Geometric construction of canonical 3D gadgets in origami extrusions
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In a series of our three previous papers, we presented several constructions of positive and negative 3D gadgets in origami extrusions which create with two simple outgoing pleats a top face parallel to the ambient paper and two side faces sharing a ridge, where a 3D gadget is said to be positive (resp. negative) if the top face of the resulting gadget seen from the front side lies above (resp. below) the ambient paper. For any possible set of angle parameters, we obtained an infinite number of positive 3D gadgets in our second paper, while we obtained a unique negative 3D gadget by our third construction in our third paper. In this paper we present a geometric (ruler and compass) construction of our third negative 3D gadgets, while the construction presented in our third paper was a numerical one using a rather complicated formula. Also, we prove that there exists a unique positive 3D gadget corresponding to each of our third negative ones. Thus we obtain a canonical pair of a positive and a negative 3D gadget. The proof is based on a geometric redefinition of the critical angles which we introduced in constructing our positive 3D gadgets. This redefinition also enables us to give a simplified proof of the existence theorem of our positive 3D gadgets in our second paper. As an application, we can construct a positive and a negative extrusion from a common crease pattern by using the canonical counterparts, as long as there arise no interferences.
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