page_title: Mathematics of paper folding
section_title: Flat folding
hierarchical_section_title: Mathematics of paper folding / Pure origami / Flat folding
caption_attribution_description: English: Created to help explain Robert Lang's third rule of origami crease patterns.
context_page_description: The art of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability and the use of paper folds to solve mathematical equations.
context_section_description: The construction of origami models is sometimes shown as crease patterns. The major question about such crease patterns is whether a given crease pattern can be folded to a flat model, and if so, how to fold them; this is an NP-complete problem. Related problems when the creases are orthogonal are called map folding problems. There are three mathematical rules for producing flat-foldable origami crease patterns:
  Maekawa's theorem: at any vertex the number of valley and mountain folds always differ by two.
  It follows from this that every vertex has an even number of creases, and therefore also the regions between the creases can be colored with two colors.
  Kawasaki's theorem: at any vertex, the sum of all the odd angles adds up to 180 degrees, as do the even.
  A sheet can never penetrate a fold.
  Paper exhibits zero Gaussian curvature at all points on its surface, and only folds naturally along lines of zero curvature.  Curved surfaces that can't be flattened can be produced using a non-folded crease in the paper, as is easily done with wet paper or a fingernail.
  Assigning a crease pattern mountain and valley folds in order to produce a flat model has been proven by Marshall Bern and Barry Hayes to be NP-complete. Further references and technical results are discussed in Part II of Geometric Folding Algorithms.
caption: Angles around a vertex
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