Improved flat-back 3D gadgets in origami extrusions completely downward compatible with the conventional pyramid-supported 3D gadgets
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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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