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

Vertex Pattern

Vertex degree (creases) 6

Angle skew 0%

Show crease angles & M/V labels

Click any crease line in the left diagram to flip it between mountain and valley.

Folding

Fold progress 0%

Flat-Foldability Checks

Mountain / Valley

0 / 0

Maekawa |M−V|=2

—

Kawasaki alt. sums

—

Fold closure error

0.0°

How it works

A single interior vertex of a flat piece of paper has N straight creases radiating from it, splitting the paper into N wedge-shaped panels. Two classic theorems govern whether such a pattern can fold completely flat:

Maekawa's theorem — at any flat-foldable vertex, the number of mountain folds M and valley folds V must differ by exactly 2: |M − V| = 2 (never 0, never 4+).
Kawasaki's theorem — label the N angles between consecutive creases a₁…a_N (they sum to 360°). The vertex is flat-foldable iff the alternating sum a₁−a₂+a₃−a₄+… = 0, equivalently the odd-indexed angles and the even-indexed angles each sum to 180°.

Both boxes above turn green only when the current pattern satisfies that theorem. The fold progress slider then rotates each panel about its shared crease using a real Rodrigues axis-angle rotation, composed panel-by-panel around the vertex and projected with a fixed oblique tilt so the paper appears to lift out of the page. The closure error is how far the last panel misses lining back up with the first at full fold — it stays near 0° only when both theorems hold, which is exactly why both are needed together to describe flat-foldability.

Note: creases here fold in lockstep proportional to the slider (a simplified synchronized animation), not the full multi-degree-of-freedom rigid-folding motion — the closure-error diagnostic is still exact for the flat-folded end state.

Mountain fold Valley fold