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تسجيل مستخدم جديدImproved flat-back 3D gadgets in origami extrusions completely downward compatible with the conventional pyramid-supported 3D gadgets
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Mamoru Doi
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An origami extrusion is a folding of a 3D object in the middle of a flat piece of paper, using 3D gadgets which create faces with solid angles. In this paper we focus on 3D gadgets which create a top face parallel to the ambient paper and two side faces sharing a ridge, with two outgoing simple pleats, where a simple pleat is a pair of a mountain fold and a valley fold. There are two such types of 3D gadgets. One is the conventional type of 3D gadgets with a triangular pyramid supporting the two side faces from inside. The other is the newer type of 3D gadgets presented in our previous paper, which improve the conventional ones in several respects: They have flat back sides above the ambient paper and no gap between the side faces; they are less interfering with adjacent gadgets so that we can make the extrusion higher at one time; they are downward compatible with conventional ones if constructible; they have a modified flat-back gadget used for repetition which does not interfere with adjacent gadgets; the angles of their outgoing pleats can be changed under certain conditions. However, there are cases where we can apply the conventional gadgets while we cannot our previous ones. The purpose of this paper is to improve our previous 3D gadgets to be completely downward compatible with the conventional ones, in the sense that any conventional gadget can be replaced by our improved one with the same outgoing pleats, but the converse is not always possible. To be more precise, we prove that for any given conventional 3D gadget there are an infinite number of improved 3D gadgets which are compatible with it, and the conventional 3D gadget can be replaced with any of these 3D gadgets without affecting any other conventional 3D gadget. Also, we see that our improved 3D gadget keep all of the above advantages over the conventional ones.
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An origami extrusion is a folding of a 3D object in the middle of a flat piece of paper, using 3D gadgets which create faces with solid angles. Our main concern is to make origami extrusions of polyhedrons using 3D gadgets with simple outgoing pleats
, where a simple pleat is a pair of a mountain fold and a valley fold which are parallel to each other. In this paper we present a new type of 3D gadgets with simple outgoing pleats in origami extrusions and their construction. Our 3D gadgets are downward compatible with the conventional pyramid-supported gadgets developed by Calros Natan as a generalization of the cube gadget, in the sense that in many cases we can replace the conventional gadgets with the new ones with the same outgoing pleats while the converse is not always possible. We can also change angles of the outgoing pleats under certain conditions. Unlike the conventional pyramid-supported 3D gadgets, the new ones have flat back sides above the ambient paper, and thus we can make flat-foldable origami extrusions. Furthermore, since our new 3D gadgets are less interfering with adjacent gadgets than the conventional ones, we can use wider pleats at one time to make the extrusion higher. For example, we prove that the maximal height of the prism of any convex polygon (resp. any triangle) that can be extruded with our new gadgets is more than 4/3 times (resp. $sqrt{2}$ times) of that with the conventional ones. We also present explicit constructions of division/repetition and negati
In our previous two papers, we studied (positive) 3D gadgets in origami extrusions which create a top face parallel to the ambient paper and two side faces sharing a ridge with two simple outgoing pleats. Then a natural problem comes up whether it is
possible to construct a `negative 3D gadget from any positive one having the same net without changing the outgoing pleats, that is, to sink the top and two side faces of any positive 3D gadget to the reverse side without changing the outgoing pleats. Of course, simply sinking the faces causes a tear of the paper, and thus we have to modify the crease pattern. There are two known constructions of negative 3D gadgets before ours, but they do not solve this problem because their outgoing pleats are different from positive ones. In the present paper we give an affirmative solution to the above problem. For this purpose, we present three constructions of negative 3D gadgets with a supporting triangle on the back side, which are based on our previous ones of positive 3D gadgets. The first two are an extension of those presented in our previous paper, and the third is new. We prove that our first and third constructions solve the problem. Our solutions enable us to deal with positive and negative 3D gadgets on the same basis, so that we can construct from an origami extrusion constructed with 3D gadgets its negative using the same pleats if there are no interferences among the 3D gadgets. We also treat repetition/division of negative 3D gadgets under certain conditions, which reduces their interferences with others.
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