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Note: The contents list below is out-of-date w.r.t. the final contents, for which see TOC.

1. Introduction
• History of Origami
• History of Origami Mathematics
• Defnitions
Piece of paper
Crease pattern
Folded state
Overlap order
Continuous folding process
Formalism: Infnitesimals?
2. Geometric Construction
• Huzita's Axioms and the Resulting Characterization of Foldability
• Regular Polygons
• Generalizing the Axioms to Solve All Polynomials
3. 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
• Map Folding: Sequence of Simple Folds
4. Origami Design
• 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: