Kawasaki's Theorem - Proof

Proof

To show that Kawaski's condition necessarily holds for any flat-folded figure, it suffices to observe that, at each fold, the orientation of the paper is reversed. Thus, if the first crease in the flat-folded figure is placed in the plane parallel to the x-axis, the next crease must be rotated from it by an angle of α₁, the crease after that by an angle of α₁ − α₂ (because the second angle has the reverse orientation from the first), etc. In order for the paper to meet back up with itself at the final angle, Kawasaki's condition must be met.

Showing that the condition is also a sufficient condition is a matter of describing how to fold a given crease pattern (that is, how to choose whether to make mountain or valley folds, and in what order the flaps of paper should be arranged on top of each other) so that it folds flat. One way to do this is to choose a number i such that the partial alternating sum

α₁ − α₂ + α₃ − ⋯ + α_{2i − 1} − α_2i

is as small as possible: either i = 0 and the partial sum is an empty sum that is also zero, or for some nonzero choice of i the partial sum is negative. Then, accordion fold the pattern, starting with angle α_{2i + 1} and alternating between mountain and valley folds, placing each angular wedge of the paper below the previous folds. At each step until the final fold, an accordion fold of this type will never self-intersect, and the choice of i ensures that the first wedge sticks out to the left of all the other folded pieces of paper, allowing the final wedge to connect back up to it.

An alternative proof of sufficiency is to consider the smallest angle α_i and the two creases on either side of it. If one of these two creases is mountain-folded and the other valley-folded, and then the resulting flap of paper is glued down onto the remaining part of the crease pattern, the result will be a crease pattern with two fewer creases, on a conical sheet of paper, that still satisfies Kawasaki's condition. Therefore, by mathematical induction, repeating this process will eventually lead to a flat folding. The base case of the induction is a cone with only two creases and two equal-angle wedges, which can obviously be flat-folded by using a mountain fold for both creases. Using this method, it can be shown that any crease pattern that satisfies Kawasaki's condition has at least 2n different choices of mountain and valley folds that all lead to valid flat foldings.