different flat foldings. Consider the smallest angle and the two creases on either side of it. Mountain-fold one of these two creases and valley-fold the other, choosing arbitrarily which fold to use for which crease. Then, glue the resulting flap of paper onto the remaining part of the crease pattern. The result of this gluing 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. There are two ways to choose which folds to use in each step of this method, and each step eliminates two creases. Therefore, any crease pattern with creases that satisfies Kawasaki's condition has at least different choices of mountain and valley folds that all lead to valid flat foldings. History In the late 1970s, Kôdi Husimi and David A. Huffman independently observed that flat-folded figures with four creases have opposite angles adding to , a special case of Kawasaki's theorem. Huffman included the result in a 1976 paper on curved creases, and Husimi published the four-crease theorem in a book on origami geometry with his wife Mitsue Husimi. The same result was published even earlier, in a pair of papers from 1966 by S. Murata that also included the six-crease case and the general case of Maekawa's theorem. The fact that crease patterns with arbitrarily many creases necessarily have alternating sums of angles adding to was discovered by Kawasaki, by Stuart Robertson, and by Jacques Justin (again, independently of each other) in the late 1970s and early 1980s. Because of Justin's contribution to the problem, Kawasaki's theorem has also been called the Kawasaki–Justin theorem. The fact that this condition is sufficient—that is, that crease patterns with evenly many angles, alternatingly summing to can always be flat-folded—may have been first stated by . Kawasaki himself has called the result Husimi's theorem, after Kôdi Husimi, and some other authors have followed this terminology as well. The name "Kawasaki's theorem" was first given to this result in Origami for the Connoisseur by Kunihiko Kasahara and Toshie Takahama (Japan Publications, 1987). credits the lower bound of on the number of different flat-foldings of a crease