part ii: paper a: flat origami joseph orourke smith college (many slides made by erik demaine)
TRANSCRIPT
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Part II: PaperPart II: Papera: Flat Origamia: Flat Origami
Joseph O’RourkeJoseph O’RourkeSmith CollegeSmith College
(Many slides made by Erik (Many slides made by Erik Demaine)Demaine)
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OutlineOutline
Single-vertex foldability Kawaskai’s Theorem Maekawa’s Theorem Justin’s Theorem
Continuous FoldabilityMap Folding (revisited)
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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
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Configuration space of piece of paper = uncountable-dim. space of all folded states
Folding motion = path in this space= continuum of folded states
Fortunately, configuration space of a polygonal paper is path-connected [Demaine, Devadoss, Mitchell, O’Rourke 2004] Focus on finding interesting folded state
FoldingsFoldings
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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
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Flat Foldings ofFlat Foldings ofSingle Sheets of PaperSingle Sheets of Paper
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Single-Vertex OrigamiSingle-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 creases:
Θ1 + Θ2 + … + Θn = 360° Mountain-valley assignment
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Single-Vertex OrigamiSingle-Vertex Origamiwithout Mountain-Valley without Mountain-Valley AssignmentAssignmentKawasaki’s Theorem:
Without a mountain-valley assignment,a vertex is flat-foldable precisely ifsum of alternate angles is 180°(Θ1 + Θ3 + … + Θn−1 = Θ2 + Θ4 + … + Θn) Tracing disk’s boundary along folded arc
moves Θ1 − Θ2 + Θ3 − Θ4 + … + Θn−1 − Θn
Should return to starting point equals 0
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Single-Vertex OrigamiSingle-Vertex Origamiwith Mountain-Valley with Mountain-Valley AssignmentAssignmentMaekawa’s Theorem:
For a vertex to be flat-foldable, need|# mountains − # valleys| = 2 Total turn angle = ±360°
= 180° × # mountains − 180° × # valleys
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Single-Vertex OrigamiSingle-Vertex Origamiwith Mountain-Valley with Mountain-Valley AssignmentAssignmentKawasaki-Justin Theorem:
If one angle is smaller than its two neighbors, the two surrounding creases must have opposite direction Otherwise, the two large angles would
collide
These theorems essentiallycharacterize all flat foldings
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Local Flat FoldabilityLocal 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 Kawasaki’s Theorem Solve a kind of matching problem to find a
valid mountain-valley assignment, if one exists
Barrier:
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Global Flat FoldabilityGlobal Flat Foldability
Testing (global) flat foldability isstrongly NP-hard [Bern & Hayes 1996]
Wire represented by “crimp” direction:
Not-all-equal 3-SAT clauseself-intersects
T
TF T
T
FT T
F T
F T
False True
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Continuous RollingContinuous Rolling
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Continuous FoldabilityContinuous Foldability
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Models of Simple FoldsModels of Simple Folds
A single line can admit several different simple folds, depending on # layers folded Extremes: one-layer or all-layers simple
fold In general: some-layers simple fold
Example in 1D:
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Simple Foldability Simple Foldability [Arkin, Bender, [Arkin, Bender, Demaine, Demaine, Mitchell, Sethia, Skiena Demaine, Demaine, Mitchell, Sethia, Skiena 2001]2001]
Alllayers
Somelayers
Onelayer
Foldedstate
1-D O(n) exp.O(n lg m)
O(n) O(n) O(n)
2-DMap
O(n) exp.O(n lg m)
O(n) O(n) Open[Edmonds]
2-D WeaklyNP-hard
WeaklyNP-hard
WeaklyNP-hard
StronglyNP-hard[B&H ‘96]
= =
=
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Open ProblemsOpen Problems
Open: Orthogonal creases on non-axis-aligned rectangular piece of paper?
Open (Edmonds): Complexity of deciding whether an m × n grid can be folded flat (has a flat folded state)with specified mountain-valley assignment Even 2xN maps open!
Open: What about orthogonal polygons with orthogonal creases, etc.?
NP-Complete?