A Computational Design Method for Tucking Axisymmetric Origami Consisting of Triangular Facets

Three-dimensional (3D) origami, which can generate a structure through folding a crease pattern on a flat sheet of paper, has received considerable attention in art, mathematics, and engineering. With consideration of symmetry, the user can efficiently generate a rational crease pattern and make the fabricated shape stable. In this paper, we focus on a category of axisymmetric origami consisting of triangular facets and edit the origami in 3D space for expanding its variations. However, it is difficult to retain the developability, which requires the sum of the angles around each interior vertex needing to equal 360 degrees, for designing origami. Intersections occur between crease lines when such a value is larger than 360 degrees. On the other hand, blank spaces (unfolded areas) emerge in the crease pattern when the value is less than 360 degrees. The former case is difficult to generate a realizable shape due to the crease lines are intersected with each other. For the latter case, however, blank spaces can be filled with crease lines and become a part of the origami through tucking. Here, we propose a computational method to add flaps or tucks on the 3D shape, which contains non-developable interior vertices, for achieving the resulting origami. Finally, on the application side, we describe a load-bearing experiment on a stool shape-like origami to demonstrate the potential usage.