Can a crease pattern be pressed perfectly flat without tearing? A 19th-century theorem answers it at a glance: at any fold point, the angles between creases, taken alternately, must each add to a straight line. Slide a crease (or tap) and watch the vertex snap in and out of flat-foldable.
Kawasaki's theorem for a single flat-foldable vertex. Going around the fold point, the crease angles alternate; the vertex can be pressed flat if and only if the alternating angles each sum to 180° (equivalently, the alternating sums are equal). Here four creases with angles [a, 90, 90, 360−a−180] give alternating sums a+90 and 270−a — equal only when a = 90°. Maekawa's theorem adds that mountain and valley folds must differ by exactly 2. Slide the angle and the vertex snaps flat only at 90°. A fail-loud self-check throws unless the alternating sums are equal exactly at the flat-foldable configuration and differ otherwise.
A single 4-crease vertex (real origami composes many vertices, and global flat-foldability is NP-hard in general); the fold is shown schematically. Kawasaki's alternating-sum condition and Maekawa's count are exact.