Kawasaki's Theorem: A Flat Fold Splits 360° Into Two 180°s

Add every other angle

Go around the point in Figure 1 and read the angles between neighbouring creases: 60°, 100°, 120° and 80°. Add every other one. The shaded pair gives 60 + 120 = 180, and the unshaded pair gives 100 + 80 = 180.

Kawasaki's theorem turns this balance into a test for one inner point, a point inside the paper where creases meet (the theorems call it an interior vertex). On a flat sheet, if the alternate angles at such a point each add up to 180°, some mountain-valley choice folds the point flat; if not, none does. Hull's survey states it as Theorem 2.1 in a zero-sum form: the angles, added and subtracted in turn, give 0. Demaine and Demaine (2001) state the 180° form for a single point.

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Move one crease 10° and the balance breaks. With creases 60°, 100°, 110° and 90° apart, the alternate sums are 170° and 190°, so no choice of mountains and valleys presses that point flat. One straight crease, like the one that halves the A4 sheet in A4 Paper Size: Built So Each Half Keeps the Same Shape, passes trivially: 180° on each side.

Why 180°: walk around the point

With an ant, Hull's survey proves only that the angles must balance. For the converse, it points to Hull's 1994 paper. Fold the point flat and let the ant walk once around it on the paper, close to the point. On the folded paper, each crease sends the ant back the way it came: 60° one way, 100° back, 120° forward and 80° back.

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Count forward as plus and back as minus. The ant ends where it started, so 60 − 100 + 120 − 80 = 0. On a flat sheet the four angles also add up to 360°, and those two facts together leave one answer: each alternate pair makes 180°. With 60°, 100°, 110° and 90°, the walk ends 20° short of its start, so the paper would have to tear or stretch.

Each crease flips the next wedge of paper over, like a mirror lying along the crease. Hull notes that the zero-sum form also holds at the tip of a paper cone, where the angles do not total 360°; the 180° form needs a flat sheet.

Angles and counts together

Kawasaki's theorem tests the angles. Maekawa's theorem tests the count: at a point that folds flat, mountains and valleys differ by exactly 2 (Hull's survey, Theorem 2.2). So such a point always has an even number of creases, and the angles pair up. On this site a mountain is a solid dark line and a valley a dashed green line, as in Flat Origami: Mountains and Valleys Differ by 2 at Inner Points.

For a choice of mountains and valleys already marked, the two tests are needed but not enough. Below, the Figure 1 point gets three mountains and one valley. The count passes (3 − 1 = 2) and the angles balance. Yet the point cannot fold flat: both creases beside the 60° angle are mountains.

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Hull's survey gives the rule for a four-crease point whose smallest angle sits between two larger ones: the two creases beside that angle must be one mountain and one valley, and the other two must match. That leaves 4 of the 16 ways to mark four creases, while 8 of the 16 pass the count. This page's search tried every stacking order of the four paper wedges for all 16 and found a flat stacking for exactly those 4: kawasakis-theorem-examples.csv (CSV; where this file comes from: source, version and license, JSON) Source data: Computed for this page by an exhaustive search over stacking orders; the four valid choices at the first point agree with Hull, arXiv:1307.1065, Section 5

Passing Kawasaki's and Maekawa's tests at every point is still no promise for a whole sheet. Hull's survey shows a two-point pattern that passes both tests at each point and still will not fold flat. Both surveys cite Bern and Hayes's 1996 proof that deciding whether a crease pattern folds flat is NP-hard: no known fast method works for every pattern. So the site's Level 1 check, which tests every point, is not a proof that a sheet folds flat (how the check levels are defined).

Who found it

Hull's survey credits Kawasaki's theorem, its Theorem 2.1, to Toshikazu Kawasaki and Jacques Justin. It cites Justin's article in British Origami (June 1986) and papers by both in the proceedings of the First International Meeting of Origami Science and Technology (Ferrara, 1989). Hull adds that the theorem appears there without proof, though he does not doubt that both authors had proofs of their own; the survey's proof of the theorem comes from his own 1994 paper. By Hull's account, Justin called the zero-sum equation the isometries condition.

Illustration

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The dates and titles below come from the survey's reference list; the last row is the survey itself.