Non-periodic folding of periodic crease patterns paves the way to novel nonlinear phenomena that cannot be feasible through periodic folding. This paper focuses on the non-periodic folding of recursive crease patterns generalized from Spidron. Although it is known that the Spidron has a 1-DOF isotropic rigid folding motion, its general kinematics and dependence on the crease pattern remain unclear. Using the kinematics of a single unit cell of the Spidron and the recursive construction of the folded state of multiple unit cells, we consider the folding of the Spidron that is not necessarily isotropic. We found that as the number of unit cells increases, the non-periodic folding is restricted and the isotropic folding becomes dominant. Then, we analyze the three kinds of isotropic folding modes by constructing 1-dimensional dynamical systems governing each of them. We show that these systems can possess different recursive natures depending on folding modes even in an identical crease pattern. Furthermore, we show their novel nonlinear nature, including the period-doubling cascade leading to the emergence of chaos.

Origami tessellations, origami whose crease pattern has translational symmetries, have attracted significant attention in designing the mechanical properties of objects. Previous origami-based engineering applications have been designed based on the “uniform-folding” of origami tessellations, where the folding of each unit cell is identical. Although “nonuniform-folding” allows for nonlinear phenomena that are impossible through uniform-folding, there is no universal model for nonuniform-folding, and the underlying mathematics for some observed phenomena remains unclear. Wavy folded states that can be achieved through nonuniform-folding of the tubular origami tessellation called waterbomb tube are an example. Recently, the authors formulated the kinematic coupled motion of unit cells within waterbomb tube as the discrete dynamical system and identified a correspondence between its quasiperiodic solutions and wavy folded states. Here, we show that the wavy folded state is a universal phenomenon that can occur in the family of rotationally symmetric tubular origami tessellations. We represent their dynamical system as the composition of the two 2D mappings: taking the intersection of three spheres and crease pattern transformation. We show the universality of the wavy folded state through numerical calculation of phase diagrams and geometric proof of the system’s conservativeness. Additionally, we present a non-conservative tubular origami tessellation, whose crease pattern includes scaling. The result demonstrates the potential of the dynamical system model as a universal model for nonuniform-folding or a tool for designing metamaterials.

Folded surfaces of origami tessellations sometimes exhibit non-trivial behaviors, which have attracted much attention. The oscillation of tubular waterbomb tessellation is one example. Recently, the authors reported that the kinematics of waterbomb tube depends on the discrete dynamical system that arises from the geometric constraints between modules and quasi-periodic solutions of the dynamical system generate oscillating configurations. Although the quasi-periodic behavior is the characteristic of conservative systems, whether the system is conservative has been unknown. In this paper, we decompose the dynamical system of waterbomb tube into three steps and represent the one-step using the two kinds of mappings between zigzag polygonal linkages. By changing parameters of the mappings and composite them, we generalize the dynamical system of waterbomb tube to that of various tubular origami tessellations and show their oscillating configurations. Furthermore, by analyzing the mapping, we give proof of the conservation of the dynamical system.

Karauchi, one of the traditional braided cords, has a circular cross-section in which the constituent strips follow the geodesic of the cylinder, spiraling from one end to the other. In this paper, we propose a structure in which the bi-axial braids of Karauchi are merged and branched through a tetrahedral curved surface composed of a tri-axial Kagome pattern. Each strip entering the tetrahedron from one vertex follows a geodesic of the tetrahedron in a spiral and always exits at another vertex. The behavior of the strips is corroborated by the geometry of the geodesics and triangular tessellations on the tetrahedron. We created an art installation by using a flat braided cord as the strips to construct the proposed curved surface structure. The geometry of this structure and the design and production of the installation are described. We also computationally model the proposed structure under the equilibrium of tension force.

Ruffled surfaces that appear in biological forms such as coral and lettuce are a great source of inspiration for architectural and furniture design. Such surfaces are produced by differential growth in which the growth rate increases from the center to the edge of the surface. We propose a mechanism based on bending-active scissors that effectively reproduce the process of differential growth through the incompatibility of in-plane shear deformation produced by scissor units. The structure can deform between a linearly folded state with rotational symmetry and a buckled surface of constant negative Gaussian curvature without rotational symmetry. First, we propose a design method for the mechanism computed from the surface of constant negative Gaussian curvature. Then, we show how to analyze the mechanism through geometric methods and elastic simulation. Then, we analyze the process of the curvature change and optimize the design variables to obtain the desired deformation process. Finally, we show the comparison between the simulated model and the physical prototypes.