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New efficient flat-back 3D gadgets in origami extrusions compatible with the conventional pyramid-supported 3D gadgets

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 negative versions of the new 3D gadgets.