Petaloid Folding of General Triangles for Combinatorial Smocking Design (Journal of Fiber Bioengineering and Informatics, 2023)
Smocking is a handicraft technique of sewing and making pleats to produce shapes over the surface of cloth, and has been widely used in clothing and fashion. However, it is not easy to combine several pieces of smocking without wrinkles on cloth, since it is necessary to satisfy constraints on angles among pleats. For instance, regular polygons have been widely used in smocking, since admissible twist angles can be easily determined in terms of the number of vertices. However, it has been difficult to utilize more generalized shapes for combinatorial smocking design. For the type of smocking which corresponds to petaloid folding in Origami, this paper proposes petaloid folding of general triangles so that combinatorial smocking without wrinkles can be realized. We derive the admissible twist angle of any triangle to realize petaloid folding in a closed form in terms of the angle and size of the triangle. Since the size of triangle can be reflected in our derivation, it is possible to produce the intended shape of smocking for the specified size of cloth. Based on the derived twist angle, we propose two classes of combinatorial smocking design by utilizing non-regular triangles and regular polygons. The proposed approach is implemented using GeoGebra, and it is validated through several pieces of handcrafted combinatorial smocking.
Keywords: Smocking; Origami; Petaloid folding; Twist angle