Origami crease patterns
Every origami model hides a crease pattern — a diagram of mountain and valley folds that encodes the entire folding sequence. Click to draw crease lines, toggle fold assignments, and test flat-foldability with Kawasaki's and Maekawa's theorems.
Fold assignment (for new creases)
Preset patterns
The mathematics of paper folding
A crease pattern is the blueprint of an origami model: every fold marked on the unfolded square. Mountain folds (the paper folds away from you, forming a ridge) are conventionally drawn as solid lines; valley folds (the paper folds toward you, forming a trough) are dashed. But not every arrangement of lines can actually be folded flat. Two theorems govern what works.
Kawasaki's theorem: the angle condition
At any interior vertex of a flat-foldable crease pattern, the alternating sum of the angles between consecutive creases must equal zero. Equivalently, the sum of alternating angles must equal 180°: if you label the angles around a vertex α1, α2, α3, …, α2n, then α1 + α3 + … = α2 + α4 + … = 180°. This is a necessary condition discovered independently by Toshikazu Kawasaki and Jacques Justin in the 1980s. It ensures that the paper can lie flat without gaps or overlaps at each vertex. Try drawing four crease lines meeting at a point — move the lines and watch whether the Kawasaki condition is satisfied.
Maekawa's theorem: the counting condition
At any interior vertex, the number of mountain folds minus the number of valley folds must equal exactly ±2. If there are M mountain folds and V valley folds, then M − V = ±2. This is Jun Maekawa's theorem, and it follows from the fact that flat-folding alternates layers above and below the paper, and the first and last layers must face the same direction. A vertex with four creases must have three mountains and one valley, or one mountain and three valleys — never two and two.
Huzita-Hatori axioms and computational origami
In 1989, Humiaki Huzita formulated six axioms describing the folds possible with a single straight fold (later a seventh was added by Koshiro Hatori). These axioms define what can be constructed by origami, just as Euclid's postulates define what can be constructed by compass and straightedge. Remarkably, origami is more powerful: it can trisect angles and double cubes — both impossible with compass and straightedge — because folding allows simultaneous alignment of points and lines, effectively solving cubic equations.
The fold-and-cut theorem
One of the most surprising results in computational origami: any shape made of straight line segments (any polygon, even disconnected or with holes) can be cut from a single sheet of paper with a single straight cut, after a finite number of folds. Erik Demaine, Martin Demaine, and Anna Lubiw proved this in 1999. The crease pattern required can be found algorithmically using the straight skeleton of the target shape. It means that the information needed to cut any straight-line figure is encodable in a single flat-folded configuration — a profound statement about the expressive power of folding.