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

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Presentation on theme: "Part II: Paper a: Flat Origami Joseph ORourke Smith College (Many slides made by Erik Demaine)"— Presentation transcript:

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

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

3 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

4 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

5 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

6 Flat Foldings of Single Sheets of Paper

7 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

8 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

9 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

10 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

11 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:

12 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

13 Continuous Rolling

14 Continuous Foldability

15 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:

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

17 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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