Universal Hinge Patterns for Folding Orthogonal Shapes

Polycube Ruby Implementation Cutting Plotter (Graphtec/CraftROBO) Folded Model PostScript Figure 8: The process by which paper origami can be constructed using the described algorithms. square was inserted as part of a row or column), and does not take into account any non-orthogonal creases made as a result of later steps. The PostScript file can be sent to a cutting plotter, which can etch the simplified composition onto a sheet of paper. Alternatively, one can use a laser cutter to score the paper. See Figure 8. Only a single cube gadget is foldable at each step when folding along visible etchings of the simplified composition, so one can simply fold cube gadgets until there are no visible etchings. If the foldings generated by the Cube Extrusion Algorithm are applied to a larger sheet of paper, it appears that the polycubes are “lying of top of” or “extruded” from the sheet. Artistically this effect is generally preferable to having a polycube closed off by a small flap. In particular, it prevents the pleats from expanding as much, and in some cases enables multiple polycubes to be extruded from a single sheet. 6 Rigid Foldability and Self-Folding Sheets The Cube Extrusion Algorithm can be used to make artistic paper origami, but to get from one folded state to the next may require curving the paper or introducing temporary nonhinge creases. This becomes an issue for controlling a self-folding programmable matter sheet, where the polygons of a hinge pattern are (nearly) rigid. It is an open question which polycubes are rigidly foldable from a particular cube gadget, though it seems through simple empirical testing that our cube gadgets fold rigidly in isolation (when making one-cube polycubes). Polycubes with more than one cube may nonetheless not be rigidly foldable, and polycubes with intermediary folded states that self-intersect are almost certainly not rigidly foldable. Our intuition suggests that the slit pattern has the broadest potential for programmable matter.