ABSTRACT
The silhouette is fully determined by the crease pattern, and so periodicity of the crease pattern does imply periodicity of the folded form. The partially folded creases in corrugations make the resulting surfaces visually interesting, usually much more so than when they are collapsed into the flat-folded state. The infinite variety of origami structures arises when we start bringing vertices together in combinations: a few, tens, hundreds, or even thousands of them in real-world objects. A periodic pattern is symmetric under translation; it is said to have translational symmetry. A periodic pattern can be characterized by its period and the direction of periodicity: two quantities that can be combined in the vector of periodicity, indicated by an arrow in the plane whose length is the period and whose orientation gives the direction of periodicity. The tiles for a doubly periodic pattern have tile lines on all four sides and must obey two sets of matching conditions.
