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Folding

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Title: 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

5
Outline1 ? 1D Linkages
  • 1D Linkages
  • Definitions and History
  • Locked chains in 3D
  • Locked trees in 2D
  • No locked chains in 2D
  • Algorithms

6
Linkages / Frameworks
  • Bar / link / edge line segment
  • Vertex / joint connection between
    endpoints of bars

Closed chain / cycle / polygon
Open chain / arc
Tree
General
7
Configurations
  • Configuration positions of the vertices that
    preserves the bar lengths
  • Non-self-intersecting No bars cross

Non-self-intersecting configurations
Self-intersecting
8
ProteinFolding
9
Protein Folding
10
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
14
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

15
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

16
Converting the Tree into a Cycle
  • Double each edge

17
Converting the Tree into a Cycle
  • But this cycle can be convexified

18
Converting the Tree into a Cycle
  • But this cycle can be convexified

19
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

20
Algorithms for 2D Chains
Connelly, Demaine, Rote (2000) ODE convex
programming
Streinu (2000) pseudotriangulations
piecewise-algebraic motions
Cantarella, Demaine, Iben, OBrien (2003) energy
21
Open1 Can Equilateral Chains Lock?
  • Does there exist an open polygonal chain embedded
    in 3D, with all links of equal length, that is
    locked?

22
Outline2 ? 2D Paper
  • Foldability
  • Crease patterns
  • Map folding

23
Modern Artistic Origami
Bat byMichael LaFosse
Mask by Eric Joisel
BlackForest CuckooClock byRobert Lang
Pangolin by Eric Joisel
24
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
25
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
26
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

27
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

28
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

29
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
30
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

31
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
33
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?

37
Simple Foldability Arkin, Bender, Demaine,
Demaine, Mitchell, Sethia, Skiena 2001



38
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?

39
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?

40
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."

41
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

42
Folding the Latin Cross
43
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?

44
Unfoldable Polygon
45
Foldability is rare
  • Lemma The probability that a random polygon of n
    vertices can fold to a polytope approaches 0 as n
    ? 1.

46
Perimeter Halving
47
Edge-to-Edge Gluings
  • Restricts gluing of whole edges to whole edges.
  • Lubiw
    ORourke, 1996

48
New Re-foldings of the Cube
49
Video
Demaine, Demaine, Lubiw, JOR, Pashchenko (Symp.
Computational Geometry, 1999)
50
Two Case Studies
  • The Latin Cross
  • The Square

51
Folding the Latin Cross
  • 85 distinct gluings
  • Reconstruct shapes by ad hoc techniques
  • 23 incongruent convex polyhedra

52
The 23 convex polyhedra foldablefrom the Latin
cross
Sasha Berkoff, Caitlin Brady, Erik Demaine,
Martin Demaine, Koichi Hirata, Anna Lubiw, Sonya
Nikolova, Joseph ORourke
53
23 Latin Cross Polyhedra
54
Foldings of a Square
  • Infinite continuum of polyhedra.
  • Connected space

55
Creases
As A varies in 0,½, the polyhedra vary
between a flat triangle and a symmetric
tetrahedron.
56
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.

57

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
59
Outline4? 3D Polyhedra
  • Edge-Unfolding Polyhedra
  • History (Dürer) Open Problem Applications
  • Evidence For
  • Evidence Against
  • Ununfoldable Polyhedra

60
Unfolding Polyhedra
  • Cut along the surface of a polyhedron
  • Unfold into a simple planar polygon without
    overlap

61
Edge Unfoldings
  • Two types of unfoldings
  • Edge unfoldings Cut only along edges
  • General unfoldings Cut through faces too

62
Commercial Software
Lundström Design, http//www.algonet.se/ludesign/
index.html
63
Albrecht Dürer, 1425
Melancholia I
64
Albrecht Dürer, 1425
Snub Cube
65
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.

68
Nonsimple Polygons
69
Andrea Mantler example
Javaview
70
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

71
Outline4? 3D Polyhedra
  • Edge-Unfolding Polyhedra
  • History (Dürer) Open Problem Applications
  • Evidence For
  • Evidence Against
  • Ununfoldable Polyhedra

72
Archimedian Solids
73
Nets for Archimedian Solids
74
Successful Software
  • Nishizeki
  • Hypergami ?
  • Javaview Unfold
  • ...

http//www.fucg.org/PartIII/JavaView/unfold/Archim
edean.html
75
Prismoids
Convex top A and bottom B, equiangular. Edges
parallel lateral faces quadrilaterals.
76
Overlapping Unfolding
77
Splay Unfolding (top view)
78
Splay Unfolding
79
Outline4? 3D Polyhedra
  • Edge-Unfolding Polyhedra
  • History (Dürer) Open Problem Applications
  • Evidence For
  • Evidence Against
  • Ununfoldable Polyhedra

80
Cube with one corner truncated
81
Sliver Tetrahedron
82
Percent Random Unfoldings that Overlap
ORourke, Schevon 1987
83
Sclickenrieder1steepest-edge-unfold
Nets of Polyhedra TU Berlin, 1997
84
Sclickenrieder2flat-spanning-tree-unfold
85
Sclickenrieder3rightmost-ascending-edge-unfold
86
Sclickenrieder4normal-order-unfold
87
Open Edge-Unfolding Convex Polyhedra (revisited)
  • Does every convex polyhedron have an
    edge-unfolding to a net (a simple, nonoverlapping
    polygon)?

88
Outline4? 3D Polyhedra
  • Edge-Unfolding Polyhedra
  • History (Dürer) Open Problem Applications
  • Evidence For
  • Evidence Against
  • Ununfoldable Polyhedra

89
Edge-Ununfoldable Orthogonal Polyhedra
Biedl, Demaine, Demaine, Lubiw, ORourke,
Overmars, Robbins, Whitesides CCCG98
90
Spiked Tetrahedron
JavaView
91
Unfoldability of Spiked Tetrahedron
(BDEKMS 99)
  • Theorem Spiked tetrahedron isedge-ununfoldable

92
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
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