra

Does every convex polyhedron have an edge-unfolding to a net (a simple, nonoverlapping polygon)?

[Shephard, 1975]

Archimedian SolidsArchimedian Solids

Nets for Archimedian SolidsNets for Archimedian Solids

Cube with one corner Cube with one corner truncatedtruncated

SclickenriederSclickenrieder11::steepest-edge-unfoldsteepest-edge-unfold

“Nets of Polyhedra”TU Berlin, 1997

SclickenriederSclickenrieder33::rightmost-ascending-edge-rightmost-ascending-edge-unfoldunfold

OpenOpen33: Edge-Unfolding Convex : Edge-Unfolding Convex PolyhedraPolyhedra

Does every convex polyhedron have an edge-unfolding to a net (a simple, nonoverlapping polygon)?

[Shephard, 1975]