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Virtual Origami

Origami attracts the minds of people all over the world. Some are interested in its geometric aspects, and others in artistic or recreational elements, so-called traditional origami. Although both origami categories rely on a single notion of paper folding, their methodologies differ due to the pursuits' objectives. We focus on traditional origami, but approach it with mathematical formalism rather than artistic values. In particular, we realize origami in a virtual space, using Mathematica's leading-edge symbolic, numeric, and graphics computation capabilities. The origami creators' interest lies more in artistic creativity than general folding rules. In contrast, origami geometers' underlying motivation is finding a few basic fold rules. Although we have such a set of rules in 2D origami geometry, we are more liberal in defining basic fold rules in traditional origami. The fold rules may be many, even open (infinite). Nevertheless, we can give some basic fold rules essential in traditional origami. We first address squash and outside-reverse folds among classical folds; these classical folds intricately combine several simple folds. One squash fold, for example, requires three simultaneous simple folds, i.e. mountain or valley folds, subjected to nontrivial constraints. We understand the 2D origami geometry to the extent that we do the 2D Euclidean geometry in terms of the constructible points in the two geometries. Namely, the set of the points constructible by Huzita–Justin's fold rules is the strict superset of intersecting points of the circles and the lines made by a straightedge and a compass, the construction tools of Euclidean geometry. We extend Huzita–Justin's rules by introducing a cut along a crease. The operations of cutting and later gluing simplify a squash fold significantly. We can apply the cut operation to other classical folds and streamline their modeling and subsequent realization in a virtual origami space. We then present the construction of a flying crane. It not only serves as a mathematical and computational description of this famous origami but also shows the effectiveness and expandability of our modeling for origami virtualization.