flatfolding

Flat-folding refers to the property of an origami crease pattern that can be folded into a flat, two‑dimensional state with all layers lying in a single plane. A crease pattern is a set of straight folds on a sheet, each designated as a mountain fold or a valley fold. A pattern is flat-foldable if there exists a sequence of valid folds that places the entire model flat without tearing or stretching the paper. The concept is central to origami mathematics and to the design of foldable structures, packaging, and deployable devices.

For a single vertex where multiple creases meet, two classical theorems describe necessary conditions for flat-foldability. Maekawa's theorem states that at such a vertex the difference between the number of mountain and valley creases is two. Kawasaki's theorem states that the sum of alternating sector angles around the vertex equals 180 degrees. These local rules provide essential checks for whether a vertex can be folded flat.

When a crease pattern contains many vertices, flat-foldability requires that every vertex be flat-foldable and that the local folds be globally compatible so that a consistent flat state exists. Even patterns with locally valid vertices may fail globally due to conflicting layer orders or overlap constraints, making flat-folding impossible in practice.

In practice, researchers use mathematical criteria and computational methods to test flat-foldability, often modeling the problem as a constraint system over mountain/valley assignments and fold sequences. The general problem can be complex, but many common patterns and design families can be analyzed with established rules and algorithms.

Historically, the theorems behind flat-folding were developed in the early 20th century by Maekawa and Kawasaki. Flat-folding remains a foundational topic in origami mathematics and informs the design of compact, deployable structures.