The least symmetric crystallographic derivative of the developable double corrugation surface: Computational design using underlying conic and cubic curves

Highlights: • A challenge in the design of functional origami tessellations is tackled using concepts from geometry and crystallography. • The least symmetric derivative of the developable double corrugation (DDC) surface is computationally designed. • The design process raises a fundamental problem in the flat-foldability of thin, quadrilateral-shaped flat sheets. • It is shown that solutions for parallelograms and convex quadrilaterals are hyperbolic and cubic curves, respectively. • The approach could be adapted and applied to the design of possible non-trivial derivatives of other origami tessellations. -- Abstract: Flat-foldable origami tessellations are a rich source of inspiration in the design of transformable structures and mechanical metamaterials. Among all such tessellations, the developable double corrugation (DDC) surface, popularly known as the Miura-ori, is perhaps the most ubiquitous origami pattern in science, engineering, and architectural design. Origami artists, designers, and researchers in various fields of science and engineering have proposed a range of symmetric variations for this pattern. While designing many such derivatives is straightforward, some of them present considerable geometric or crystallographic challenges. In general, the problem of finding flat-foldable derivatives for a given origami tessellation is more challenging for less symmetric descendants. This paper studies the existence and design of the least symmetric derivative of the Miura fold pattern with minimal unit cell enlargement in the longitudinal direction. The course of this study raises a fundamental problem in the flat-foldability of quadrilateral-shaped flat sheets on fold lines through their vertices. An analytical solution to this general problem is presented along with solutions for the special cases of convex quadrilaterals.