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Part II: Paper a: Flat Origami Joseph ORourke Smith College (Many slides made by Erik Demaine)

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Outline zSingle-vertex foldability yKawaskais Theorem yMaekawas Theorem yJustins Theorem zContinuous Foldability zMap Folding (revisited)

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Foldings zPiece of paper = 2D surface ySquare, or polygon, or polyhedral surface zFolded state = isometric embedding yIsometric = preserve intrinsic distances (measured along paper surface) yEmbedding = no self- intersections except that multiple surfaces can touch with infinitesimal separation Flat origami crane Nonflat folding

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zConfiguration space of piece of paper = uncountable-dim. space of all folded states zFolding motion = path in this space = continuum of folded states zFortunately, configuration space of a polygonal paper is path-connected [Demaine, Devadoss, Mitchell, ORourke 2004] y Focus on finding interesting folded state Foldings

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Structure of Foldings zCreases in folded state = discontinuities in the derivative zCrease pattern = planar graph drawn with straight edges (creases) on the paper, corresponding to unfolded creases zMountain-valley assignment = specify crease directions as or Nonflat folding Flat origami crane

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Flat Foldings of Single Sheets of Paper

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Single-Vertex Origami zConsider a disk surrounding a lone vertex in a crease pattern (local foldability) zWhen can it be folded flat? zDepends on yCircular sequence of angles between creases: Θ 1 + Θ 2 + … + Θ n = 360° yMountain-valley assignment

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Single-Vertex Origami without Mountain-Valley Assignment zKawasakis Theorem: Without a mountain-valley assignment, a vertex is flat-foldable precisely if sum of alternate angles is 180° (Θ 1 + Θ 3 + … + Θ n1 = Θ 2 + Θ 4 + … + Θ n ) yTracing disks boundary along folded arc moves Θ 1 Θ 2 + Θ 3 Θ 4 + … + Θ n1 Θ n yShould return to starting point equals 0

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Single-Vertex Origami with Mountain-Valley Assignment zMaekawas Theorem: For a vertex to be flat-foldable, need |# mountains # valleys| = 2 yTotal turn angle = ±360° = 180° × # mountains 180° × # valleys

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Single-Vertex Origami with Mountain-Valley Assignment zKawasaki-Justin Theorem: If one angle is smaller than its two neighbors, the two surrounding creases must have opposite direction yOtherwise, the two large angles would collide zThese theorems essentially characterize all flat foldings

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Local Flat Foldability zLocally flat-foldable crease pattern = each vertex is flat-foldable if cut out = flat-foldable except possibly for nonlocal self-intersection zTestable in linear time [Bern & Hayes 1996] yCheck Kawasakis Theorem ySolve a kind of matching problem to find a valid mountain-valley assignment, if one exists yBarrier:

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Global Flat Foldability zTesting (global) flat foldability is strongly NP-hard [Bern & Hayes 1996] zWire represented by crimp direction: Not-all-equal 3-SAT clause self-intersects T T FT T F TT FT F T FalseTrue

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Continuous Rolling

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Continuous Foldability

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Models of Simple Folds zA single line can admit several different simple folds, depending on # layers folded yExtremes: one-layer or all-layers simple fold yIn general: some-layers simple fold zExample in 1D:

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Simple Foldability [Arkin, Bender, Demaine, Demaine, Mitchell, Sethia, Skiena 2001] == =

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Open Problems zOpen: Orthogonal creases on non-axis- aligned rectangular piece of paper? zOpen (Edmonds): Complexity of deciding whether an m × n grid can be folded flat (has a flat folded state) with specified mountain-valley assignment yEven 2xN maps open! zOpen: What about orthogonal polygons with orthogonal creases, etc.? NP-Complete?

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