Part II: Paper a: Flat Origami

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Part II Papera Flat Origami
Joseph ORourke Smith College (Many slides made
by Erik Demaine)
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Outline

• Single-vertex foldability
• Kawaskais Theorem
• Maekawas Theorem
• Justins Theorem
• Continuous Foldability
• Map Folding (revisited)

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Foldings

• 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

Nonflat folding
Flat origami crane
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Foldings

• 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, ORourke 2004
• ? Focus on finding interesting folded state

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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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Flat Foldings ofSingle Sheets of Paper
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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 Origamiwithout Mountain-Valley
Assignment

• Kawasakis TheoremWithout a mountain-valley
  assignment,a vertex is flat-foldable precisely
  ifsum of alternate angles is 180(T1 T3
  Tn-1 T2 T4 Tn)
• Tracing disks boundary along folded arcmoves T1
  - T2 T3 - T4 Tn-1 - Tn
• Should return to starting point ? equals 0

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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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Single-Vertex Origamiwith Mountain-Valley
Assignment

• Kawasaki-Justin TheoremIf 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 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 Kawasakis Theorem
• Solve a kind 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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Continuous Rolling
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Continuous Foldability
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Models 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 Arkin, Bender, Demaine,
Demaine, Mitchell, Sethia, Skiena 2001



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Open 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?