Home▸Drawing with Maths▸Origami Crease-Pattern Folding Lab: Maekawa & Kawasaki Theorems (2D)

Origami Crease-Pattern Folding Lab: Maekawa & Kawasaki Theorems (2D)

2D single-vertex origami crease-pattern lab: build a mountain/valley fold pattern around a vertex, watch Maekawa's theorem (|M-V|=2) and Kawasaki's theorem (alternating angle sums = 180°) checked live, then drag the fold-angle slider to fold the flat pattern toward its 3D shape via real Rodrigues-rotation projection.

Drawing with Maths2DModerate60 FPS⇄ 3D version

This 2D companion to the 3D origami box-folding animation strips origami down to its core mathematics: a single flat-foldable vertex. Build a crease pattern by clicking creases to flip them between mountain and valley, change the vertex degree and skew the sector angles, and watch Maekawa's theorem and Kawasaki's theorem — the two conditions that decide whether a paper vertex can ever fold flat — evaluated live. Dragging the fold-progress slider then rotates every panel around its crease with a real Rodrigues axis-angle rotation, projected obliquely so the pattern visibly lifts off the page, and a closure-error readout shows numerically why a pattern that fails either theorem never quite comes back together.

⚙ Under the hood

2D single-vertex origami crease-pattern lab: build a mountain/valley fold pattern around a vertex, watch Maekawa's theorem (|M-V|=2) and Kawasaki's theorem (alternating angle sums = 180 degrees) checked live, then drag the fold-progress slider to fold the flat pattern toward its 3D shape via a real Rodrigues-rotation projection.

origami mathematicscrease patternmaekawa theoremkawasaki theoremflat-foldabilitycomputational origami

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install