Folding%20

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Folding Unfolding in Computational
GeometryIntroduction
Joseph ORourke Smith College (Many slides made
by Erik Demaine)
2
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

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Characteristics

• Tangible
• Applicable
• Elementary
• Deep
• Frontier Accessible

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Outline

• Topics
• 1D Linkages
• 2D Paper
• 3D Polyhedra

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Lectures Schedule
Sunday 730-830 0 Introduction and Overview
Monday 900-950 1 Part Ia Linkages and Universality
Monday 1000-1050 2 Part Ib Pantographs and Pop-ups
Monday 130-230 Discussion
Monday 240-330 3 Part Ic Locked Chains
Monday 340-430 4 Part IIa Flat Origami
Tuesday 900-950 5 Part IIb One-Cut Theorem
Tuesday 1000-1050 6 Part IIIa Folding Polygons to Polyhedra
Tuesday 130-230 Discussion
Tuesday 240-330 7 Part IIIb Unfolding Polyhedra to Nets
Tuesday 340-430 Guest Lecture Jane Sangwine-Yeager
Wednesday 900-950 8 Part Id Protein Folding Fixed-angle Chains
Wednesday 1000-1050 9 Part Ie Unit-Length Chains Locked?
Thursday 900-950 10 Part IIc Skeletons, Roofs, Medial Axis
Thursday 1000-1050 11 Part IId Medial Axis Models
Friday 900-950 12 Part IIIc Cauchys Rigidity Theorem
Friday 1000-1050 13 Part IIId Bellows, Volume, Reconstruction
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Outline Tonight

• Topics
• 1D Linkages
• 2D Paper
• 3D Polyhedra
• Within each
• Definitions
• One application
• One open problem

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Outline1 ? 1D Linkages

• Definitions
• Configurations
• Locked chain in 3D
• Fixed-angle chains
• Application Protein folding
• Open Problem unit-length locked chains?

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Linkages / Frameworks

• Link / bar / edge line segment
• Joint / vertex connection between
  endpoints of bars

Closed chain / cycle / polygon
Open chain / arc
Tree
General
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Configurations

• Configuration positions of the vertices that
  preserves the bar lengths

• Non-self-intersecting No bars cross

Non-self-intersecting configurations
Self-intersecting
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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

?
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Canonical Configurations

• Chains Straight configuration

• Polygons Convex configurations

• Trees 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 TreesBiedl, Demaine, Demaine, Lazard,
Lubiw, ORourke, 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

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Can Chains Lock?

• Can every chain, with universal joints, be
  straightened?

Chains Straightened?
2D Yes
3D No some locked
4D beyond Yes
Polygonal Chains Cannot Lock in 4D. Roxana
Cocan and J. O'RourkeComput. Geom. Theory Appl.,
20 (2001) 105-129.
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Open1 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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ProteinFolding
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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 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 1e?
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Outline2 ? 2D Paper

• Definitions
• Foldings
• Crease patterns
• Application Map Folding
• Open Problem Complexity of Map Folding

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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
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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 as? or ?

Nonflat folding
Flat origami crane
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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

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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

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Easy?
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Hard?
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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
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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
1
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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
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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
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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?

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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?

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Outline3 ? 3D Polyhedra

• Edge-Unfolding
• Definitions
• Cut tree spanning tree
• Net
• Applications Manufacturing
• Open Problem Does every polyhedron have a net?

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Unfolding Polyhedra

• Cut along the surface of a polyhedron

• Unfold into a simple planar polygon without
  overlap

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Edge Unfoldings

• Two types of unfoldings
• Edge unfoldings Cut only along edges
• General 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.

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

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Commercial Software
Lundström Design, http//www.algonet.se/ludesign/
index.html
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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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Archimedian Solids
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Nets for Archimedian Solids
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Cube with one corner truncated
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Sclickenrieder1steepest-edge-unfold
Nets of Polyhedra TU Berlin, 1997
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Sclickenrieder3rightmost-ascending-edge-unfold
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Open3 Edge-Unfolding Convex Polyhedra

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

Shephard, 1975