In this paper we give a comprehensive introduction to the mathematical definitions and ideas used to analyze origami. An origami is defined as a two-dimensional plane projected into three dimensions by rotation about a set of creases. We represent this rotation using matrices, and discuss the implied restrictions on the movement of origami structures. Flat foldability conditions such as Kawasaki's theorem determine whether local crease patterns can be folded at, and rigid foldability conditions determine their range of motion. We extend one existing at foldability theorem to include a broader class of structures. Origami has inspired designs for many engineering applications, such as efficient storage and impact damping. Metallic origami structures are useful for impact damping because they absorb energy by metallic deformation and can return to their original shapes. We create a design sketch for earthquake damping using origami, and outline how one might use the mathematical principles discussed in this paper to find an optimal design for this application.
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