Folding and Unfolding in Computational Geometry

Part II: Paper

Last Update:

All material in these pages is Copyright 2006 by Erik D. Demaine and Joseph O'Rourke.

Note: The contents list below is out-of-date w.r.t. the final contents, for which see TOC.

13. Introduction
    • History of Origami
    • History of Origami Mathematics
    • Defnitions
      Piece of paper
      Crease pattern
      Folded state
      Overlap order
      Continuous folding process
      Formalism: Infnitesimals?
14. Geometric Construction
    • Huzita's Axioms and the Resulting Characterization of Foldability
    • Regular Polygons
    • Generalizing the Axioms to Solve All Polynomials
15. Foldability
    • Single-Vertex Foldability
      • Flat Foldings
      • 3D Foldings
    • Continuous Single-Vertex Foldability
    • Local Flat Foldability is Easy
    • Global Flat Foldability is Hard
      • All-Positive Not-All-Equal 3-Satisfiability
      • Reduction Overview
      • Wire
      • Not-All-Equal Clause
      • Splitting and Routing
      • Putting It Together
      • Overlap Order from Valid Mountain-Valley Assignment
    • Continuous Foldability
      • Introduction
      • Definitions
        • Folded States: Overview
        • Well-Behaved Folded States
        • Folded States: Isometry
        • Folded States: Order
        • Folded States: Noncrossing
      • Folding Motions
      • Rolling between Flat-Folded States
      • Unfurling on f(P)
    • One-Dimensional Origami
      • Equivalence of Folded States and Crimps
      • Connection to Linkages
    • Map Folding: Sequence of Simple Folds
16. Origami Design
    • Silhouettes and Gift Wrapping
      • Strip Folding
      • Hamiltonian Triangulation
      • Seam Placement
    • Tree Method
      • Origami bases
      • Uniaxial bases
    • One Complete Straight Cut
      • History
      • Result
      • Straight-Skeleton Method
        • Straight Skeleton
        • Perpendiculars
        • Strange Behavior
          Spiraling
          Density
        • Corridors
        • Folded State for Linear Corridors
        • Circular Corridors
        • Disk-Packing Method
        • Parallel Offset
        • Disk Packing
        • Decomposition into Triangles and Quadrangles
        • Molecules
        • Gluing Molecules
          Molecule tree
          Folding the molecule tree
          Sewing the cut edges
        • Generalizations
    • Flattening Polyhedra
      • Connection to Part III: Models of Folding
        • Open Problem:
      • Connection to Fold-and-Cut Problem
      • Solution via Disk Packing
        • Open Problem:
        • Open Problem:
      • Partial Solution via Straight Skeleton
        • Open Problem:
    • Silhouettes and Gift Wrapping
    • Tessellations
17. Other
    • Disk Hiding
    • Min-Crease-Thickness Foldings of Polyominoes