Folding

1
Folding Unfolding in Computational Geometry
Joseph ORourke Smith College
(Erik Demaine) MIT
2
Folding and Unfoldingin Science

• Linkages
• Robotic arms
• Proteins
• Paper
• Airbags
• Spacedeployment
• Polyhedra
• Sheet metal

BRL
5-meter lens (2x Hubble)
Hyde
Touch-3D
3
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

4
Outline

• 1D Linkages
• 2D Paper
• 3D Polyhedra
• a. Folding Polygon ? Polyhedra
• b. Unfolding Polyhedra ? Polygon

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Outline1 ? 1D Linkages

• 1D Linkages
• Definitions and History
• Locked chains in 3D
• Locked trees in 2D
• No locked chains in 2D
• Algorithms

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Linkages / Frameworks

• Bar / link / edge line segment
• Vertex / joint 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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ProteinFolding
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Protein Folding
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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

?
11
Canonical Configurations

• Chains Straight configuration

• Polygons Convex configurations

• Trees Flat configurations

12
What Linkages Can Lock?Schanuel Bergman,
early 1970s Grenander 1987 Lenhart
Whitesides 1991 Mitchell 1992

• Can every chain be straightened?
• Can every polygon be convexified?
• Can every tree be flattened?

13
Locked 3D Chains Cantarella Johnston 1998
Biedl, Demaine, Demaine, Lazard, Lubiw, ORourke,
Overmars, Robbins, Streinu, Toussaint, Whitesides
1999

• Cannot straighten some chains
• Idea of proof
• Ends must be far away from the turns
• Turns must stay relatively close to each other
• ? Could effectively connect ends together
• Hence, any straightening unties a trefoil knot

Sphere separates turns from ends
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Locked 3D Chains Cantarella Johnston 1998
Biedl, Demaine, Demaine, Lazard, Lubiw, ORourke,
Overmars, Robbins, Streinu, Toussaint, Whitesides
1999

• Double this chain
• This unknotted polygon cannot be convexified by
  the same argument
• Several locked hexagons are also known

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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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Converting the Tree into a Cycle

• Double each edge

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Converting the Tree into a Cycle

• But this cycle can be convexified

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Converting the Tree into a Cycle

• But this cycle can be convexified

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TheoremConnelly, Demaine, Rote 2000

• For any family of chains and polygons,there is a
  motion that
• Makes the chains straight
• Makes the polygons convex
• Except Chains or polygons contained within a
  cycle might not be straightened or convexified

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Algorithms for 2D Chains
Connelly, Demaine, Rote (2000) ODE convex
programming
Streinu (2000) pseudotriangulations
piecewise-algebraic motions
Cantarella, Demaine, Iben, OBrien (2003) energy
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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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Outline2 ? 2D Paper

• Foldability
• Crease patterns
• Map folding

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Modern Artistic Origami
Bat byMichael LaFosse
Mask by Eric Joisel
BlackForest CuckooClock byRobert Lang
Pangolin by Eric Joisel
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Foldings

• Piece of paper 2D surface
• Square, or polygon, or polyhedral surface
• Folded state isometric embedding
• Isometric preserve intrinsic distances
• Embedding no self-intersections,
  exceptmultiple 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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Single-Vertex Origami

• Consider a disk surrounding a lone vertex in a
  crease pattern (local foldability)
• When can it be folded flat?
• Depends on
• Circular sequence of angles between creasesT1
  T2 Tn 360
• Mountain-valley assignment

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Single-Vertex Origamiwith Mountain-Valley
Assignment

• Maekawas TheoremFor a vertex to be
  flat-foldable, need mountains - valleys 2
• Total turn angle 360 180 mountains -
  180 valleys

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Local Flat Foldability

• Locally flat-foldable crease pattern each
  vertex is flat-foldable if cut out
  flat-foldable except possibly for nonlocal
  self-intersection
• Testable in linear time Bern Hayes 1996
• Check local conditions
• Solve a type of matching problem to find a valid
  mountain-valley assignment, if one exists
• Barrier

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Global Flat Foldability

• Testing (global) flat foldability isstrongly
  NP-hard Bern Hayes 1996
• Wire represented by crimp direction

False
True
T
F
T
T
F
T
T
T
F
T
F
T
self-intersects
Not-all-equal 3-SAT clause
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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

32
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
34
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
35
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
36
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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Simple Foldability Arkin, Bender, Demaine,
Demaine, Mitchell, Sethia, Skiena 2001



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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

• Folding Polygons
• Alexandrovs Theorem
• Algorithms
• Edge-to-Edge Foldings
• Examples
• Foldings of the Latin Cross
• Foldings of the Square
• Open Problems
• Transforming shapes?

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Aleksandrovs Theorem (1941)

• For every convex polyhedral metric, there exists
  a unique polyhedron (up to a translation or a
  translation with a symmetry) realizing this
  metric."

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Alexandrov Gluing (of polygons)

• Uses up the perimeter of all the polygons with
  boundary matches
• No gaps.
• No paper overlap.
• Several points may glue together.
• At most 2? angle at any glued point.
• Homeomorphic to a sphere.
• Aleksandrovs Theorem ? unique polyhedron

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Folding the Latin Cross
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Folding Polygons to Convex Polyhedra

• When can a polygon fold to a polyhedron?
• Fold close up perimeter, no overlap, no gap
• When does a polygon have an Aleksandrov gluing?

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Unfoldable Polygon
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Foldability is rare

• Lemma The probability that a random polygon of n
  vertices can fold to a polytope approaches 0 as n
  ? 1.

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Perimeter Halving
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Edge-to-Edge Gluings

• Restricts gluing of whole edges to whole edges.
• Lubiw
  ORourke, 1996

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New Re-foldings of the Cube
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Video
Demaine, Demaine, Lubiw, JOR, Pashchenko (Symp.
Computational Geometry, 1999)
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Two Case Studies

• The Latin Cross
• The Square

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Folding the Latin Cross

• 85 distinct gluings
• Reconstruct shapes by ad hoc techniques
• 23 incongruent convex polyhedra

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The 23 convex polyhedra foldablefrom the Latin
cross
Sasha Berkoff, Caitlin Brady, Erik Demaine,
Martin Demaine, Koichi Hirata, Anna Lubiw, Sonya
Nikolova, Joseph ORourke
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23 Latin Cross Polyhedra
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Foldings of a Square

• Infinite continuum of polyhedra.
• Connected space

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Creases
As A varies in 0,½, the polyhedra vary
between a flat triangle and a symmetric
tetrahedron.
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Nine Combinatorial Classes of Polyhedra foldable
from a Square

• Five nondegenerate polyhedra
• Tetrahedra.
• Pentahedra 5 vertices and a single quadrilateral
  face,
• Pentahedra 6 vertices and three quadrilateral
  faces (and all other faces triangles).
• Hexahedra 5-vertex, 6-triangle polyhedra with
  vertex degrees (3,3,4,4,4).
• Octahedra 6-vertex, 8-triangle polyhedra with
  all vertices of degree 4.
• Four flat polyhedra
• A right triangle.
• A square.
• A 1 ? ½ rectangle.
• A pentagon with a line of symmetry.

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Dynamic Web page
58
Open3 Fold/Refold Dissections

• Can a cube be cut open and unfolded to a polygon
  that may be refolded to a regular tetrahedron?

M. Demaine 98
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Outline4? 3D Polyhedra

• Edge-Unfolding Polyhedra
• History (Dürer) Open Problem Applications
• Evidence For
• Evidence Against
• Ununfoldable Polyhedra

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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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Commercial Software
Lundström Design, http//www.algonet.se/ludesign/
index.html
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Albrecht Dürer, 1425
Melancholia I
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Albrecht Dürer, 1425
Snub Cube
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Open Edge-Unfolding Convex Polyhedra

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

Shephard, 1975
66
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

67
Cut Edges (revisited)

• 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.

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Nonsimple Polygons
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Andrea Mantler example
Javaview
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Cut edges strengthening

• Lemma The cut edges of an edge unfolding of a
  convex polyhedron to a single, connected piece
  form a spanning tree of the 1-skeleton of the
  polyhedron.
• Bern, Demaine, Eppstein, Kuo, Mantler, ORourke,
  Snoeyink 01

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Outline4? 3D Polyhedra

• Edge-Unfolding Polyhedra
• History (Dürer) Open Problem Applications
• Evidence For
• Evidence Against
• Ununfoldable Polyhedra

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Archimedian Solids
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Nets for Archimedian Solids
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Successful Software

• Nishizeki
• Hypergami ?
• Javaview Unfold
• ...

http//www.fucg.org/PartIII/JavaView/unfold/Archim
edean.html
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Prismoids
Convex top A and bottom B, equiangular. Edges
parallel lateral faces quadrilaterals.
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Overlapping Unfolding
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Splay Unfolding (top view)
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Splay Unfolding
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Outline4? 3D Polyhedra

• Edge-Unfolding Polyhedra
• History (Dürer) Open Problem Applications
• Evidence For
• Evidence Against
• Ununfoldable Polyhedra

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Cube with one corner truncated
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Sliver Tetrahedron
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Percent Random Unfoldings that Overlap
ORourke, Schevon 1987
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Sclickenrieder1steepest-edge-unfold
Nets of Polyhedra TU Berlin, 1997
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Sclickenrieder2flat-spanning-tree-unfold
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Sclickenrieder3rightmost-ascending-edge-unfold
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Sclickenrieder4normal-order-unfold
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Open Edge-Unfolding Convex Polyhedra (revisited)

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

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Outline4? 3D Polyhedra

• Edge-Unfolding Polyhedra
• History (Dürer) Open Problem Applications
• Evidence For
• Evidence Against
• Ununfoldable Polyhedra

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Edge-Ununfoldable Orthogonal Polyhedra
Biedl, Demaine, Demaine, Lubiw, ORourke,
Overmars, Robbins, Whitesides CCCG98
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Spiked Tetrahedron
JavaView
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Unfoldability of Spiked Tetrahedron
(BDEKMS 99)

• Theorem Spiked tetrahedron isedge-ununfoldable

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Open4 Fewest Nets

• For a convex polyhedron of n vertices and F
  faces, what is the fewest number of nets (simple,
  nonoverlapping polygons) into which it may be cut
  along edges?

• F

• F
• Simplicial polyhedra F/2
• Simple polyhedra (2/3)(F-2)

• F
• Simplicial polyhedra F/2
• Simple polyhedra (2/3)(F-2)
• (2/3)F for large F