creasepattern

A crease pattern is a diagram that shows all the crease lines produced when an origami model is formed from a sheet of paper. The pattern records where the paper is creased and how those creases will be folded, but does not prescribe a single sequence of steps. In common practice, lines are drawn as solid for mountain folds and as dashed for valley folds, with colors used to distinguish layers or pre-creases.

Crease patterns are used both in design and analysis. A designer may derive a pattern from a planned folding sequence, or generate a pattern that encodes a target shape and then determine a valid folding order. A single crease pattern can yield multiple models depending on the assignment of mountain and valley folds, and on how the layers are collapsed.

Flat-foldability constraints apply to crease patterns. At each vertex, angle conditions such as Kawasaki's theorem and Maekawa's theorem govern whether the pattern can be folded flat. Kawasaki's theorem states that at a vertex, the sum of alternating sector angles around the vertex equals 180 degrees. Maekawa's theorem states that the difference between the numbers of mountain and valley creases meeting at a vertex is two. These rules help determine if a given pattern is physically realizable and guide the creation of workable origami designs.

In modern origami, crease patterns are studied in both artistic and mathematical contexts. They are drawn by hand or generated with software, and are key to understanding and reproducing complex models as well as to exploring the computational aspects of origami design.